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+XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE 
+FIPS Compliant Quantum Secure Communication 
+using Quantum Permutation Pad  
+Abstract—Quantum computing has entered fast development 
+track since Shor’s algorithm was proposed in 1994. Multi-cloud 
+services of quantum computing farms are currently available. One 
+of which, IBM quantum computing, presented a road map 
+showing their Kookaburra system with over 4158 qubits will be 
+available in 2025. For the standardization of Post-Quantum 
+Cryptography or PQC, the National Institute of Standards and 
+Technology or NIST recently announced the first candidates for 
+standardization with one algorithm for key encapsulation 
+mechanism (KEM), Kyber, and three algorithms for digital 
+signatures. NIST has also issued a new call for quantum-safe 
+digital signature algorithms due June 1, 2023. This timeline shows 
+that FIPS-certified quantum-safe TLS protocol would take a 
+predictably long time. However, ‘steal now, crack later’ tactic 
+requires protecting data against future quantum threat actors 
+today. NIST recommended the use of a hybrid mode of TLS 1.3 
+with its extensions to support PQC. The hybrid mode works for 
+certain cases but FIPS certification for the hybridized 
+cryptomodule might still be required. This paper proposes to take 
+a nested mode to enable TLS 1.3 protocol with quantum-safe data, 
+which can be made available today and is FIPS compliant. We 
+discussed the performance impacts of the handshaking phase of 
+the nested TLS 1.3 with PQC and the symmetric encryption phase. 
+The major impact on performance using the nested mode is in the 
+data symmetric encryption with AES. To overcome this 
+performance reduction, we suggest using quantum encryption 
+with a quantum permutation pad for the data encryption with a 
+minor performance reduction of less than 10%. 
+Keywords—quantum communication, quantum encryption, 
+quantum decryption, quantum security, secure communication, 
+QPP, FIPS, TLS 1.3.  
+I. INTRODUCTION  
+Peter Shor proposed his celebrated quantum algorithm in 
+1994 [1], which solves the NP-hard problem of prime integer 
+factorization in polynomial time.  At its beginning, quantum 
+computing, 
+especially 
+universal 
+gate-based 
+quantum 
+computing, experienced a slow development phase for about 
+two decades. In 2019, Arute et al. from Google claimed 
+Quantum Supremacy with their 53-qubits Sycamore processor 
+[2]. This marked the start of global quantum computing race. 
+Since IBM released their 5-qubit quantum computer for public 
+access with Qiskit tool in 2017, IBM recently announced their 
+433-qubit quantum computer and plan to double its qubits every 
+year for 2023 to reach over 4,000-qubits by 2025, outlined in 
+their development roadmap [3].  
+The fundamental shift from classical computing to quantum 
+computing is the shift in computing algebra from the classical 
+Boolean algebra to linear algebra, used in quantum computing. 
+That is, from classical logic gates, implemented in CPU, to 
+quantum logic gates to be implemented in QPU. That means, 
+quantum computing indicates a quadratic speedup of classical 
+computers or �2��2, where � denotes the number of qubits of a 
+QPU. This exponential computing power would break today’s 
+RSA-2048 in 10 seconds if a fault tolerant quantum computer 
+reaches 4099 qubits, while a classical computer would take 300 
+trillion years. Mosca predicted that there is a “1/2 chance of 
+breaking RSA-2048 by 2031” [4].  The year from quantum 
+computing to break classical public key RSA has been called 
+the Year to Quantum threat or Y2Q.  
+Symmetric cryptography such as the well-known Advanced 
+Encryption Standard or AES also suffered quadratic speedup of 
+the best quantum attack on the key space using Grover’s 
+algorithm proposed by Grover in 1996 [5]. That requires the 
+key length to be doubled in comparison with the equivalent 
+classical security level. For example, the classical AES-128 
+would be replaced by AES-256 for quantum security.  
+It has been well-understood that the upcoming quantum 
+computing systems will destroy the foundation of classical 
+public key infrastructure or PKI for both key establishment 
+such as RSA, Diffie-Hellman or elliptical curve Diffie-Hellman 
+and digital signature such as Digital Signature Algorithm or 
+DSA. National Institute of Standards and Technology or NIST 
+has announced its standardization process of Post-Quantum 
+Cryptography or PQC in November of 2017. In July 2022 [6], 
+NIST announced the lattice-based Kyber [7] to be its first 
+standardized key encapsulation mechanism or KEM. And 
+lattice-based Dillithium [8] Falcon [9], as well as hash-based 
+SPHINCS+ [10] to be first standardized digital signature 
+schemes. Other KEM candidates BIKE [11], Classic McEliece 
+[12], HQC [13], and SIKE [14] moved to the 4th round to be 
+considered further. NIST has also announced its reopening 
+Alex He  
+Quantropi Inc. 
+Ottawa, Canada 
+[email protected] 
+Dafu Lou 
+Quantropi Inc. 
+Ottawa, Canada 
+[email protected] 
+Eric She 
+DLS Technology Corporation  
+Ottawa, Canada 
+[email protected] 
+Shangjie Guo 
+FinQ Tech Inc.  
+College Park, Maryland 
+[email protected] 
+Hareesh Watson   
+DLS Technology Corporation  
+Ottawa, Canada 
+[email protected] 
+Sibyl Weng   
+DLS Technology Corporation  
+Ottawa, Canada 
+[email protected] 
+Maria Perepechaenko 
+Quantropi Inc. 
+Ottawa, Canada 
+[email protected] 
+ 
+Randy Kuang 
+Quantropi Inc. 
+Ottawa, Canada 
+[email protected] 
+ORCID: 000-0002-5567-2192-
+0002-5567-2192 
+OORCID: RCID iD: 0000-0002-
+
+submissions for digital signature standardization due Jaune 
+2023 [15].   
+In 2022 some PQC algorithms such as Rainbow (a digital 
+signature scheme based on Multivariate Public Key 
+Cryptography or MPKC) [16] and Supersingular Isogeny 
+Diffie-Hellman protocol or SIDH [17, 18] proved to be 
+vulnerable to classical attacks. Another very interesting 
+cryptoanalysis was reported in the late 2022 by Wenger et al. 
+[19]. The team used transformers (a deep learning model) to 
+develop an attack on certain lattice-based schemes. They 
+noticed that the basic equation system for Learning With Error 
+or LWE can be expressed as linear regression used in machine 
+learning. If trained transformers, especially combined with 
+quantum computational advantage, can solve the Short Vector 
+Problem, then PQC algorithms face an enormous challenge.  
+Recall that most of the PQC standard schemes are lattice-based. 
+This issue is especially concerning since estimation on Y2Q has 
+not taken the power of quantum machine learning into account. 
+Some recent novel PQC algorithms were proposed by 
+Kuang, Perepechaenko, and Barbeau for KEM [20, 21] and 
+digital signature [22, 23, 24, 25], based on NP-complete 
+Modular Diophantine Equation Problem or MDEP. These 
+novel schemes share a foundation which we call Multivariate 
+Polynomial Public Key or MPPK. MPPK is built on two vector 
+spaces: a linear multivariate vector space { ��, … , �� } 
+containing noise variables used for obscurity and polynomial 
+vector space { 1, ��, … , ���� } containing secret message 
+variable.  MPPK offers small parameter sizes at the level of 
+hundreds of bytes for key, ciphertext, and signature and 
+outperforms NIST finalists in key generation, encryption, 
+decryption, as well as signing and signature verification. They 
+could be considered as generic PQC algorithms for a wide range 
+of devices and systems.  
+NIST recommended using a hybrid scheme for quantum 
+resistant TLS 1.3 [26]. By leveraging the keyShare extension of 
+TLS 1.3, one can combine a NIST-approved classical key 
+establishment algorithm such as ECCDH in TLS 1.3 with one 
+or more PQC KEM algorithms and a NIST-approved digital 
+signature algorithm such as DSA with a list of PQC digital 
+signature algorithms for a chain signatures.  This hybrid scheme 
+offers crypto agility as the ongoing process of PQC 
+standardization and cryptanalysis. However, this hybrid 
+scheme brings two potential limitations: 1) the hybrid TLS 1.3 
+may still require FIPS certification although its core 
+cryptomodule is certified in classical TLS 1.3. In general, the 
+FIPS certification comes at a cost for both time and money, 
+although it would be possible to receive the certificate; 2) The 
+certified hybrid TLS 1.3 must be integrated with an application 
+for a quantum resistant service. But in some cases, this 
+integration may be difficult for some applications running 
+inside web browsers.   
+This paper proposes a new hybrid scheme by nesting PQC 
+inside classical TLS 1.3, creating nested TLS 1.3, to overcome 
+the limitations in the above hybrid scheme. The nested TLS 1.3 
+does not require a new FIPS certification because it does not 
+change the existing certified TLS. Moreover, it supports TLS 
+1.3 as well as any certified TLS such as TLS 1.2 or even earlier. 
+One drawback of the nested scheme is that it may reduce the 
+performance of the encryption and decryption operations 
+because the transmitted data would be encrypted twice if AES 
+is used for both encryptions. To minimize the performance 
+impact, we propose to use quantum encryption with a Quantum 
+Permutation Pad algorithm or QPP [27, 28]. The QPP will be 
+used to encrypt the raw data first, producing a quantum-
+encrypted message used as a TLS 1.3 message to be encrypted 
+with AES.    
+II. FIPS COMPLIANT QUANTUM ENCRYPTION WITH QPP 
+In this section, we first introduce the concept of a nested TLS 
+1.3 protocol, that uses quantum-safe cryptosystems. The nested 
+TLS 1.3 allows for a smooth transition from classical era to 
+quantum era while maintaining FIPS compliance. We then 
+discuss the nested TLS 1.3 handshake process. Next, we 
+consider the symmetric data encryption with QPP for 
+considerations of both performance and security.  
+ 
+A. Nested TLS 1.3 with PQC in TLS Handshaking Proccess 
+The proposed nested TLS 1.3 is illustrated in Fig. 1. The 
+figure illustrates an Open Systems’ Interconnection model 
+(OSI-model) consisting of 7 layers: physical, data link, internet, 
+transport, 
+session, 
+presentation, 
+and 
+application. 
+The 
+functionality of each layer is illustrated in [29]. On the right-
+hand side, we mark the corresponding layers in TCP/IP model.  
+The existing TLS cryptomodule is in the transport layer, while 
+the nested TLS is in the application layer. White arrows denote 
+a FIPS certified TLS 1.3 for clientHello request from the client 
+and serverHello response from the server to establish a shared 
+session for session data encryption and decryption. The NIST-
+Approved key agreement protocol, ECCDH, is used for forward 
+secrecy in the existing TLS. The RSA algorithm is excluded 
+from the key agreement protocol. RSA, DSA, and ECDSA are 
+paired with hash functions for digital signature in the existing 
+TLS.  The current conventional FIPS certified TLS 
+cryptomodule will be vulnerable to quantum computing attacks 
+once the fault tolerable quantum computers are available. 
+However, the "steal now crack later" tactics are already in use, 
+meaning all encrypted data today is at risk. Immediate action is 
+imperative to protect data against future quantum threat actors. 
+If sensitive information requires to remain secret for over 10 
+years, then it would not be wise to wait for the FIPS certified 
+TLS 1.3 with quantum resistance. 
+The proposed nested TLS 1.3 with PQC cryptographic 
+modules is independent from the FIPS certified TLS 
+cryptomodule in a sense that packets from the nested TLS 1.3 
+become data packets for the FIPS certified TLS. Since the outer 
+classical TLS cryptomodule is not altered, the nested TLS 1.3 
+does not violate the FIPS certification. This solution can be 
+considered as a promising FIPS compliant TLS 1.3 for quantum 
+security. The nested TLS 1.3 can be used to turn “steal now, 
+crack later” into “steal now, safe forever”. 
+Nested TLS 1.3 is based on the Open Quantum Safe or OQS 
+OpenSSL to support PQC KEM algorithms such as the NIST 
+finalists Kyber and Saber, as well as MPPK [20] and 
+Homomorphic Polynomial Public Key further evolved from 
+MPPK [30]. For the digital signatures, nested TLS 1.3 supports 
+PQC digital signature algorithms such as NIST finalists Falcon, 
+
+Dilithium, Rainbow, as well as MPPK/DS to be submitted to 
+NIST for standardization in 2023 [22].  
+ 
+B. Performance of a Nested TLS 1.3 Handshake 
+We tested the performance of a nested TLS1.3 handshake 
+in a local machine on a 16-core Intel®Core™ i7-10700 CPU at 
+2.90 GHz system for all the measured primitives. Fig. 2 
+illustrates the performance of TLS 1.3 handShake for each pair 
+of KEM and digital signature schemes in terms of NIST 
+security levels. In general, Rainbow digital signature 
+demonstrates the worst performances for TLS handshake with 
+about 500 connections/second at all NIST security level I, 65 
+connections/second at all NIST security level III, and 30 
+connections/second at all NIST security level V.  
+ 
+By pairing MPPK/DS with Saber, Kyber, MPPK KEM, and 
+HPPK, the MPPK/DS scheme outperforms digital signature 
+schemes Falcon and Dilithium over 30% at security level I, over 
+35% at security level III, and 40% at security level V, 
+respectively. On the other hand, pairing MPPK KEM and 
+HPPK with NIST digital signature finalists Falcon and 
+Dilithium, as well as MPPK/DS, the pairs of MPPK KEM with 
+MPPK/DS and HPPK with MPPK/DS outperform NIST 
+finalists Falcon and Dilithium. HPPK pairing with MPPK/DS 
+demonstrates a slightly better performance than MPPK KEM 
+pairing with MPPK/DS, about 10% for all three security levels.   
+Fig. 2 also demonstrates that the average TLS 1.3 
+handShake can be completed at sub-million second. For 
+example, MPPK/DS paired with MPPK KEM and HPPK would 
+establish about 5000 TLS 1.3 connections per second which 
+gives 0.2 ms/connection. In an actual cloud environment, the 
+performance would be reduced due to the network latency. A 
+typical network latency is at 10 ms level, so a sub-million 
+second processing time contribute no impact on a practical TLS 
+1.3 handShake. That means, a nested TLS 1.3 inside the 
+existing TLS cryptomodule would not impact the overall 
+performance if we consider the fact that there is only one 
+handshaking per session.      
+ 
+C. Nested TLS 1.3 with PQC in Symmetric Encryption 
+After the handshake process of a nested TLS is complete, 
+communication peers establish a shared session key for 
+symmetric encryption during the session. Undoubtedly, the 
+NIST-Approved AES-256 can be used for data encryption in 
+the nested TLS, and the produced ciphertext would be 
+Quantum Encryption 
+Encryption 
+1. ClientHello 
+
+
+ 
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+ 
+2. ServerHello   
+
+
+ 
+Nested TLS 1.3 
+1. ClientHello 
+
+
+ 
+2. ServerHello   
+
+
+ 
+Existing TLS  
+Figure 21. Illustration of nested TLS with PQC and quantum 
+encryption in OSI and TCP/IP models. The colorful OSI model is 
+taken from literature [29]. The TCP/IP model is indicated on the right-
+hand side. The white arrows refer to the existing certified TLS 1.x and 
+green arrows denote the nested TLS 1.3 with quantum resistant 
+cryptographic modules. 
+Figure 12. TLS 1.3 handshake performances (connections/second) are 
+illustrated in terms of PQC KEM algorithms paired with PQC digital 
+signature algorithms. The x-axis is marked with KEM schemes and y-
+axis represents handshaking connections per second. Digital signature 
+schemes are listed in the legends. NIST security level I, III, and V are 
+associated with the top, middle, and bottom graphs respectively. 
+
+Connectior
+2000
+1000
+66
+65
+66
+65
+0
+Saber
+Kyber768
+MPPKL 416
+HPPKL 214
+Key ExchangeMechanism
+TLs 1.3 handshake performance of different asymmetric algorithms - security level 5
+Signature algorithm
+second
+4188
+4000
+Falcon-1024
+3584
+Dilithium5
+3025
+per
+3000
+2893
+Rainbow-V-Classic
+2506
+Connections
+2432
+2239
+2225
+MPPKDS 623
+2089
+2000
+1000
+30
+30 
+31
+30
+0
+FireSaber
+Kyber1024
+MPPKL 417
+HPPKL_215
+KeyExchangeMechanismTLS1.3handshakeperformanceofdifferentasymmetricalgorithms-securitylevel1
+5219
+Signaturealgorithm
+5000
+4762
+4709
+Falcon-512
+Dilithium2
+persecond
+4000
+3852
+3983
+3943
+3849
+Rainbow-l-Classic
+3782
+MPPKDS_223
+3000
+3047
+Connections
+2000
+1000
+529
+477
+480
+508
+0
+LightSaber
+Kyber512
+MPPKL415
+HPPKL_213
+KeyExchangeMechanism
+TLs 1.3 handshake performance of different asymmetric algorithms - security level 3
+5000
+5037
+persecond
+Signature algorithm
+4591
+4277
+Dilithium3
+4000
+Rainbow-lll-Classic
+3319
+3140
+MPPKDS 423
+3000
+2778
+2776
+SUP
+P
+Et
+Frgn
+da
+ho
+on
+on
+abl
+CO
+UD
+CT
+ep
+min
+RP,
+ca
+dr
+ne
+2
+e
+.HnSession7.Application
+tion4.Transport6:Data-
+Google
+Networkprocessto application
+7.Application
+DNS,WWW/HTTP,P2P,EMAIL/POP,SMTP,Telnet,FTP
+Presentation
+Datarepresentationand encryption
+Recognizingdata:HTML,DOC,JPEG,MP3,AvI,Sockets
+Session
+Interhostcommunication
+SessionestablishmentinTCP,SiP,RTP,RPC-Namedpipes
+End-to-endconnectionsandreliability
+4.Transport
+TCP,UDP,SCTP,SSL,TLS
+Pathdeterminationandlogicaladdressing
+3.Network
+IP,ARP,IPseCICMP.IGMP.OSPF
+Router
+MAC
+MAC
+Physicaladdressing
+2.Data Link
+Ethernet,8o2.11,MAC/LLC,VLAN,ATM,HDP,FibreChannel
+FrameRelay,HDLC,PPP,Q.921,TokenRing
+Media,signal,and binarytransmission
+1.Physical
+RS-232,RI45.V.34.100BASE-TX,SDH.DSL,802.11encrypted again by the outer FIPS certified AES-256 with its 
+own session key. In this case, the session keys may be 
+potentially obtained by attackers using “Steal now, crack later” 
+strategy, and later decrypted using quantum attacking 
+mechanisms. However, if the nested TLS uses quantum-safe 
+algorithms for encryption, then vulnerability of the outer 
+session does not influence the security of transmitted data to a 
+great extent. That is, if the nested TLS 1.3 with PQC establishes 
+a secure session key for session data encryption, then the 
+attacker with quantum resources will not be able to decrypt the 
+data encrypted in the nested TLS layer even if they were able 
+to decrypt the data encrypted in the outer classical layer. This 
+nested TLS 1.3 with PQC stops the “steal now, crack later” and 
+offers the “steal now, safe forever” services. 
+If AES-256 is used for encryption in conventional outer and 
+nested layer, that would cause the overall performance to drop 
+by 50%. However, there is no requirement to use AES-256 for 
+the nested layer encryption. Under the consideration of the FIPS 
+compliance, the inner data encryption does not need to be the 
+NIST-Approved and FIPS certified, we can choose quantum 
+encryption with Quantum Permutation Pad or QPP [28], 
+implemented classically with permutation matrices. Unlike 
+AES-256 encryption with 14 rounds, QPP encryption follows 
+the same way as quantum gate operations or matrix vector 
+multiplications. QPP encryption is bijective transformation so 
+typical pre-randomization and dispatch techniques would be 
+applied before the gate operations to avoid statistic patterns 
+appearing in the ciphertext. Quantum encryption with QPP 
+demonstrates excellent performance in both encryption and 
+decryption, being over 10x faster than AES-256 [31, 32, 33, 34] 
+Using QPP algorithm for encryption in the inner layer does not 
+impact the performance as greatly as AES-256. The drop in 
+performance with QPP is less than 10%. At this cost we can 
+offer “steal now, safe forever” services. In addition, quantum 
+encryption with QPP has been implemented into IBM quantum 
+computers and compiled into 2-qubit and 3-qubit quantum 
+circuits [35, 36, 37]. 
+For the detailed discission of quantum encryption with QPP, 
+please refer to [27-34]. Here we briefly summarize classical or 
+quantum implementation of QPP as follows shown in Fig. 3: 
+1. Choose the number of n-bits or n-qubits used to 
+generate the permutation gates, for example, � �
+4, 8.  
+2. Choose the number of gates to be used, M. M=64 
+for � �  8  in the digital QKD implementation 
+[38]. It can be reduced to 8 permutation gates with 
+� � 4.  
+3. Session key is first expanded for M permutation 
+gates and then mapped to a set of permutation 
+gates through the Init module where the Fisher-
+Yates shuffling algorithm. 
+4. The session key is also used to seed a 
+cryptographic pseudo random number generator or 
+PRNG 
+5. The pseudo random number generated by PRNG 
+is used to pre-randomize the plaintext m with XOR 
+and then dispatch the randomized data to the 
+indexed permutation gate for encryption and then 
+the ciphertext c is output accordingly. 
+6.  At the receiving side, the process is symmetric to 
+the encryption side, but the permutation gate must 
+be reversed or transposed. The pseudo random 
+number is first used to dispatch the ciphertext to 
+the indexed permutation for decryption and after 
+then derandomize with the pseudo random 
+number, the original plaintext m is obtained.  
+From Fig. 3, we can see that quantum encryption with QPP 
+only consists of three steps to complete its encryption: 
+randomization to remove any statistic bias, dispatching to the 
+indexed permutation gate/matrix, and then gate operation 
+applied to the randomized plaintext. Overall, the entire 
+encryption process may take at most the process time of a single 
+round in AES encryption. That is why QPP would be 10x faster 
+than AES.  Therefore, the nested TLS 1.3 with PQC only has a 
+minor impact on the communication performance, less than 
+10%. 
+If the attackers apply the “steal now, crack later” strategy 
+and wait for the quantum computers to break the public key for 
+the session key, then they can decrypt the outer AES encryption 
+and obtain the quantum encrypted ciphertext. The quantum 
+encrypted ciphertext is then requires to be cracked. However, 
+QPP as well as PQC algorithms and any other quantum-safe 
+algorithms are designed to withstand classical and quantum 
+attacks.  
+We have performed randomness analysis on ciphertext 
+produced by QPP. Table 1 illustrates the randomness analysis 
+of very biased English character files and ciphertext encrypted 
+with QPP using ENT test tool. ENT randomness test tool is very 
+sensitive to bit and byte level bias, especially detected in the 
+Chi Square values.  ENT outputs six reports on their entropy 
+per 8 bits, Chi Square value, p-value, arithmetic mean, Monte 
+Carlo �, and serial correlation. Table 1 shows that the total 
+biased English plaintext is encrypted with QPP into ciphertexts 
+which demonstrate excellent randomness, especially the Chi 
+Square value 233.2 with a p-value 0.83. The acceptable p-value 
+a good randomness is from 0.01 to 9.99. The ciphertext also 
+demonstrated excellent value for arithmetic mean 127.49, 
+Monte Carlo � = 3.14198164, and finally the serial correlation 
+at 9.3�10��.  
+Init 
+PRNG 
+�
+⬚
+⋯
+⬚
+⋮
+⋱
+⋮
+⬚
+⋯
+⬚
+� 
+�
+⬚
+⋯
+⬚
+⋮
+⋱
+⋮
+⬚
+⋯
+⬚
+� 
+…….. 
+�
+⬚
+⋯
+⬚
+⋮
+⋱
+⋮
+⬚
+⋯
+⬚
+� 
+�
+⬚
+⋯
+⬚
+⋮
+⋱
+⋮
+⬚
+⋯
+⬚
+� 
+Dispatcher 
+QPP 
+Key   
+m 
+c 
+Figure 3. Quantum encryption with QPP is illustrated with the session 
+key, plaintext data, and ciphertext together with related modules. 
+
+ Indeed, the nested TLS 1.3 with PQC offers FIPS 
+compliant solution with a quantum encryption component. We 
+this this solution as a good strategy for transition from classical 
+security to quantum security without waiting for NIST 
+standardization and FIPS certification to complete necessary 
+processes. Crypto agility is essential nowadays, since PQC and 
+other algorithms are novel and might have undiscovered 
+attacks, especially as quantum computing matures. So, 
+whenever a specific algorithm is found to be vulnerable, the 
+algorithm can be easily removed from the nested cryptomodule 
+and replaced with a new one in a convenient and quick manner. 
+When all PQC KEMs and digital signature algorithm are 
+standardized and the FIPS certification is required, then the 
+whole TLS 1.3 cryptomodule would be used for FIPS 
+certification. Once it is FIPS certified, the nested TLS 1.3 
+automatically becomes certified quantum resistant TLS 1.3 
+cryptomodule to replace the classical TLS 1.3.  With this, we 
+feel confident to turn the “steal now, crack later” into “Steal 
+now, safe forever.”  
+ Table 1. ENT testing is tabulated for statistically biased plaintext 
+inputs and ciphertext encrypted with QPP, together with their ideal 
+values. 
+ 
+III. CONCLUSION 
+In this work, we propose a FIPS compliant TLS 1.3 solution 
+with a nested quantum-secure TLS component. This solution 
+makes seamless transition and mitigation from classical 
+security to quantum security, that does not require any wait time 
+for standardization and certification. Given that the 
+standardization and FIPS certification process might take 10 
+years, adversaries can use this time to take advantage of the 
+“steal now, crack later” strategy. The proposal of the nested 
+TLS 1.3 with quantum-safe component could turn the “steal 
+now, crack later” into “steal now, safe forever”, while 
+preserving FIPS certified outer TLS layer. Therefore, this 
+proposed work is critical for protecting sensitive data today 
+with long shelf life against future quantum threats, which may 
+impact sectors including public health, insurance, genetics, 
+retirement. Any symmetric algorithm can be used in the nested 
+TLS layer. However, to overcome performance impact in 
+symmetric encryption, we suggest using quantum encryption 
+with QPP to further enhance the data security even with the 
+successful crack the outer public key with quantum computer, 
+the inner TLS 1.3 is still secure. In the future, we plan to build 
+the nested TLS 1.3 with PQC and test its real performance in a 
+cloud environment in comparison with normal TLS 1.3 with 
+PQC.  
+ACKNOWLEDGMENT 
+We acknowledge that Ryan Toth provided the OQS 
+OpenSSL diagram shown in Fig. 2. The overall performance of 
+TLS 1.3 hand shaking with MPPK/DS and MPPK KEM, as 
+well as HPPK would be published separately. 
+ 
+We acknowledge that the image in Fig.1 has been originally 
+taken from [29]. The image is licensed under the Creative 
+Commons Attribution 4.0 International [39]. We have made 
+modifications to the original image available in its original form 
+in [29]. 
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+

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+1
+Sample-Efficient Unsupervised Domain Adaptation
+of Speech Recognition Systems:
+A case study for Modern Greek
+Georgios Paraskevopoulos Student Member, IEEE, Theodoros Kouzelis, Georgios Rouvalis, Athanasios
+Katsamanis Member, IEEE, Vassilis Katsouros Member, IEEE, Alexandros Potamianos Fellow, IEEE
+Abstract—Modern speech recognition systems exhibits rapid
+performance degradation under domain shift. This issue is
+especially prevalent in data-scarce settings, such as low-resource
+languages, where diversity of training data is limited. In this work
+we propose M2DS2, a simple and sample-efficient finetuning
+strategy for large pretrained speech models, based on mixed
+source and target domain self-supervision. We find that including
+source domain self-supervision stabilizes training and avoids
+mode collapse of the latent representations. For evaluation, we
+collect HParl, a 120 hour speech corpus for Greek, consisting
+of plenary sessions in the Greek Parliament. We merge HParl
+with two popular Greek corpora to create GREC-MD, a test-
+bed for multi-domain evaluation of Greek ASR systems. In our
+experiments we find that, while other Unsupervised Domain
+Adaptation baselines fail in this resource-constrained environ-
+ment, M2DS2 yields significant improvements for cross-domain
+adaptation, even when a only a few hours of in-domain audio
+are available. When we relax the problem in a weakly supervised
+setting, we find that independent adaptation for audio using
+M2DS2 and language using simple LM augmentation techniques
+is particularly effective, yielding word error rates comparable to
+the fully supervised baselines.
+Index Terms—Unsupervised Domain Adaptation, Automatic
+Speech Recognition, Multi-Domain Evaluation, Greek Speech
+I. INTRODUCTION
+Automatic Speech recognition (ASR) models have matured
+to the point where they can enable commercial, real-world
+applications, e.g., voice assistants, dictation systems, etc., thus
+being one of machine learning’s success stories. However,
+the performance of ASR systems rapidly deteriorates when
+the test data domain differs significantly from the training
+data. Domain mismatches can be caused by differences in
+the recording conditions, such as environmental noise, room
+reverberation, speaker and accent variability, or shifts in the
+target vocabulary. These issues are extenuated in the case of
+low-resource languages, where diversity in the training data
+is limited due to poor availability of high-quality transcribed
+audio. Therefore, specialized domain adaptation approaches
+need to be employed when operating under domain-shift.
+Unsupervised Domain Adaptation (UDA) methods are of
+special interest, as they do not rely on expensive annotation
+G. Paraskevopoulos is with the Graduate School of ECE, National Technical
+University of Athens, Athens, Greece
+G. Paraskevopoulos, T. Kouzelis, G. Rouvalis, A. Katsamanis, V. Katsouros
+are with the Institute for Speech and Language Processing, Athena Research
+Center, Athens, Greece
+A. Potamianos is with the Faculty of ECE, National Technical University
+of Athens, Athens, Greece
+of domain-specific data for supervised in-domain training.
+In contrast to supervised approaches, where the existence of
+labeled data would allow to train domain-specific models,
+UDA methods aim to leverage data in the absense of labels
+to improve system performance in the domain of interest [1],
+[2]. In the context of speech recognition the importance of
+UDA is extenuated, as the transcription and alignment pro-
+cess is especially expensive and time-consuming. Adaptation
+methods have been explored since the early days of ASR,
+at different levels of the system and different deployment
+settings [3]. UDA has been used to improve the robustness
+of ASR on a variety of recording conditions including far-
+field speech, environmental noise and reverberation [4], [5],
+[6]. Furthermore, UDA has been used for speaker adaptation,
+and to improve performance under speaker, gender and accent
+variability [7], [8]. UDA has also been employed for multilin-
+gual and cross-lingual ASR, in order to improve ASR models
+for low-resource languages [9], adapt to different dialects [10],
+and even train speech recognition systems for endangered
+languages [11].
+Classical speech adaptation techniques involve feature-
+based techniques, e.g., speaker normalization [12], feature-
+based approaches [13]–[15], or multi-condition training [16].
+Generally, traditional approaches require some knowledge
+about the target domain, and the domain mismatch, e.g.,
+regarding the noise and reverberation variability [17], and
+require specific engineering for each adaptation scenario.
+Modern ASR pipelines, increasingly rely on end-to-end
+neural networks, e.g., [18], [19], or large pretrained models
+with self-supervised objectives [20], [21]. The key approaches
+employed for UDA of end-to-end ASR models can be grouped
+in three categories, namely, teacher-student learning [10],
+domain adversarial training [22], and target domain self-
+supervision [23]. The benefit of these techniques is that they
+do not require any special knowledge about the source or
+the target domain. This makes end-to-end UDA approaches
+versatile and able to be utilized in a larger array of adaptation
+scenarios. In particular, adaptation through self-supervision
+has been shown to be a robust, simple and efficient technique
+for adaptation of state-of-the-art speech models [24].
+Here, we leverage in-domain self-supervision to propose
+the Mixed Multi-Domain Self-Supervision (M2DS2) finetun-
+ing strategy, enabling sample-efficient domain adaptation of
+wav2vec2 [20] based speech recognition models, even when
+available in-domain data are scarce. Our key contributions are
+arXiv:2301.00304v1  [cs.CL]  31 Dec 2022
+
+2
+TABLE I
+SUMMARY OF RELATED WORKS ON UNSUPERVISED DOMAIN ADAPTATION FOR ASR.
+Work
+Method
+Model
+Adaptation Setting
+Language
+[23], [25], [26]
+Teacher-Student
+Hard and soft labels
+Conformer RNN-T [27]
+Transformer CTC
+RNN-T [19]
+News speech, Voice search, Far-field,
+Telephony, YouTube
+English
+[4], [5]
+Teacher-Student
+Soft labels
+TDNN-LSTM [28]
+Noise, Far-field
+English
+[29]
+Teacher-Student
+Hard and soft labels
+NiN-CNN [30]
+Dialects
+Children speech
+Japanese
+[31]
+Teacher-Student
+Soft labels
+Streaming RNN-T [32]
+Multilingual
+English,
+Brazilian Portuguese,
+Russian,
+, Turkish,
+Nordic/Germanic
+[6], [33], [34]
+Domain Adversarial Training
+TDNN Kaldi [35], [36]
+DNN-HMM
+DNN-HMM
+Noise, Channel
+English
+[37]
+Domain Adversarial Training
+RNN-CTC [38]
+Far-field
+English
+[8], [39]
+Domain Adversarial Training
+TDNN Kaldi
+RNN-T
+Accent
+Mandarin
+[7], [40]
+Domain Adversarial Training
+DNN-HMM
+CNN-DNN
+Speaker, Gender,
+Accent
+English
+[9]
+Domain Adversarial Training
+DSN [41]
+Multilingual
+Hindi, Sanskri
+[24], [42]
+Continual Pre-Training
+wav2vec2 [20]
+Audiobooks, Accents,
+Ted Talks, Telephony,
+Crowd-sourced, Parlamentary speech
+English
+[43]
+Continual Pre-Training
+wav2vec2
+Cross-lingual
+Korean
+[11], [44]
+Continual Pre-Training
+XLSR-53 [21]
+wav2vec2
+Low resource languages
+Ainu
+Georgian, Somali,
+Tagalog, Farsi
+organized as follows:
+1) Inspired by recent advances on UDA for Natural Lan-
+guage Processing systems [45], we propose a finetuning
+strategy for speech models, where the self-supervised
+objective is based on a contrastive loss in Section III.
+Contrary to prior works, who leverage only in-domain
+self-supervision, we find that in this contrastive setting
+this leads to mode-collapse of the latent representations,
+and mixed source and target domain self-supervision
+is essential. We demonstrate this empirically in Sec-
+tion VII-B.
+2) We collect and curate HParl, the largest publicly avail-
+able1 speech corpus for Greek, collected from plenary
+sessions in the Greek Parliament between 2018 and
+2022. We establish a data collection, pre-processing
+and alignment pipeline that can be used for continuous
+data integration, as the parliamentary proceedings get
+regularly uploaded. We provide a detailed description of
+our data collection process and the dataset statistics in
+Section IV-A. HParl is merged in Section IV with two
+popular Greek corpora (Logotypografia and Common-
+Voice) to create GREC-MD, a testbed for multi-domain
+evaluation of ASR systems in Greek.
+3) We demonstrate that, while other baselines fail at UDA
+in our resource-constrained setting, M2DS2 can improve
+model performance in the target domain in multiple
+adaptation scenarios in Section VII. Specifical emphasis
+is given in the sample efficiency of our approach in Sec-
+1We plan to release this version of HParl under the CC BY-NC 4.0 license
+upon publication. The other corpora used in this work are available through
+their respective distributors.
+tion VII-A, where we demonstrate successful adaptation
+even when we reduce the available in-domain data.
+4) When we relax the problem to a weakly supervised
+adaptation setting, where some in-domain text is avail-
+able but the pairing between audio and text is unknown,
+we find that M2DS2 can be effectively combined with
+simple N-gram adaptation techniques to get compara-
+ble performance with the fully supervised baseline in
+Section VIII. Furthermore we find that a simple text
+augmentation approach, based on perplexity filtering of a
+large corpus can produce strong adaptation results, even
+for small amounts of in-domain text.
+Additionally, we provide a formulation of the UDA problem
+for ASR in Section II-A and link prior works to this formu-
+lation in Sections II-B, II-C and II-D. We provide detailed
+experimental settings for reproducibility in Section V, and
+an upper-bound estimation for UDA performance with fully
+supervised finetuning in Section VI.
+II. BACKGROUND
+We start by formally defining the Unsupervised Domain
+Adaptation (UDA) problem. Initially, we formulate the prob-
+lem in a classification setting and then we extend it for
+speech recognition. We then provide an overview of different
+adaptation approaches in the literature, and link each approach
+to the UDA problem formulation. Table I presents a summary
+of the key adaptation settings and applications that are ex-
+plored in the literature. We see, that a relatively small amount
+of methods, and their variants, is used to address multiple
+real-world ASR problems, for example, cross-lingual, accent,
+speaker and noise adaptation. Furthermore, while the majority
+
+3
+of the works focus on the English language, there is an effort
+to explore other popular languages, e.g., Mandarin, and under-
+resourced languages, e.g., Ainu, Somali etc.
+A. Problem Definition
+Formally, the problem of UDA can be defined as follows.
+Let X ⊆ Rn be a real-valued space that consists of n-
+dimentional feature vectors x ∈ X, and Y a finite set of
+labels y ∈ Y , i.e., Y = {1, 2, . . . , L}. Furthermore, assume
+two different distributions, i.e., the source domain distribution
+S(x, y) and the target domain distribution T (x, y), defined on
+the cartesian product X × Y .
+The goal is to train a model that learns a mapping between
+feature vectors xT to their respective labels yT for samples
+drawn from the target distribution (xT , yT ) ∼ T .
+At training time we have access to samples from the source
+distribution S(x, y) and the marginalized target distribution
+T (x), i.e., no target labels are provided. We define the training
+dataset D as the concatenation of the source and target training
+sets, D = (DS, DT ). DS and DT are defined as sequences of
+tuples, i.e.,
+DS = {(xi, yi) | (xi, yi) ∼ S(x, y), 1 ≤ i ≤ N}
+DT = {(xi, ∅) | xi ∼ T (x), 1 ≤ i ≤ M},
+(1)
+where we draw N samples from S(x, y) and M samples
+from T (x). Finally, we augment tuples in D with a domain
+indicator function:
+D = {(xi, y′
+i, 1i) | 1 ≤ i ≤ N + M}
+1i =
+�
+0
+if xi ∼ S(x),
+1
+if xi ∼ T (x).
+y′
+i =
+�
+yi
+if xi ∼ S(x),
+∅
+if xi ∼ T (x).
+(2)
+1) Unsupervised (Acoustic) Adaptation for ASR: The above
+definition can be directly extended in the case of speech
+recognition, with some modifications. In detail, we modify
+the feature space X, to be the set of (finite) sequences of
+real-valued feature vectors (xk)k∈N\{∞} ∈ X ⊆ (Rn)∗.
+Furthermore, the label space Y is modified to be the set
+of sequences (ym)m∈N\{∞}, where Y
+= ({1, 2, . . . , L})∗
+contains finite-length sequences over a finite lexicon. For
+CTC training we make the assumption that k > m for any
+sample (xk, ym), i.e., feature sequences are longer than their
+respective label sequences [46]. The rest of the definitions need
+no modifications.
+2) Unsupervised (Language) Adaptation for ASR: Adapta-
+tion for ASR systems can also be performed at the language
+level, i.e., the label space. In this setting, we assume that
+the target domain samples are drawn from the marginalized
+target distribution T (y). The target dataset DT now consists of
+tuples in the form (∅, yi), where yi is the label word sequence
+(ym)m∈N\{∞} for the i-th sample.
+3) Weakly supervised Adaptation for ASR: The last setting
+we explore is the case were both audio and language in-
+domain samples are available, but the mapping between them
+is unknown. This situation can be encountered in real-world
+settings, e.g., in the case in-domain audio and text are collected
+independently. For example consider the case where audio
+clips from news casts are collected, along with contemporary
+newspaper articles. Another example is the case where long
+audio clips alongside with transcriptions are available, but no
+fine-grained time alignments2. In this case the target domain
+samples are drawn independently from the marginalized dis-
+tributions T (x) and T (y), and the target dataset DT consists
+of tuples in the form (xi, ∅) and (∅, yi).
+B. Teacher-Student Models
+Teacher-Student learning or self-training, is one of the
+earliest methods in semi-supervised learning [47]–[49]. The
+key idea is to reduce the problem of unsupervised learning
+of the task at hand in the target domain to a supervised one.
+The general methodology is to train a teacher model gS using
+the labeled data in the source domain DS, and then use this
+for inference on the target domain to produce pseudolabels
+ˆyi = gS(xi), xi ∼ T (x). The target domain dataset DT is
+augmented with these silver labels, to contain tuples (xi, ˆyi).
+Finally, a student model gT is trained in a supervised fashion,
+using the augmented DT or a combination of DS and DT .
+This process is usually repeated, with the student model
+serving as the teacher model for the next iteration, until no
+further improvement is observed. More recently, soft target
+Teacher-Student learning has been explored for ASR [26],
+[31], [50], where the KL divergence between the teacher and
+student output label distributions is used as the loss function.
+Being trained only on the source domain data the teacher
+model is susceptible to error propagation. Filtering is a com-
+monly used technique to achieve the right balance between
+the size of the target domain used for training the student
+model and the noise in the pseudolabels. Confidence scoring
+based on the likelihood is usually applied, discarding those
+utterances for which the hypothesized labels are untrustworthy
+[51]. In [25] dropout is used to measure the model uncertainty.
+The agreement between model predictions with and without
+dropout are used for confidence scoring. In [23] a multi-task
+training objective with a confidence loss is applied to minimise
+the binary cross entropy between the estimated confidence and
+the binary target sequence. In order to learn more robust and
+generalizable features from the teacher model, Noisy Student
+Training (NST) has been proposed in [52]. The teacher models
+generates pseudolabels for DT while the student models are
+trained on a heavily augmented version of DT [52]. In [52],
+[53] the augmentation of the input target data is performed
+with SpecAugment [54], while in [29] a spectrum frequency
+augmentation is performed.
+In [4] Teacher-Student learning with soft labels is introduced
+for ASR to tackle noisy, far-field, and children speech. In
+2While a fully supervised in-domain dataset can be constructed in this
+case using long / forced alignment methods, this is not a focal point for the
+experimental part of this work.
+
+4
+[5], this approach is extended for LF-MMI based models and
+used for noisy, far-field and bandwidth adaptation. In [29] a
+weighted sum of hard and soft target cross entropy losses
+is used for Japanese dialects and children speech adaptation.
+Ramabhadran et al. [31] propose a self-adaptive distillation,
+and a method for distilling from multiple teachers that is
+applied across several multilingual ASR systems for different
+language groups. A comparison between soft and hard targets
+for RNN-T models [19] showed that soft targets perform better
+when both the teacher and student models have the same
+architecture. Otherwise, hard targets are superior [50].
+C. Domain Adversarial Training
+Domain Adversarial Training (DAT) was initially introduced
+for image classification [55]. The key idea is to train a
+model that learns deep features that solve the task at hand
+in the source domain, while being invariant with respect
+to the domain shift. Concretely, the model is trained end-
+to-end using a combination of the supervised task loss Lt,
+learned on DS, and the domain discrimination loss La, i.e.,
+L = Lt − αLa. The loss La is binary cross-entropy, trained
+for domain discrimination using the tuples (xi, 1i). Notice
+the − sign in the loss indicates adversarial learning, i.e., the
+model should learn features that cannot discriminate between
+domains, while solving the task.
+In [6] DAT is employed for noise adaptation on a noise
+corrupted version of WSJ [56] as the target dataset. Using the
+Aurora-4 [57] dataset which has labels associated to the noise
+type, Serdyuk et al. [33] train an adversarial noise classifier. In
+[8] and [39] DAT is utilized for accent adaptation for Mandarin
+and English respectively. Anoop C.S. et al. [9] propose DAT,
+to address the scarcity of data in low-resource languages which
+share a common acoustic space with a high-resource language,
+namely Sanskrit and Hindi. They empirically demonstrate the
+effectiveness of adversarial training, presenting experiments
+with and without the reversal of the domain classification loss.
+D. Leveraging In-domain Self-supervision
+These lines of work have roots in Natural Language Pro-
+cessing tasks [45], [58], and explore domain adaptation by
+leveraging the in-domain data DT for self-supervised learning.
+The core focus is domain adaptation of large pre-trained
+models, e.g., [59], and self-supervision is achieved by use
+of the pre-training self-supervised loss Ls. This process can
+either take part in stages, via continual pre-training [58], or by
+constructing a multitask objective L = Lt + αLs, as in [45].
+Continual Pre-Training (CPT) has been explored for adap-
+tation of ASR models. Robust wav2vec2 [24] explores the
+effectiveness of CPT for domain adaptation, indicating the
+importance of utilizing unlabeled in-domain data. In CASTLE
+[42], CPT is combined with an online pseudolabeling strategy
+for domain adaptation of wav2vec2. Cross-dataset evaluation
+for popular English speech corpora indicates that CPT helps
+to reduce the error rate in the target domain. In [43] and [11]
+CPT is utilized for cross-lingual adaptation of wav2vec2 for
+Korean and Ainu respectively. Notably for Ainu, which is an
+endagered language, CPT has resulted in significant system
+Fig. 1. Target-domain adaptation through self-supervision. In the left we see
+the general pre-training stage of XLSR-53 using the self-supervised loss Ls.
+General pre-training is performed on 56, 000 hours of audio in 53 languages.
+In the right, we see the proposed domain-adaptive finetuning stage, where the
+speech recognition task is learned using transcribed source domain data, while
+adaptation to the target domain is performed by including the self-supervised
+loss over (audio-only) source and target domain data
+improvement. DeHaven and Jayadev [44] compare CPT and
+pseudolabeling for adapting XLSR-53 to four under-resourced
+languages, i.e., Georgian, Somali, Tagalog and Farsi. They find
+that both approaches yield similar improvements, with CPT
+being the more computationally efficient approach.
+While CPT yields significant improvements in a variety of
+tasks, one common theme in these works is the assumption
+of hundreds or thousands of hours of available in-domain
+data, mostly from online resources, e.g., YouTube. This can be
+infeasible when we consider more niche adaptation settings,
+or possible privacy concerns, e.g., how would one collect
+1000 hours of psychotherapy sessions in Greek? In this work,
+we explore domain adaptation methods in a more resource-
+constrained environment.
+III. DOMAIN ADAPTATION THROUGH MULTI-DOMAIN
+SELF-SUPERVISION
+The proposed approach is based on end-to-end adaptation of
+a large pre-trained speech model during the finetuning phase,
+by including in-domain self-supervision. We extend UDALM
+[45], that has shown promise for NLP tasks, for adaptation of
+wav2vec2 based acoustic models, and specifically XLSR. We
+focus on the problem of UDA in the context of a low-resource
+language, i.e., Greek. The key finding of our exploration is
+that straight-forward extension of UDALM, i.e., by using only
+target domain self-supervision, underperforms in this setting,
+and use of both source and target domain data is essential for
+successful adaptation. In this section, first, we will present
+a quick overview of the XLSR-53 training procedure, and
+then we are going to outline the proposed domain adaptation
+approach, which is shown in Fig. 1.
+A. XLSR-53
+XLSR-53 [21] is a massively pre-trained speech model,
+trained on 56, 000 hours of multilingual speech, covering 53
+languages. The model is based on wav2vec2 [20], which is
+composed of a multi-layer convolutional feature encoder, that
+
+General Pretraining
+Finetuning
+Ls
+LCTC
+Ls
+Masked Transformer
+Masked Transformer
+XLSR
+XLSR
+MLS, CommonVoice and BABEL
+Source Domain
+Target Domain
+56.0000 of speech data from 53 languages5
+TABLE II
+THE GREC-MD CORPUS. WE CAN SEE THE DURATION OF EACH SPLIT IN H O U R S:M I N U T E S:S E C O N D S FORMAT, AS WELL AS THE NUMBER OF
+SPEAKERS FOR EACH OF THE SUB-CORPORA.
+Dataset
+Domain
+Speakers
+Train
+Dev
+Test
+Total Duration
+HParl
+Public (political) speech
+387
+99:31:41
+9:03:33
+11:12:28
+119:47:42
+CV
+Crowd-sourced speech
+325
+12:16:17
+1:57:44
+1:59:19
+16:13:20
+Logotypografia
+News casts
+125
+51:58:45
+9:08:35
+8:59:22
+70:06:42
+Total
+-
+713
+163:46:43
+20:09:52
+22:11:44
+206:08:19
+extracts audio features zt from the raw audio, and a trans-
+former context encoder that maps the latent audio features to
+the output hidden states ct. Each latent feature zt corresponds
+to 25 ms of audio with stride 20 ms. A contrastive objective Lc
+is used for pre-training. For this, product quantization [60] is
+applied to the features zt, and then a discrete approximation of
+zt is obtained by sampling from a Gumbel-softmax distribution
+[61], to obtain discrete code vectors qt, organized into G = 2
+codebooks with V
+= 320 vocabulary entries each. The
+contrastive loss aims to identify the correct code vector for
+a given time step, among a set of distractors Qt, obtained
+through negative sampling from other timesteps. To avoid
+mode collapse, a diversity loss Ld is included by maximizing
+the entropy over the averaged softmax distribution over the
+code vector entries ¯pg. The total loss is:
+Ls = −log
+es(zt,qt)
+�
+˜q∼Qt es(zt,˜q)
+�
+��
+�
+Contrastive Loss
+Diversity Loss
+�
+��
+�
+− 1
+GV
+G
+�
+g=1
+V
+�
+v=1
+¯pg,vlog(¯pg,v)
+(3)
+B. Domain Adaptive finetuning for Contrastive Learning of
+Speech Representations
+Fig. 1 shows the proposed finetuning process. The key
+intuition is that we want the model to synergistically learn
+the task at hand (in our case ASR), while being adapted to
+the target domain by in-domain self-supervision. In the left
+we see the general pre-training stage of XLSR-53, which is
+pre-trained on 56K hours of multilingual audio corpora using
+the contrastive pre-training objective. In the right we see the
+proposed finetuning stage, which is inspired by [45].
+During finetuning we form a mixed objective function:
+L = LCT C(xs, ys) + αLs(xs) + βLs(xt),
+(4)
+where (xs, ys) ∼ S(x, y), xt ∼ T (x), LCT C is the CTC
+objective function, optimized using transcribed source domain
+data, and Ls is the contrastive loss from Eq. (3). We scale the
+contribution of each term using hyper-parameters α and β.
+Note that contrary to [45], who use only in-domain self-
+supervision, we leverage both source and target domain sam-
+ples for the mixed self-supervision. We find that this is essen-
+tial in our case to avoid mode collapse, i.e., the model using
+only a few of the available discrete code vectors. Simultaneous
+self-supervision on both the source and target data alleviates
+mode collapse by anchoring the target code vector space to
+have a similar structure as the source code vectors.
+Hence we refer to this approach as Mixed Multi-Domain
+Self-Supervision (M2DS2).
+IV. THE GREC-MD CORPUS
+For our experiments we compose a speech corpus for the
+Greek language, that is suitable for multi- and cross-domain
+evaluation. The GREC-MD corpus contains 206 hours of
+Greek speech. Audio is segmented into individual utterances
+and each utterance is paired with its corresponding tran-
+scription. Table II summarizes the included sub-corpora, as
+well as the train, development and test splits. The dataset is
+constructed with three core principles in mind:
+1) Data Volume: We collect the largest publicly available
+speech recognition corpus for the Greek language, able
+to scale to hundreds of hours of transcribed audio.
+2) Temporal Relevance: Language changes over time. We
+aim at an up-to-date corpus that encompasses the latest
+terms and topics that appear in daily speech.
+3) Multi-Domain Evaluation: Single domain evaluation
+can lead to misleading estimations of the expected
+performance for ASR models. For example, state-of-
+the-art ASR models [27] achieve under 5% Word Error
+Rate (WER) on Librispeech [62] test sets, but this is
+an over-estimation of system performance in the field.
+This is extenuated when considering different acoustic
+conditions or terminology. We consider multi-domain
+evaluation essential when developing and deploying
+real-world ASR models.
+To satisfy the first two points, we collect data from a public,
+continuously updated resource, i.e., the Hellenic Parliament
+Proceedings, where recordings of the parliamentary sessions
+are regularly uploaded. The benefit of using this resource is the
+straight-forward collection of a continuously growing, multi-
+speaker corpus of transcribed audio that is always up-to-date,
+as the parliamentary discussions revolve around current affairs.
+We refer to this corpus as HParl. For the multi-domain evalua-
+tion, we merge HParl with two publicly available corpora, that
+have different acoustic and language characteristics. We refer
+to the merged, multi-domain corpus as GREC-MD. In this
+Section, we will describe the collection and curation process
+of HParl, and present the relevant statistics for the experiments.
+TABLE III
+PLENARY SESSIONS INCLUDED IN HPARL. THE HOURS COLUMN REFERS
+TO THE RAW (UNSEGMENTED) HOURS OF COLLECTED AUDIO.
+Start date
+End date
+#Sessions
+Hours
+15-02-2022
+01-03-2022
+10
+55
+18-01-2019
+01-02-2019
+10
+52
+28-03-2019
+10-05-2019
+20
+108
+10-12-2018
+21-12-2018
+10
+88
+
+6
+Fig. 2. Overview of the Hellenic Parliament Chamber. The chamber has an
+amphitheatrical shape and can accomodate approximately 400 − 450 people.
+The positions of the key speakers, i.e., current speaker and the parliament
+president are annotated in the image.
+A. Collection and Curation of HParl
+Modern technological advances allow for more direct gov-
+ernment transparency, through the commodification of storage
+and internet speeds. In this spirit, the records of plenary ses-
+sions of the Hellenic Parliament are made publicly available,
+for direct access through a webpage3. The available video
+recordings date back to 2015. For each plenary session, a
+video recording is uploaded, along with a full transcription
+that is recorded verbatim, and in real time by the parlia-
+ment secretaries. For the creation of HParl, we build a web-
+crawler that can traverse and download the video recordings,
+along with the transcriptions from the official website. The
+collection process is parallelized over multiple threads, and
+parameterized by a range of dates and, optionally, a target
+corpus size in GB or in hours. For this version of HParl, we
+collect the plenary sessions in four date ranges, as described in
+Table III. The majority of the collected sessions are from 2019,
+but we also include sessions from 2018 and 2022 to include
+coverage of different topics. The individual components of the
+HParl curation pipeline are: Audio Pre-processing, Text Pre-
+processing, Alignment, Post-processing, and dataset Splitting.
+1) Audio Pre-processing: Fig. 2 shows the layout of the
+Hellenic Parliament Chamber. Plenary sessions mainly take
+place in this room, or in the secondary House Chamber that
+has similar setup but is smaller in size. Because of the room
+and microphone characteristics, the captured audio in the
+video streams contains reverberation, due to sound reflections.
+We employ a light preprocessing pipeline, by passing the
+input video streams through FFmpeg, and converting them to
+monophonic, lossless audio format at 16000 Hz sampling rate.
+The resulting audio is not passed through any de-reverberation
+or speech enhancement software. The resulting audio files have
+a minimum, average and maximum duration of 6 minutes, 6
+hours and 16 hours respectively.
+2) Text Pre-processing: The text files contain full, word-
+by-word transcription of the speeches and questions asked by
+members of the audience, as well as extra annotations made
+by the parliament secretaries. Some annotations are relevant,
+3https://www.hellenicparliament.gr/en/
+i.e., the speaker name, while others are plain text descriptions
+of events happening during the session and need to be filtered
+out (e.g., “The session is interrupted for a 15 minute break”).
+We use a rule-based system, based on regular expressions,
+that filters the unnecessary information, keeping only the
+transcriptions and the speaker names. The speaker labels are
+created by transliterating their names and roles from Greek
+to Greeklish using the “All Greek to Me!” tool [63]. Text is
+lower-cased and normalized to remove multiple whitespaces.
+The result is a text file containing the raw transcriptions, and
+a mapping from speaker labels to their respective text parts.
+3) Aligment and Segmentation: The primary challenge of
+exploiting the plenary sessions for ASR purposes is the length
+of the plenary recordings, as their durations vary from 6
+minutes to 16 hours in length. However, data samples used to
+train ASR are generally less than 30 seconds long. Computa-
+tional challenges have limited the length of training utterances
+for HMM-GMM models [64], and continue to do so in the
+contemporary neural network models. Therefore, we need to
+segment the sessions into smaller pieces more suitable for ASR
+training. A second challenge is posed by mismatches between
+audio and transcripts. Parliamentary proceedings do not fully
+capture everything that is said during the parliamentary ses-
+sions, and do not account for speech disfluencies.
+In order to obtain smaller, clean segments, that are suit-
+able for ASR training we follow the segmentation procedure
+proposed by [65]. Initially the raw recordings are segmented
+into
+30 second segments and the transcriptions are split
+into smaller segments of approximately 1000 words called
+documents. Each segment is decoded using a seed acoustic
+model trained on the Logotypografia corpus [66] and a 4-
+gram biased LM trained on the corresponding transcription
+of each recording. The best path transcript of each segment
+is obtained and paired with the best matching document via
+TF-IDF similarity. Finally each hypothesis is aligned with the
+transcription using Smith-Waterman alignment [67] to select
+the best matching sub-sequence of words. The above method
+yields a list of text utterances, with their corresponding start
+and end times in the source audio files. The procedure yields
+120 hours of useable segmented utterances out of the original
+303 hours of raw audio, or a ratio of 39.6%.
+4) Post-processing: After the segments are extracted, we
+filter out extremely short segments (less than 2 words).
+Moreover, the iterative alignment algorithm may replace some
+intermediate words with a <spoken-noise> tag. When this
+tag is inserted, we match the surrounding text with the raw
+transcriptions and re-insert the missing words. Furthermore,
+we match each segment to its corresponding speaker label.
+Segments without a speaker label are discarded. Lastly, speak-
+ers are associated to their gender based on name suffixes, using
+a simple, Greek language-specific, rule: Speaker names which
+end in a(α), h(η), w(ω) or is(ις) are classified as female, while
+the rest as male. We format the segments, speaker and gender
+mappings in the standard folder structure used by the Kaldi
+speech recognition toolkit [36].
+5) Data Splitting: We provide an official train - devel-
+opment - test split. The development set contains 3 plenary
+sessions, one from 2018, one from 2019 and one from 2022,
+
+Current Speaker
+Parliament President7
+resulting to 9 hours of segmented speech. Similarly, the test
+set contains one session from each year, resulting to 11 hours
+of segmented speech. The rest 99 hours of segmented speech
+are assigned to the training set.
+B. Including corpora from different domains
+We merge HParl with two publicly available corpora to
+create GREC-MD for multi-domain evaluation.
+1) Common Voice: Common Voice (CV) [68] is a crowd-
+sourced, multi-lingual corpus of dictated speech, created by
+Mozilla. The data collection is performed by use of a web
+app or an iPhone app. Contributors are presented with a
+prompt and are asked to read it. The prompts are taken from
+public domain sources, i.e., books, wikipedia, user submitted
+prompts and other public corpora. The maximum prompt
+length is 15 words. A rating system is built into the plat-
+form, where contributors can upvote or downvote submitted
+<audio,transcript> pairs. A pair is considered valid, if
+it receives two upvotes. Speaker independent train, develop-
+ment and test splits are provided. The dataset is open to the
+research community, released under a permisFsive Creative
+Commons license (CC0). In this work, we use version 9.0
+of CV, accessed on April 27, 2022. We keep only the valid
+utterances, i.e., 16 hours of speech from 325 contributors
+(19 − 49 years old, 67% male / 23% female).
+2) Logotypografia: Logotypografia [66] is one of the first
+corpora for Large Vocabulary Continuous Speech Recognition
+in Greek. The dataset contains 33, 136 newscast utterances, or
+72 hours of speech. The utterances were collected from 125
+speakers (55 male, 70 female), who were staff of the popular
+“Eleftherotypia” newspaper in Greece, under varied acoustic
+conditions. Approximately one third of the utterances were
+collected in a sound proof room, one third in a quiet room and
+the last third in an office room. The average utterance duration
+is 7.8 seconds. The transcriptions contain several speech and
+non-speech events (e.g., <cough>), lower-cased Greek words
+and stress marks. Numbers are expanded to full words. We
+use the whole dataset, and perform light preprocessing in
+the transcriptions, by discarding the annotated events and
+punctuation.
+We hence refer to each dataset by the abbreviations: HParl:
+HP, CommonVoice: CV, Logotypografia: LG.
+V. EXPERIMENTAL SETTINGS
+For our experiments we use the following hyper-parameter
+settings, unless explicitly stated otherwise. For model training,
+we use AdamW optimizer [69] with learning rate 0.0003. We
+apply warmup for the first 10% of the maximum training
+steps, and a linear learning rate decay after that. Models
+are finetuned for a maximum of 10000 steps. For speech
+recognition training, we make use of the Connectionist Tem-
+poral Classification (CTC) loss [70], optimized using the
+available transcribed data in each scenario. Validation runs
+every 500 steps on the development set, and early stopping
+is employed on the development CTC loss with patience 5.
+Batch size is set to 8 during finetuning for all scenarios,
+except for M2DS2. In the case of M2DS2 we create mixed
+batches of size 12, containing 4 transcribed source domain
+samples and 8 unlabeled target domain samples and train
+for 10, 000 CTC updates. For memory reasons we split the
+mixed batches in mini-batches of 4 and interleave them during
+model training. Gradients are accumulated over 3 interleaved
+batches. For the self-supervised objective, we create masks
+of maximum timestep length 10, with masking probability
+0.4. We weigh the contributions of the source and target
+domain contrastive objectives, and bring them to the same
+order of magnitude as the CTC loss, by setting α = 0.01 and
+β = 0.02. The convolutional feature encoder is kept frozen
+for all experiments. Our code is based on the huggingface 4
+implementation of XLSR. For all experiments we resample
+the audio files to 16 kHz and downsample to single channel
+audio. We exclude utterances in the training set that are longer
+than 12 seconds. All experiments are run on a single NVIDIA
+RTX 3090 GPU, with mixed precision training.
+For the Language model training, we create a large corpus
+for the Greek language using a subset of the Greek part of CC-
+Net [71] (approximately 11 billion tokens) and combine it with
+1.5 billion tokens from the Greek version of Wikipedia and the
+Hellenic National Corpus (HNC) [72]. During preprocessing,
+we remove all punctuation and accents, deduplicate lines and
+convert all letters to lowercase. We will refer to this corpus as
+the Generic Greek Corpus (GGC). We train a 4-gram language
+model on GGC using KenLM [73] and prune bigrams, trigrams
+and four-grams with counts less than 3, 5 and 7 respectively.
+We incorporate the n-gram LMs at inference time using the
+pyctcdecode framework5. We use language model rescoring
+over a beam search decoder with 13 beams.
+The evaluation metric is the Word Error Rate (WER) over
+the target test set. For assessing the adaptation effectiveness we
+also report the relative WER improvement over the unadapted
+baseline in appropriate scenarios, which is defined in Eq. (5).
+We refer to this metric as Relative Adaptation Improvement
+(RAI) for the rest of this paper:
+RAI = −WERadapted − WERunadapted
+WERunadapted
+× 100%
+(5)
+The minus sign is included, so that RAI takes negative
+values when the adaptation fails, i.e., when WERunadapted <
+WERadapted.
+TABLE IV
+ASR PERFORMANCE OF XLSR-53 OVER THE THREE CORPORA FOR FULLY
+SUPERVISED IN-DOMAIN FINETUING (WER)
+Dataset
+LM
+No LM
+4g GGC
+HP
+26.21
+15.64
+CV
+29.33
+9.52
+LG
+31.94
+26.45
+VI. SUPERVISED IN-DOMAIN TRAINING
+In the first set of experiments, we explore the performance
+of supervised finetuning of XLSR-53 for each domain. This
+4https://huggingface.co/docs/transformers/
+5https://github.com/kensho-technologies/pyctcdecode
+
+8
+TABLE V
+M2DS2 PERFORMANCE USING GREEDY DECODING FOR UDA BETWEEN HP, CV, AND LG. A → B INDICATES THAT A IS THE SOURCE DOMAIN AND B IS
+THE TARGET DOMAIN. (G) INDICATES GREEDY DECODING. (LM) INDICATES BEAM SEARCH WITH LM RESCORING. WE REPORT THE WER ON THE
+TARGET TEST SET, AS WELL AS THE RAI (%) OVER THE SO (UNADAPTED) BASELINE. WER: LOWER IS BETTER. RAI: HIGHER IS BETTER.
+Method
+SO (G)
+CPT (G)
+PSL (G)
+M2DS2 (G)
+SO (LM)
+CPT (LM)
+PSL (LM)
+M2DS2 (LM)
+Setting
+WER
+WER
+RAI
+WER
+RAI
+WER
+RAI
+WER
+WER
+RAI
+WER
+RAI
+WER
+RAI
+HP → CV
+55.9
+59.68
+−6.8
+55.3
+1.2
+52.95
+5.3
+25.26
+26.44
+−4.7
+24.24
+4.0
+18.35
+27.4
+HP → LG
+48.65
+52.63
+−8.2
+57.68
+−18.6
+58.99
+−21.3
+30.34
+32.27
+−6.4
+39.32
+−29.6
+32.58
+−7.4
+LG → CV
+59.57
+66.43
+−13.4
+81.90
+−39.8
+51.31
+12.4
+25.96
+31.51
+−21.4
+52.05
+−100.5
+17.30
+33.4
+LG → HP
+62.13
+67.51
+−8.7
+71.46
+−15.0
+60.09
+3.3
+31.48
+31.58
+−0.3
+45.36
+−44.1
+31.36
+0.4
+CV → LG
+69.55
+71.12
+−2.3
+71.34
+−2.6
+63.40
+8.8
+50.80
+52.40
+−3.2
+48.68
+4.2
+36.93
+27.3
+CV → HP
+70.72
+73.83
+−4.4
+78.05
+−10.4
+68.70
+2.9
+52.09
+52.18
+−0.2
+54.82
+−5.2
+41.88
+19.6
+will give an upper bound estimation for UDA performance.
+We finetune XLSR-53 on CV, HP and LG (separately) and
+perform in-domain evaluation on the respective test sets.
+Results are summarized in Table IV. The first row indicates the
+performance of greedy decoding, while in the second row we
+report the performance of the beam search decoder, rescored
+using the scores of the 4-gram GGC language model. We
+observe that the greedy decoding performance is under 30
+WER for both HP and CV, while for LG we achieve ∼ 32
+WER. This makes sense, as LG is the most diverse dataset,
+with respect to the included acoustic conditions. Furthermore,
+we observe that the incorporation of a language model results
+in an impressive WER reduction on CV, followed by HP and
+then LG. While CV includes relatively simple phrases with
+common vocabulary, HP and LG contain more specialized
+terminology.
+VII. UNSUPERVISED DOMAIN ADAPTATION USING
+IN-DOMAIN AUDIO
+Here, we evaluate the effectiveness of M2DS2 for UDA.
+We compare with three baselines:
+1) Source Only Training (SO): We perform supervised
+finetuning of XLSR-53 (CTC) using only the source-
+domain data, and run decoding on the target domain
+test set. No in-domain data are used for adaptation.
+2) Continual Pre-Training (CPT): We perform a pre-
+training phase using the loss in Eq. (3) on the target
+domain train set, to create adapted versions of XLSR.
+Pre-training is run for 20000 steps with batch size
+4. Only the audio is used, without transcriptions. The
+adapted checkpoints are then finetuned by use of CTC
+loss on the source domain transcribed data. Evaluation
+is performed on the target test set.
+3) Pseudolabeling (PSL): We finetune XLSR-53 using the
+source domain data with CTC loss. Then we run infer-
+ence on the source model, to extract silver transcriptions
+for the target domain training set. We use the silver
+transcriptions for supervised finetuning on the target
+domain.
+In Table V we compare M2DS2 with the SO, CPT and
+PSL baselines for six adaptation scenarios, i.e., cross dataset
+evaluation between the three datasets in GREC-MD. The left
+half corresponds to greedy decoding, while for the right half
+we use the 4-gram LM trained on GGC. First, we observe
+the SO model performance. The SO models are the finetuned
+Fig. 3. Performance of M2DS2 (blue line) for the LG → CV setting, when
+reducing the amount of available target samples to 50%, 25%, and 10% of
+the original dataset (horizontal axis). SO performance is indicated with the
+orange line. Vertical axis: WER, Horizontal Axis: target audio percentage
+(100% → 0%)
+models from Table IV, evaluated in out-of-domain settings.
+We see that out-of-domain evaluation results in a large perfor-
+mance hit, e.g., while in the CV9 → CV9 in-domain setting
+we achieve 29.33 WER, in the CV9 → HP out-of-domain
+setting we get 69.55 WER. This confirms that for real-world
+ASR tasks, multi-domain evaluation is of essence. Second, we
+observe that in most adaptation scenarios both CPT and PSL
+fail to surpass the SO (unadapted) baseline. In the case of CPT,
+we hypothesize that is due to the relatively data constrained
+version of our setting. In the best-case scenario, we have 99
+hours of available target domain audio, which is not enough
+to perform a discrete CPT stage. Note that most of works in
+the literature use ∼ 1000 hours of target audio for CPT. In
+the case of PSL, the poor performance is due to the quality
+of the silver labels created by the seed model. While the
+performance would improve with more elaborate approaches
+(e.g., confidence filtering), in challenging adaptation scenarios
+PSL approaches are limited by the SO model’s performance.
+Lastly, we observe that M2DS2 is the only approach among
+our baselines that manages to achieve a positive RAI in most
+adaptation scenarios, by consistently outperforming the SO
+baseline by significant margins. This is exaggerated when
+we include a LM during inference. One exception in this
+pattern is the HP → LG scenario, where the SO baseline
+achieves the best performance. We attribute this to the fact that
+we performed minimal hyper-parameter tuning during model
+development.
+
+65
+60
+WER
+55
+50
+100%
+90%
+80%
+70%
+60%
+50%
+40%
+30%
+20%
+10%
+0%
+Percentage of In-Domain Audio Data9
+A. The sample efficiency of M2DS2
+One key observation in the literature, and in our experiments
+is that CPT requires a large amount of un-transcribed target
+domain audio. This raises the question, can we leverage self-
+supervision for domain adaptation in data constrained settings?
+In Fig. 3 we evaluate the performance of M2DS2, when
+we reduce the amount of target domain audio. Specifically
+we focus on the scenario of LG → CV. The full training
+corpus of CV contains 12 hours of audio. We train M2DS2
+with 50%, 25% and 10% of the available samples, or 6, 3
+and 1.2 hours of audio respectively, and plot the resulting
+WER on the target (CV) test set. In all cases, the full source
+(LG) training corpus is used. We observe that M2DS2 achieves
+lower WER than the SO baseline, even with only 3 hours of
+target domain audio. While CPT can suffer from catastrophic
+forgetting, as most multi-stage training approaches, M2DS2
+avoids this issue, being a single-stage approach with a mixed
+task-specific and self-supervised objective. This provides a
+promising avenue for adaptation, when collection of in-domain
+recordings is expensive, or infeasible.
+(a) Only target domain self-supervision
+(b) Target and source domain self-supervision
+Fig. 4. T-SNE scatter plots of code vectors extracted from M2DS2 without
+source domain self-supervision (top) and with source domain self-supervision
+(bottom) for LG (red) and CV (teal)
+B. The importance of Multi-Domain Self-Supervision
+In Section III-B we argue that it is essential to include both
+source and target domain data for the self-supervised objective
+of M2DS2. To illustrate the effect of this approach, we train
+two versions of M2DS2 for the LG → CV scenario. For the
+TABLE VI
+LANGUAGE ADAPTATION OF THE M2DS2 LG → CV MODEL, USING
+BIASED AND AUGMENTED LMS. WE USE THE VARIANT OF THE MODEL
+TRAINED WITH 3 HOURS OF IN-DOMAIN AUDIO. WE VARY THE AMOUNT
+OF IN-DOMAIN TEXT DATA FROM 752K TOKENS TO 38K TOKENS.
+Biased LM
+Augmented LM
+100%
+11.22
+12.84
+50%
+15.13
+15.05
+25%
+20.84
+16.64
+10%
+27.75
+18.47
+5%
+33.04
+19.31
+Baseline (M2DS2 + Generic LM)
+20.7
+Fig. 5. Language-only adaptation for LG → HP using the SO model finetuned
+on LG. In-domain text data range from 11M tokens (left) to 110K tokens
+(right). Blue/dashed: Baseline with generic LM. Purple/circles: Biased LM.
+Orange/diamonds: Augmented LM.
+first version we set α = 0.01, while for the second we set
+α = 0, removing the second term of Eq. (4). We extract the
+code vectors for the first 100 samples of both LG and CV, and
+flatten them across the time steps , resulting to 60000 × 768
+code vectors corresponding to individual timesteps. We plot
+these code vectors using T-SNE [74] in Fig. 4 for both models.
+We see that when we do not include the source domain self-
+supervision, the code vector space collapses in a few tight
+clusters, and most audio segments correspond to just a few
+code vectors. This is a visual clue that indicates the mode
+collapse problem. When we include the source domain term,
+we see that the that the code vector space has more structure,
+and coverage of the space is more complete, both for CV
+(target domain) and LG (source domain). Experimentally we
+train M2DS2 with α = 0 for all source / target domain pairs
+and we find that the mode collapse is destructive for target
+domain performance. During our experiments we got WER in
+the range 80−99, indicating failure to converge to acceptable
+solutions across all scenarios. The simple inclusion of both
+source and target domain self supervision stabilizes training,
+avoids mode collapse and leads to successful unsupervised
+adaptation between domains.
+VIII. UNSUPERVISED AND WEAKLY SUPERVISED
+LANGUAGE ADAPTATION
+When small amounts of in-domain textual data are avail-
+able, simple N-gram LM adaptation techniques can be very
+effective. In this brief set of experiments, we first explore
+the unsupervised language adaptation setting, where no in-
+
+.
+:
+:·
+.
+！
+.
+.
+.0
+C
+·
+.
+.
+.
+.
+.
+.
+008
+:
+.80
+·
+.
+o43
+41
+39
+37
+WER
+35
+33
+31
+29
+100%
+90%
+80%
+70%
+60%
+50%
+40%
+30%
+20%
+10%
+0%
+Percentage of In-Domain Text Data10
+TABLE VII
+CLOSING THE GAP BETWEEN SO TRAINING AND FULLY SUPERVISED
+TRAINING FOR THE LG → CV ADAPTATION SCENARIO USING M2DS2,
+WITH VARYING AMOUNTS OF AVAILABLE UNPAIRED IN-DOMAIN AUDIO
+AND TEXT. (U): UNSUPERVISED ACOUSTIC OR LANGUAGE ADAPTATION.
+(W): WEAKLY SUPERVISED ADAPTATION.
+Method
+#Audio (h)
+#Tokens
+LM
+WER
+SO (U)
+-
+-
+N/A
+59.57
+M2DS2 (U)
+3
+-
+N/A
+57.31
+M2DS2 (U)
+12
+-
+N/A
+51.31
+SO (U)
+-
+-
+Generic
+25.96
+SO (U)
+-
+38, 632
+Augmented
+24.67
+SO (U)
+-
+751, 953
+Augmented
+20.46
+M2DS2 (U)
+3
+-
+Generic
+20.7
+M2DS2 (U)
+12
+-
+Generic
+17.3
+M2DS2 (W)
+3
+38, 632
+Augmented
+19.31
+M2DS2 (W)
+12
+38, 632
+Augmented
+16.29
+M2DS2 (W)
+3
+751, 953
+Augmented
+12.84
+M2DS2 (W)
+12
+751, 953
+Augmented
+10.61
+Supervised
+12
+751, 953
+Generic
+9.52
+Supervised
+12
+751, 953
+Augmented
+7.94
+domain audio is used, and then we relax the problem to
+the weakly supervised setting, where M2DS2 is combined
+with the adapted N-Gram LMs. These settings are described
+in Sections II-A2 and II-A3 respectively. We explore two
+approaches for LM adaptation: biased LMs, and in-domain
+data augmentation. To create biased LMs, we train a 4-gram
+LM on the available in-domain data. Then we replace the
+generic LM trained on GGC. For LM data augmentation we
+follow a perplexity filtering approach similar to [71]. We first
+train a biased LM using available target domain text, and
+then use it to calculate the perplexity of each line in the
+GGC corpus. We keep the 10% of the lines with the lowest
+perplexity. Then we train a 4-gram LM on the augmented “in-
+domain” corpus and use it for inference.
+Fig. 5 shows the performance of the SO LG → HP model
+with biased and augmented LMs, as we reduce the amount
+of available in-domain text data from 100% to 1% of the
+in-domain transcriptions (11B tokens to 110K tokens respec-
+tively). As a baseline we include the LG → HP SO model in
+combination with the generic LM trained on GGC. We observe
+that the use of biased LMs can lead to successful adaptation,
+when an adequate amount of in-domain text data is available.
+On the other hand the LM augmentation approach results to
+successful augmentation, even with very small amounts of in-
+domain text.
+In Table VI we see the results of LM adaptation, combined
+with the M2DS2 LG → CV model. To demonstrate the sample
+efficiency of the approach, we use the variant that was trained
+using only 25% of the target domain audio (3 hours). We
+compare with M2DS2 combined with the 4-gram GGC LM for
+inference. We draw similar conclusions, i.e., use of biased LMs
+performs well for sufficient text data. When we use augmented
+LMs we can leverage very small amounts of in-domain text.
+IX. DISCUSSION & CONCLUSIONS
+In this work, we have explored Unsupervised and Weakly
+Supervised Domain Adaptation of ASR systems in the con-
+text of an under-resourced language, i.e., Greek. We focus
+on domain adaptation through in-domain self-supervision for
+XLSR-53, a state-of-the-art multilingual ASR model. Specif-
+ically, we adopt a mixed task and self-supervised objective,
+inspired from NLP, and show that using only in-domain self-
+supervision can lead to mode collapse of the representa-
+tions created by the contrastive loss of XLSR-53. Therefore,
+we propose the use of mixed task and multi-domain self-
+supervision, M2DS2, where the contrastive loss leverages both
+the source and target domain audio data. For evaluation we
+create and release HParl, the largest to-date public corpus
+of transcribed Greek speech (120 hours), collected from the
+Greek Parliamentary Proceedings. HParl is combined with two
+other popular Greek speech corpora, i.e., Logotypografia and
+CommonVoice, for multi-domain evaluation.
+In our experiments, we find that while most UDA baselines
+fail in our low-resource setting, the proposed mixed task
+and multi-domain self-supervised finetuning strategy yields
+significant improvements for the majority of adaptation sce-
+narios. Furthermore, we focus our ablations on showcasing
+the sample efficiency of the proposed finetuning strategy,
+and demonstrating the necessity of including both source
+and target domain data for self-supervision. Finally, we show
+that M2DS2 can be combined with simple language model
+adaptation techniques in a relaxed weakly supervised setting,
+where we achieve significant performance improvements with
+a few hours of in-domain audio and a small, unpaired in-
+domain text corpus.
+More concretely, in Table VII we present a summary of
+the discussed unsupervised and weakly supervised adaptation
+combinations, for different amounts of available in-domain
+audio and text. Note that for the weakly supervised scenarios,
+the in-domain audio and text are unpaired. We see, that when
+no in-domain data are available, including an n-gram LM
+trained on large corpora is recommended. Furthermore, when
+in-domain audio is available, following a mixed multi-domain
+finetuning strategy using M2DS2 can yield significant WER
+reductions, even for a few hours of audio. When small amounts
+of in-domain text is available, using a corpus augmentation
+strategy, e.g., perplexity filtering, can produce adapted LMs
+and yield small improvements to the final WER. In the case
+of sufficient amounts of unpaired in-domain text and audio,
+independent adaptation of XLSR-53 using the audio data and
+the n-gram LM using the text data can yield comparable
+performance with a fully supervised finetuning pipeline.
+X. FUTURE WORK
+In the future we plan to explore the effectiveness of the
+proposed adaptation strategy for other languages, and different
+adaptation settings, e.g., accent or cross-lingual adaptation.
+Of special interest is the investigation of the effectiveness
+of our approach for endagered languages, e.g., Pomak. Fur-
+thermore, we plan to explore the combination of in-domain
+self-supervision, when combined with other popular UDA
+techniques, e.g., teacher student models, adversarial learning,
+and data augmentation approaches. On the language adaptation
+side, we plan to explore multi-resolution learning, which has
+
+11
+shown promise for ASR [75], and investigate more elaborate
+end-to-end weakly supervised adaptation methods. Finally, we
+plan to expand our study in a multimodal setting, where both
+audio and video are available, e.g., lip reading.
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+MNRAS 000, 1–7 (2023)
+Preprint 30 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Formation of asymmetric arms in barred galaxies
+P. Sánchez-Martín,1⋆ C. García-Gómez,2 J. J. Masdemont3 and M. Romero-Gómez4
+1 IUMA, Universidad de Zaragoza, Dept. de Matemática Aplicada, Pedro Cerbuna, 12, 50009 Zaragoza, Spain
+2D.E.I.M, Universitat Rovira i Virgili, Avd. Països Catalans, 26, 43007 Tarragona, Spain
+3IEEC & Universitat Politècnica de Catalunya, Dept. de Matemàtiques, Diagonal 647, E08028 Barcelona, Spain
+4Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, E08028 Barcelona, Spain
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We establish a dynamical mechanism to explain the origin of the asymmetry between the arms observed in some barred disk
+galaxies, where one of the two arms emanating from the bar ends is very well defined, while the second one displays a ragged
+structure, extending between its ridge and the bar. To this purpose, we study the invariant manifolds associated to the Lyapunov
+periodic orbits around the unstable equilibrium points at the ends of the bar. Matter from the galaxy center is transported along
+these manifolds to the periphery, forming this way the spiral arms that emanate from the bar ends. If the mass distribution in
+the galaxy center is not homogeneous, because of an asymmetric bar with one side stronger than the other, or because of a
+non-centered bulge, the dynamics about the two unstable Lagrange points at the ends of the bar will not be symmetric as well.
+One of their invariant manifolds becomes more extended than the other, enclosing a smaller section and the escaping orbits on it
+are fewer and dispersed in a wider region. The result is a weaker arm, and more ragged than the one at the other end of the bar.
+Key words: galaxies: kinematics and dynamics – galaxies: structure – galaxies: spiral
+1 INTRODUCTION
+A high fraction of disk galaxies appear to be barred (74% to 85%
+in automatic classifications or 36% to 63% with alternative meth-
+ods (Lee et al. 2019)). These authors also suggest that strongly
+barred galaxies, classified as SBs, are preponderant in late-type
+galaxies. Most of these strongly barred galaxies appear to be asym-
+metric under a simple visual inspection. They usually present a
+strong bar and two spiral arms emanating from the bar ends, and
+frequently some inner rings as well. However, a detailed visual in-
+spection of these galaxies, available in the STScI Digitized Sky Sur-
+vey or in the Sloan Digital Sky Survey (SDSS) (Bundy et al. 2015),
+reveals that some barred galaxies exhibit important asymmetries in
+their spiral structure. While one of the two spiral arms is long and
+strongly defined, the second arm shows a ragged structure, and the
+matter is distributed irregularly between the ridge of the spiral arm
+and the bar. A non-symmetric distribution of matter in the galaxy
+disc can be a natural outcome, since disc galaxies may be formed
+through a combination of secular evolution and violent events, in-
+cluding smooth accretion, disc instabilities and minor and major
+mergers (e.g. Tonini et al. 2016). In Fig. 1 we show some examples
+of galaxies showing asymmetric discs.
+In this paper we study the dynamic response of an asymmetric
+mass distribution on the orbital structure of barred galaxies, mod-
+elling the asymmetry of the central parts by a slightly off-centered
+bulge which, viewing the galaxy as a whole, represents a simple but
+realistic model of an asymmetric bar. In this scenario, the Lagrange
+⋆ E-mail: [email protected]
+Figure 1. Images of some strongly barred galaxies showing the asymmetric
+spiral arm patterns in the R-filter from the STScI Digitized Sky Survey. In the
+top row we show the galaxies PGC 05849 (NGC 0613), PGC 12412 (NGC
+1300) and in the bottom one PGC 15941 (NGC 1672) and PGC 54849 (NGC
+5921).
+invariant points located at the bar ends will present also an asym-
+metric orbital structure, as initially shown in Colin & Athanassoula
+(1989). The new step in this work is to additionally study the unsta-
+ble periodic orbits around these Lagrange points and the associated
+unstable manifolds which provide escape routes for orbits which
+can transport material from the central regions to the outer parts of
+the discs (Romero-Gómez et al. 2006; Sánchez-Martín et al. 2016).
+© 2023 The Authors
+arXiv:2301.11385v1  [astro-ph.GA]  26 Jan 2023
+
+.PGC 05849(R)
+PGC 12412 (R)
+PGC 15941 (R)
+PGC 54849 (R)2
+P. Sánchez-Martín et al.
+These unstable manifolds are the backbones of the spiral arms ema-
+nating from the bar ends. In the case of asymmetric Lagrange points,
+however, the unstable manifolds display also important differences.
+The orbits following one of these manifolds are close together and
+can explain the presence of a strong spiral arm. On the other hand,
+the second manifold has a more open orbital structure, with its es-
+caping orbits dispersed in a wider zone, causing the ragged struc-
+ture of the second arm. Thus, the orbital structure of the unstable
+Lagrange points is able to explain the asymmetric phenomenology
+present in some of the strongly barred galaxies.
+The paper is organized as follows: in Section 2 we quantify the
+level of asymmetry in asymmetric barred galaxies using a two-
+dimensional Fourier transform method. In Sections 3 and 4 we de-
+scribe the asymmetric barred galaxy model used in this work and
+show the orbital analysis and invariant manifolds, respectively. Dif-
+ferent approaches are summarized and conclusions are given in Sec-
+tion 5.
+2 ANALYSIS OF THE BAR ASYMMETRY
+In this paper, we model the galactic asymmetric mass distribution
+using a classical galactic model with three symmetric components
+(disc, bar and bulge) but displacing slightly the bulge from the
+galaxy centre. This results in a total mass distribution biased to-
+wards one side of the bar. Many barred galaxies show some kind
+of asymmetries in their inner mass distribution with one side of the
+bar stronger than the other. An example is the galaxy NGC 1300,
+studied by Patsis et al. (2010). They generated a numerical model
+of the potential using K images of the galaxy. The resultant model
+was clearly asymmetric, containing a bar with a stronger arm. This
+model was used to study the orbits associated with this asymmetric
+barred structure. Many barred galaxies show similar asymmetries.
+Some of these galaxies are shown in Fig. 1. In order to quantify
+these asymmetries, we can analyze in detail the mass distribution of
+one of the galaxies showing this phenomenology, namely the galaxy
+PGC 70419 (NGC 7479). For this purpose, we use a galaxy image
+from the SDSS survey in the infrared z-filter. Assuming a constant
+M/L ratio, the light distribution is a good tracer of the mass distribu-
+tion in the disk. The image is previously cleaned from background
+stars and then deprojected using the FFT method (García-Gómez et
+al. 2004) giving a position angle (PA) of 38◦ and an inclination angle
+of 45◦.
+The image is then decomposed in its Fourier components using a
+technique first introduced by Considère & Athanassoula (1982); Iye
+et al. (1982), and further developed by García-Gómez et al. (2017).
+For deprojected image of the galaxy I(u,θ), where u = ln(r), we cal-
+culate the two-dimensional Fourier transform defined as
+A(p,m) =
+� umax
+umin
+� 2π
+0
+I(u,θ)ei(pu+mθ) dθdu
+(1)
+Where m is the azimuthal frequency, associated with the multiplicity
+of the structures, i.e., the number of arms while p is the radial fre-
+quency, associated to the pitch angle of the structure i, through the
+relation
+p = −
+m
+tan(i)
+(2)
+In this way, the m = 1 spectrum contains the spiral components with
+no symmetry, the m = 2 spectrum the components with a periodicity
+of π radians or bisymmetric signals, and so on for the rest of the m
+frequencies. Each of the azimuthal components m = 1,2,... can be
+further decomposed in its radial components using a Gaussian fit to
+the modulus and keeping the phase constant as follows:
+| A(p,m) |=
+Ng
+�
+j=1
+C j exp−(p− pj)2
+2σ2
+j
+.
+(3)
+In this relation, pj represents the central frequency of the Gaussian,
+σj its dispersion, and C j its amplitude. The number of Gaussians
+used in each fit, Ng, will depend on the complexity of the spectrum.
+In the upper panel of Fig. 2 we show the deprojected SDSS galaxy
+image of NGC 7479 using the z-filter. Note that one of its arms is
+very pronounced, while the opposite arm appears diffuse. On the left
+of the middle and lower panels of this figure we show the modulus
+of the Fourier spectrum of the m = 1 component which is associ-
+ated to the asymmetries in the light distribution. The modulus of the
+Fourier components are normalized to the modulus of the stronger
+component, which in this case is the m = 2 containing the bisymmet-
+ric signals of the strong bar and the spiral arms. The m = 1 spectrum
+shown here contains the spiral components responsible for the asym-
+metries in the mass distribution. The relative low values of the m = 1
+in this scale shows that the mass asymmetry is a second order effect
+for this galaxy. In red we superpose two of the Gaussian components
+into which this signal is decomposed. These Gaussian components
+can be transformed back to obtain the density distribution associated
+to each particular spiral mode. The density distributions of these spi-
+ral components are presented in the respective right panels in green,
+superposed on the galaxy image. The isocontours come from the
+normalized density corresponding to the Gaussian components and
+show that the asymmetries are related to the southern end of the
+bar and the strong arm emanating from its end. This shows that the
+galaxy has an asymmetric mass distribution, biased to the southern
+part of the galaxy.
+3 CHARACTERISTICS OF THE GALACTIC MODEL
+The equations of motion of the classical galactic model (see e.g.
+Pfenniger 1984; Skokos et al. 2002; Romero-Gómez et al. 2006;
+Sánchez-Martín et al. 2016) describe the movement of a particle in
+a gravitational potential φ . In the rotating frame, the equations of
+motion are described by
+¨r = −2(Ωp × ˙r)−Ωp ×(Ωp ×r)−∇φ,
+(4)
+where r = (x, y, z) is the position of the particle, φ is the total grav-
+itational potential of the system and Ωp is the angular velocity of
+the bar around the z-axis, Ωp = (0,0,Ω). The origin of the reference
+frame is located at the center of mass of the system and the frame is
+aligned with the main axis of the bar.
+The potential model φ used in this paper is a combination of three
+analytical components: an axisymmetric Miyamoto-Nagai disc with
+potential φd (Miyamoto & Nagai 1975), an ellipsoid Ferrers bar with
+potential φb (Ferrers 1877) and a bulge structure represented by a
+Plummer spheroid potential φbl (Plummer 1911). The total potential
+is the addition of these three components, φ = φd +φb +φbl.
+The disc potential is described by the equation
+φd = −
+GMd
+�
+R2 +(A+
+�
+B2 +z2)2
+,
+(5)
+were R2 = x2 + y2 is the cylindrical coordinate radius of the poten-
+tial in the disc plane, and z is the vertical distance over the disk
+component. The parameters G, Md, A and B denote the gravitational
+MNRAS 000, 1–7 (2023)
+
+Formation of asymmetric arms in barred galaxies
+3
+Figure 2. Analysis of the m = 1 component of the SDSS galaxy image of
+NGC 7479 in the z-filter. The upper panel shows the deprojected image of the
+galaxy. On the left of the middle and lower panels we show the modulus of
+the m = 1 component of the Fourier spectrum, and with red-dashed lines two
+Gaussian components fitted to this modulus. In the respective right panels,
+we show in green the density distribution associated to each of these single
+components superimposed on the galaxy image.
+constant, the disc mass and the shape of the disc, respectively. Tak-
+ing A = 0 the potential becomes the Plummer potential. The bar is
+modelled by an ellipsoid with density function
+ρ =
+�
+ρ0(1−m2)nh,
+m ≤ 1
+0,
+m > 1
+(6)
+where m2 = x2/a2 + y2/b2 + z2/c2, a (semi-major axis), b (interme-
+diate axis) and c (semi-minor axis) determine the shape of the bar,
+nh is the homogeneity degree of the mass distribution (nh = 2 in our
+work) and ρ0 is the density at the origin (ρ0 = 105
+32π
+GMb
+abc if nh = 2,
+where Mb is the bar mass).
+The unit of length considered is the kpc, the time unit is ut =
+2 × 106 yr, Ω is in [ut]−1, and the mass unit is uM = 2 × 1011M⊙,
+where M⊙ denotes the mass of the Sun. G stands for the gravitational
+constant.
+In our model, we select a disc radius of A = 3 kpc, height B = 1 kpc
+and mass giving a value of GMd = 0.52 kpc3/u2
+t . The dimensions of
+the bar are a = 6 kpc, b = 1.5 kpc, c = 0.4 kpc, and its mass is such
+that GMb = 0.4 kpc3/u2
+t . The Plummer bulge with a radius B = 1 kpc
+and GMbl is set to have around 15% of the mass of the bar GMb, in
+order that G(Md + Mb + Mbl) = 1. The bar pattern speed is fixed as
+Ω = 0.0633 [ut]−1 (∼ 30.97 km/s/kpc).
+In the rotating reference frame aligned with the main axis of the
+bar, the equations of motion given in Eq.(4) are written as the fol-
+lowing dynamical system:
+���������
+¨x = 2Ω ˙y+Ω2 x−φx
+¨y = −2Ω ˙x+Ω2 y−φy
+¨z = −φz .
+(7)
+The Jacobi first integral of Eq.(4) (which can be regarded as the
+energy in the rotating frame) is given by
+CJ(x,y,z, ˙x, ˙y, ˙z) = −(˙x2 + ˙y2 + ˙z2)+Ω2 (x2 +y2)−2φ,
+(8)
+and the effective potential is defined by φeff = φ− 1
+2 Ω2 (x2 +y2).
+The goal of this paper is to analyze which is the effect of an asym-
+metric distribution of mass in the central parts of the galaxy on the
+external spiral structure. To introduce this asymmetry, we displace
+the bulge potential along the x-axis (main axis of the bar) towards the
+equilibrium point placed at the right end of the bar. The bulge center
+is then located at (xd,0,0) using several values for the displacement
+xd (in kpc), namely: (0,0,0), (0.5,0,0), (1,0,0) and (1.5,0,0). We
+move the center of our potential model to the resulting center of
+mass of the system. The plot of the equal density contours of the
+resulting models are shown in Fig. 3. Note that the displacement of
+the bulge along the main axis of the bar creates an asymmetry in the
+central part of the model and around the libration points L1 and L2.
+The rotation curves of the model, defined as V2
+rot = r dφ
+dr , where the
+potential is φ = φd +φb +φbl with the selected values of the parame-
+ters and for the above explained positions of the bulge displacement,
+are shown in Fig. 4. The resulting rotation curve is reasonably flat in
+the outer parts, and displays only minor differences in the position
+of the maximum.
+4 DYNAMICS OF THE MODELS
+The solutions of ∇φeff = 0 in rotating coordinates give five La-
+grangian equilibrium points of the model (Li, i = 1,...,5). Points L1
+and L2 are linearly unstable points and lie on the x-axis at the ends of
+the bar. Point L3 is linearly stable and it is placed on the origin of co-
+ordinates in the case of a symmetric model. Points L4 and L5 are also
+linearly stable and located out of the x-axis. A detailed explanation
+of the dynamics around these points can be found in Athanassoula
+et al. (1983); Romero-Gómez et al. (2006); Sánchez-Martín et al.
+(2016).
+The regions where φeff > CJ are forbidden regions for a star of en-
+ergy CJ. In the plane, these regions are delimited by the zero velocity
+curves, which are defined by the level surfaces φeff = CJ intersected
+with z = 0. In Fig. 5 we show the zero velocity curves corresponding
+to an energy slightly above of that of the equilibrium point, CJ,Li +δ,
+for the models with off-centered bulges in Fig. 4, i.e., setting the val-
+ues xd = 0, 0.5, 1, 1.5. These zero velocity curves limit the inner and
+outer regions in the galaxy. In the symmetric case (xd = 0), the en-
+ergy of both equilibrium points L1 and L2 is the same, CJ,L1 = CJ,L2.
+For the asymmetric cases, as xd grows CJ,L2 becomes smaller than
+CJ,L1, which makes the zero velocity curves related to L1 more open
+at the opposite point.
+Particular attention is given in this work to the unstable points
+L1 and L2. They are surrounded by planar and vertical families of
+Lyapunov periodic orbits, which are unstable in the neighborhood of
+the equilibrium point. The relevant family for the transport of matter
+between the inner and outer regions of the galaxy is the planar family
+(Romero-Gómez et al. 2009). Fig. 6 shows the (x, y) projection of
+the planar family for the models with xd = 0 (solid black line) and
+xd = 1.5 (dotted red line). The family around L2 is displayed at the
+left panel and that around L1 at the right one. In the symmetric case
+(xd = 0) both families coincide, in the asymmetric one (xd = 1.5) the
+family around L2 is smaller than the one around L1.
+These critical points are characterized by the superposition of a
+saddle and two harmonic oscillations in the rotating frame. Conse-
+quently, for a given Jacobi constant, stable and unstable invariant
+MNRAS 000, 1–7 (2023)
+
+PGC70419
+ (z)
+NGC 7479
+m=1
+A=0.11
+g=
+0.57
+A
+m=1
+A=
+0.09
+1.86
+=6
+0.86
+A
+-10
+0
+10
+p4
+P. Sánchez-Martín et al.
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+Figure 3. Isodensity curves of the potential φ = φd +φb +φbl. Equilibrium points of the system are marked with a cross. The Ferrers bar and the Plummer bulge
+are outlined by dotted black curves. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0) and at (1.5,0,0).
+0
+2
+4
+6
+8
+10
+R [kpc]
+0
+50
+100
+150
+Vrot  [km/s]
+Figure 4. Rotation curve of the potential φ = φd + φb + φbl for the bulge
+centered at (0,0,0) (in blue), (0.5,0,0) (in red), (1,0,0) (in magenta) and
+(1.5,0,0) (in green).
+manifolds emanate from the periodic Lyapunov orbit around each
+point. The stable manifold is defined as the set of orbits that asymp-
+totically tend to the periodic orbit forward in time, and the unstable
+manifold consists of those orbits which depart asymptotically from
+the periodic orbit. These latter manifold drive the escape orbits that
+are responsible for the visible trajectories, in the form of arms and
+rings. Fig. 7 shows the invariant manifolds associated to L1 and L2
+for the set of models where xd = 0, 0.5, 1, 1.5. The effect of an asym-
+metric mass distribution, modelled by the displacement of the bulge,
+makes the exterior manifold that emanates from L2 to differ from the
+invariant manifold associated with L1.
+The transit orbits trapped inside the manifolds are in charge of
+the transfer of matter, from the inner to the outer regions delimited
+by the zero velocity curves (Gidea & Masdemont 2007; Romero-
+Gómez et al. 2006; Sánchez-Martín et al. 2018). As the dynamics of
+our system (4) takes place in a four dimensional phase space when
+we consider orbits with z = ˙z = 0 (the plane z = 0 is invariant), the
+intersection of the trajectories of the inner branch of the stable in-
+variant manifold of a Lyapunov orbit with the hyperplane S defined
+by the section x = 0 in phase space gives a closed curve in the (y, ˙y)
+projection. Given a pair (y, ˙y) and an energy level, this lets us to de-
+fine a state on S by selecting (0,y,0, ˙x, ˙y,0), where ˙x is determined
+by the fixed energy level of the Lyapunov orbit and the orientation
+of the crossing, taking into account Eq. (8),
+˙x =
+�
+−˙y2 +Ω2y2 −2φ(0,y,0)−(CJ,Li +δ),
+(9)
+with CJ,Li +δ the energy of the Lyapunov orbit around the point Li,
+i = 1,2, slightly above of that of the equilibrium point. The forward
+in time integration of initial conditions corresponding to (y, ˙y) points
+inside the closed curve establishes the trajectories of the particles
+confined inside the invariant manifold that transit from the inner to
+the outer region.
+This procedure enables us to quantify the amount of matter trans-
+ferred inside each invariant manifold associated to any energy level
+and, consequently, from each spiral arm of the galaxy. Fig. 8 repre-
+sents the (y, ˙y) projection of the intersection with the hyperplane S
+of the invariant manifolds arising from three orbits with different Ja-
+cobi constants in the Lyapunov family around L2 (top), and the same
+corresponding orbits for L1 (bottom). The three closed curves are
+displayed with different colors, from red to yellow, according to the
+increasing Jacobi constant. The initial conditions located inside each
+of the closed curves are marked with crosses of the same color as the
+curve. From left to right, the figure displays the (y, ˙y) projection for
+the models with bulge center ranging from (0,0,0) to (1.5,0,0). The
+main feature to notice in this figure is the narrowing and stretching
+of the curves as the bulge moves away from the L2 equilibrium point.
+This constriction marks the difference between initial conditions for
+the escape orbits related with L2 and those related to L1. The initial
+conditions emanating from near L2 are more extended along the y-
+axis, resulting in a dispersion in space of the orbits enclosed by the
+manifold. As we integrate forward in time these initial conditions,
+we obtain fewer escape orbits, and dispersed in a wider region, in
+comparison to those emanating near L1. So, the spiral arm defined
+by these orbits becomes more spread out and less bright. The result
+of these integrations is exhibited in Fig. 9 where it can be appreci-
+ated how the orbits inside the invariant manifolds develop the arms.
+The orbits corresponding to the L1 point are more concentrated as
+the density of the bar increases in the region close to L1.
+5 DISCUSSION AND CONCLUSIONS
+The goal of this work is to analyze the relation between the asym-
+metric arms observed in certain barred galaxies and the mass distri-
+bution in the central part of the galaxy. We propose and show that
+there is a strong correlation between their asymmetries. The mod-
+els used to study this fact consist of a superposition of a bar and an
+off-centered bulge. The displacements of the bulge along the main
+axis of the bar introduce an increasing asymmetry in the central part
+of the model. When the center of mass of the galaxy is displaced
+along the major axis of the bar, the zero velocity curves of the ef-
+fective potential become asymmetric: the one at the opposite side
+of the dispacement becomes more open (see Fig. 5). We show that
+this asymmetry between the zero velocity curves in their opening
+is carried to the orbits trapped by invariant manifolds around the
+Lagrangian equilibrium points L1 and L2. This orbits become asym-
+MNRAS 000, 1–7 (2023)
+
+Formation of asymmetric arms in barred galaxies
+5
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+-5
+0
+5
+x
+-5
+0
+5
+y
+Figure 5. Zero velocity curves for a Jacobi constant slightly above to that of L1 (blue) and for one slightly above to that of L2 (red). Equilibrium points of
+the system are marked with a cross. The Ferrers bar and the Plummer bulge are outlined by dotted black curves. From left to right: Bulge centered at (0,0,0),
+(0.5,0,0), (1,0,0) and at (1.5,0,0).
+-7
+-6.5
+-6
+-5.5
+x
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+y
+5.5
+6
+6.5
+7
+x
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+y
+Figure 6. Lyapunov family of periodic orbits around Li, i = 1, 2, for a range
+of values of the Jacobi constant in (CJ,Li, CJ,Li +10−4). Solid black line: (x, y)
+projection of the family around L2 (left) and L1 (right) for the symmetric
+model with bulge centered at (0,0,0), CJ,Li = −0.2338, i = 1, 2. Dotted red
+line: (x, y) projection of the family around L2 (left) and L1 (right) for the
+asymmetric model with bulge centered at (1.5,0,0), CJ,L2 = −0.2358, CJ,L1 =
+−0.2331.
+metric as well, with the arm at the opposite side of the displacement
+wider and more spread out. We quantify this effect by computing the
+fraction of orbits trapped by the manifolds. The procedure consists
+in intersecting the manifold with a hyperplane and, inside the result-
+ing closed curve in the (y, ˙y) projection, obtaining the set of points
+with the same energy of the manifold as initial conditions that char-
+acterize the escaping orbits.
+Indeed, the above closed curve turns out to be the most relevant
+feature in order to predict the asymmetry of the arms. When the
+bulge is displaced along the main axis of the bar, causing an asym-
+metry in the density distribution of the model, the closed curves
+around the equilibrium point at the end of the less dense side of
+the bar, narrow. This leads to a smaller measure of states in phase
+space that become initial conditions for escaping orbits. Moreover,
+these closed curves are more stretched, making the distribution of
+the points inside them more spread out in space. Both aspects are
+responsible for the resulting ragged and dispersed spiral arms in the
+less dense end of the bar, while the arm associated with the denser
+end becomes brighter and well defined.
+The dynamics of off-centered bars have been studied using ana-
+lytical models (Colin & Athanassoula 1989), showing how the La-
+grangian points vary as a function of the displacement with respect
+to the center of mass of the galaxy. Łokas (2021) digs into the
+barred galaxies in the IllustrisTNG simulations to check how com-
+mon asymmetric and off-centered bars are and study its possible ori-
+gin, concluding that asymmetric bars are persistent in time and this
+asymmetric may be due to the interaction with a companion galaxy
+or due to the disc itself being asymmetric. In either case, no further
+development has been proposed to link the asymmetry in the bar
+with the one-armed dominant spiral structure.
+As shown in Fig. 1, this is a quite common phenomenon in barred
+galaxies. In particular, the dynamics of NGC 1300 have been studied
+in detail by (Patsis et al. 2010), including an orbital analysis. The iso-
+contours of a smooth K-band image (see their Fig. 1) show a clear
+asymmetric bar distribution leading to an asymmetry in the spiral
+arms, which is reproduced by the different orbital models. Other ex-
+amples of one armed dominated galaxies with an asymmetric bar
+may be: NGC 4027 (Phookun et al. 1992), the density contours
+show an asymmetric and off-centered bar leading to a mass distri-
+bution dominated by an m = 1 mode, though a weak counter-part is
+also clear; the Large Magellanic Cloud (de Vaucouleurs & Freeman
+1972), showing a rotational asymmetry, which is confirmed by more
+recent studies (e.g. Jiménez-Arranz et al. 2022; Niederhofer et al.
+2022, and references therein).
+To sum up, we show that asymmetric arms are a common feature
+in barred galaxies and that there is a clear correlation between arm
+asymmetry and the displacement of the center of mass caused by an
+asymmetric central density distribution. Escaping orbits trapped in
+the invariant manifolds of an asymmetric bar distribution are asym-
+metric and the limiting case would be to have only one armed barred
+spiral.
+ACKNOWLEDGEMENTS
+P.S.M. thanks the Spanish Ministry of Economy grants PID2020-
+117066GB-I00
+and
+PID2021-123968NB-I00.
+J.J.M.
+thanks
+MINECO-FEDER for the grant PID2021-123968NB-I00 and
+the Catalan government grant 2017SGR-1049. M.R.G. thanks
+the Spanish Ministry of Science grant MICIU/FEDER RTI2018-
+095076-B-C21, and the Institute of Cosmos Sciences University
+of Barcelona (ICCUB, Unidad de Excelencia "María de Maeztu")
+grant CEX2019-000918-M.
+Funding for the Sloan Digital Sky Survey IV has been provided
+by the Alfred P. Sloan Foundation, the U.S. Department of En-
+ergy Office of Science, and the Participating Institutions. SDSS
+(www.sdss.org) acknowledges support and resources from the Cen-
+ter for High-Performance Computing at the University of Utah.
+DATA AVAILABILITY
+The data underlying this article are available in the article.
+MNRAS 000, 1–7 (2023)
+
+6
+P. Sánchez-Martín et al.
+-10
+0
+10
+x
+-10
+-5
+0
+5
+10
+y
+-10
+0
+10
+x
+-10
+-5
+0
+5
+10
+y
+-10
+0
+10
+x
+-10
+-5
+0
+5
+10
+y
+-10
+0
+10
+x
+-10
+-5
+0
+5
+10
+y
+Figure 7. Unstable invariant manifolds associated to the Lyapunov periodic orbits of L1 and L2. The position of the equilibrium points is marked with crosses.
+The bar and the bulge are outlined by dotted black curves. The reference system is marked with a solid black line and the center of the bar with a dotted magenta
+line. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0) and at (1.5,0,0).
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+2.8
+y
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+1.6
+1.8
+2
+2.2
+2.4
+2.6
+y
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+y
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+y
+0.1
+0.15
+0.2
+0.25
+0.3
+-2.6
+-2.4
+-2.2
+-2
+y
+-0.06
+-0.04
+-0.02
+0
+-2.8
+-2.6
+-2.4
+-2.2
+y
+-0.06
+-0.04
+-0.02
+0
+-3
+-2.8
+-2.6
+-2.4
+y
+-0.06
+-0.04
+-0.02
+0
+-3
+-2.8
+-2.6
+-2.4
+y
+-0.06
+-0.04
+-0.02
+0
+Figure 8. (y, ˙y) projection of the intersection of the plane S with the stable manifolds associated to three orbits with different Jacobi constant of the Lyapunov
+family around L2 (top) and L1 (bottom). The intersection of each manifold is in a color, from red to yellow, according to the energy of the manifold. Initial
+conditions distributed inside each curve are marked with a cross with the same color as the curve. From left to right: Bulge centered at (0,0,0), (0.5,0,0), (1,0,0)
+and (1.5,0,0). Note that the axis limits are different but their scale and range length are constant in each row.
+Figure 9. Orbits resulting from the integration of the initial conditions of Fig. 8. Equilibrium points marked in red. Bar and bulge outlined by dotted black
+curves. The reference system is marked with a solid black line and the center of the bar with a dotted magenta line. From left to right: Bulge centered at (0,0,0),
+(0.5,0,0), (1,0,0) and at (1.5,0,0).
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+

1NE4T4oBgHgl3EQfaAwh/content/tmp_files/2301.05060v1.pdf.txt

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+arXiv:2301.05060v1  [cs.CR]  12 Jan 2023
+Evaluating the Fork-Awareness of Coverage-Guided Fuzzers
+Marcello Maugeri1
+a, Cristian Daniele2
+b, Giampaolo Bella1
+c, and Erik Poll2
+d
+1Department of Maths and Computer Science, University of Catania, Catania, Italy
+2Department of Digital Security, Radboud University, Nijmegen, The Netherlands
+[email protected], [email protected], [email protected], [email protected]
+Keywords:
+Fuzzing, Fork, Security Testing, Software Security
+Abstract:
+Fuzz testing (or fuzzing) is an effective technique used to find security vulnerabilities. It consists of feeding a
+software under test with malformed inputs, waiting for a weird system behaviour (often a crash of the system).
+Over the years, different approaches have been developed, and among the most popular lies the coverage-based
+one. It relies on the instrumentation of the system to generate inputs able to cover as much code as possible.
+The success of this approach is also due to its usability as fuzzing techniques research approaches that do
+not require (or only partial require) human interactions. Despite the efforts, devising a fully-automated fuzzer
+still seems to be a challenging task. Target systems may be very complex; they may integrate cryptographic
+primitives, compute and verify check-sums and employ forks to enhance the system security, achieve better
+performances or manage different connections at the same time. This paper introduces the fork-awareness
+property to express the fuzzer ability to manage systems using forks. This property is leveraged to evaluate 14
+of the most widely coverage-guided fuzzers and highlight how current fuzzers are ineffective against systems
+using forks.
+1
+Introduction
+In the last years, plenty of fuzzers have been devel-
+oped to deal with sophisticated software and nowa-
+days it is extremely common that network systems
+employ forks to deal with different connections at the
+same time. This leads to 1) the need to devise accurate
+and ad-hoc fuzzers and 2) the need to evaluate these
+fuzzer according to their ability to cope with such ad-
+vanced systems.
+Unfortunately,
+as
+pointed
+out
+in
+[Hazimeh et al., 2020],
+it is not easy to bench-
+mark all of them since the fuzzers are very different
+from each other. Metzman et al. faced this problem
+by devising FuzzBench [Metzman et al., 2021], an
+open-source service for the evaluations of state-
+less fuzzers.
+Later, Natella and Pham presented
+ProFuzzBench
+[Natella and Pham, 2021],
+which
+similarly to FuzzBench provides a service to evaluate
+stateful fuzzers.
+Although FuzzBench includes a sample of real
+word programs and ProFuzzBench includes dif-
+a
+https://orcid.org/0000-0002-6585-5494
+b
+https://orcid.org/0000-0001-7435-4176
+c
+https://orcid.org/0000-0002-7615-8643
+d
+https://orcid.org/0000-0003-4635-187X
+ferent network systems (i.e.
+systems that often
+employ forks to deal with multiple connections
+[Tanenbaum, 2009]), they do not evaluate the ability
+of the fuzzers to cope with programs that use forks.
+Despite forks representing the only way to create
+a new process [Tanenbaum, 2009], experimental re-
+sults have shown that current fuzzers cannot deal with
+forked processes.
+The existing approach merely relies on code mod-
+ifications to remove the forks. Unfortunately, this ap-
+proach goes against the willingness to reduce manual
+work and improve automation during a fuzzing cam-
+paign. [Boehme et al., 2021].
+In this work, we explore and classify the limita-
+tions current fuzzers exhibit in front of forking pro-
+grams.
+In summary, this paper:
+1. devises a novel property capturing the ability of
+fuzzers to deal with forks appropriately;
+2. evaluates 14 coverage-guided fuzzers based on
+this property;
+3. proposes possible improvements to the current
+state-of-the-art and future directions.
+The paper is organised as follows. Section 2 de-
+scribes the relevant background, Section 3 presents
+
+our contributions to knowledge, Section 4 shows the
+existing approaches that try to cope with the fork
+problem and, eventually, Section 5 discuss the results
+and propose possible future directions.
+2
+Background
+2.1
+Fuzz testing
+Fuzzing is an automated testing technique pioneered
+by Miller et al. [Miller et al., 1990] in 1990 to test
+UNIX utilities. As outlined in Figure 1, coverage-
+guided fuzzing is composed at least of seed selection,
+input generation and system execution.
+1) Seeds selection. The user must provide some
+input messages (seeds) representative of some usual
+inputs for the system.
+2) Input generation. The core of every fuzzer is
+the generation of slightly malformed input messages
+to forward to the software under test. A fuzzer is as
+efficient as the generated inputs are able to break the
+system. According to the approach used to generate
+the messages, the fuzzers may be classified into:
+• dumb: generate random strings (as the first fuzzer
+[Miller et al., 1995] did);
+• dumb mutational: blindly mutate seed messages
+provided by the user;
+• grammar-based: leverage the grammar of the sys-
+tem to craft the input messages;
+• smart mutational (often called evolutionary): re-
+quire a sample of inputs and leverage feedback
+mechanisms to craft system-tailored messages.
+An example of feedback mechanisms is the code
+coverage feedback, explored in Section 2.2.
+3) System execution. Each execution of the fuzzer
+involves three components:
+• Bugs detector: it reports eventual bugs. The ma-
+jority of the bugs detectors only report crashes,
+however for many systems, also a weird deviation
+from the happy flow of the protocol may represent
+significant security issues;
+• Hangs detector:
+it detects program execution
+hangs;
+• Code coverage detector: as further explained in
+Section 2.2, the code coverage represents one of
+the feedbacks the fuzzer leverages to improve the
+quality of the input messages.
+2.2
+Coverage-Guided Fuzzing
+Smart mutational fuzzers use feedback mecha-
+nisms to steer the generation of the messages.
+Different
+types
+of
+feedback
+mechanisms
+exist
+[Shahid et al., 2011], and often different terms are
+used to express the same idea. To avoid further noise,
+in this work we use the term code coverage to express
+the lines of code that are reached by a specific mes-
+sage.
+Code coverage fuzzers need to recompile the code
+with ad-hoc compilers (e.g. the AFL compiler) to in-
+strument the code and obtain run-time information.
+AFL [Zalewski, 2017], for example, instruments
+the code to fill a bitmap that represents the lines of
+the code covered by the inputs.
+Later, it uses this bitmap to assign a higher score
+to messages able to explore previously unseen lines of
+code.
+Start
+Seeds selection I
+Input
+generation
+System execution
+Hangs
+detector
+Bugs
+detector
+Code
+coverage
+detector
+Report O
+End
+Figure 1: Coverage-guided fuzzing process
+
+2.3
+Inter-Process Communication
+Operating systems provide system calls to perform
+different tasks (e.g. writing and reading files, access-
+ing hardware services, creating and executing new
+processes). On UNIX systems, new processes are cre-
+ated by using the fork system call [Tanenbaum, 2009].
+In short, the first process, called parent process, gen-
+erates a clone, called child process, that is an exact
+copy of the parent process. After the fork, file de-
+scriptors and registers are duplicated, thus a change
+in one of the processes does not affect the other one.
+Also, the parent and child process will follow sepa-
+rate execution paths.
+3
+Our contribution
+This paper aims to understand how the state-of-the-
+art coverage-guided fuzzers deal with software under
+tests containing forks.
+It was not obvious to come up with a way to com-
+pare and contrast the various tools.
+We devised a
+novel property, the fork awareness, that must be sat-
+isfied when a fuzzer deals with forks effectively and
+efficiently. As we shall see below, fork awareness
+rests upon three aspects representing the ability to
+deal with child processes.
+Also, we evaluate the novel property over the
+most widely used fuzzers from two benchmark frame-
+works, reaching a total of 14 evaluated tools, 11
+drawn from FuzzBench and 3 from ProFuzzBench.
+3.1
+Fork-awareness
+Abstractly, fork awareness insists that every fuzzer
+should address the child process as the parent one.
+During the system execution, the system monitor
+should detect bugs or hangs regardless of their loca-
+tion and the coverage should be measured also in the
+child process. This is formalised through Definition 1.
+Definition 1. A coverage-guided fuzzer is fork-aware
+if it can detect bugs and hangs and measure coverage
+in the same way for both the child and the parent’s
+branch.
+The three aspects in this definition are called:
+[C.1] Child bugs detection: any anomaly is reported
+also if it occurs in child processes;
+[C.2] Child hangs detection: any infinite hang is re-
+ported also if it occurs in child processes;
+[C.3] Child code coverage: code coverage is measured
+also for child processes.
+3.2
+Example challenges
+We wrote three simple C programs to use as chal-
+lenges for the fuzzers, namely to test whether the
+fuzzers satisfy the aspects given above.
+a) Bugs detection challenge:
+1
+if(fork()==0){ //Child process
+2
+raise(SIGSEGV); //Simulated crash
+3
+} else { //Parent process
+4
+wait(NULL); //Waiting child
+5
+//termination
+6
+}
+The snippet sends a SIGSEGV signal to simulate
+a bug in the child process. This signal is used to
+report a segmentation fault, i.e. a memory access
+violation, which is common in programs written
+in low-level languages. The fuzzer must detect
+this bug also after the parent’s termination.
+b) Hangs detection challenge:
+1
+if(fork()==0){ //Child process
+2
+while(1){ ; } //Simulation of
+3
+//blocking code
+4
+}
+The snippet simulates an infinite loop in the child
+process. The fuzzers must report processes still
+in execution after the loop and must kill child
+processes at the end of the fuzzing campaign,
+avoiding pending process executions.
+c) Code coverage challenge:
+1
+pid_t pid = fork();
+2
+if(pid==0){ //Child process
+3
+if(data %2 == 0){ do_something(); }
+4
+else { do_something(); }
+5
+if(data %3 == 0){ do_something(); }
+6
+else { do_something(); }
+7
+if(data %5 == 0){ do_something(); }
+8
+else { do_something(); }
+9
+if(data %7 == 0){ do_something(); }
+10
+else { do_something(); }
+11
+}
+12 else { //Parent process
+13
+wait(NULL); //Waiting child
+14
+//termination
+15
+}
+This snippet simulates a child with several branches.
+A fuzzer must cover and consider every child’s
+branches.
+We run the 14 fuzzers over these challenges and
+organised the results in Table 1. We noticed that none
+of the fuzzers succeeded through all three challenges.
+
+Fuzzer
+Based on
+Monitor technique
+Bugs
+Detection
+(C1)
+Hangs
+Detection
+(C2)
+Code
+coverage
+(C3)
+AFL [Zalewski, 2017]
+-
+POSIX signals
+×
+×
+✓
+AFL++ [Fioraldi et al., 2020]
+AFL
+POSIX signals
+×
+×
+✓
+AFLFast [Bohme et al., 2017]
+AFL
+POSIX signals
+×
+×
+✓
+AFLSmart [Pham et al., 2021]
+AFL
+POSIX signals
+×
+×
+✓
+Eclipser [Choi et al., 2019]
+AFL
+POSIX signals
+×
+×
+✓
+FairFuzz [Lemieux and Sen, 2018]
+AFL
+POSIX signals
+×
+×
+✓
+lafintel [Besler and Frederic, 2016]
+AFL
+POSIX signals
+×
+×
+✓
+AFLnwe1
+AFL
+POSIX signals
+×
+×
+✓
+AFLNet [Pham et al., 2020]
+AFL
+POSIX signals
+×
+×
+✓
+MOpt-AFL [Lyu et al., 2019]
+AFL
+POSIX signals
+×
+×
+✓
+StateAFL [Natella, 2022]
+- AFL
+- AFLNet
+POSIX signals
+×
+×
+✓
+LibFuzzer2
+-
+- UBSAN
+- ASAN
+- MSAN
+✓
+×
+✓
+Entropic [Bohme et al., 2020]
+LibFuzzer
+- UBSAN
+- ASAN
+- MSAN
+✓
+×
+✓
+Honggfuzz3
+-
+ptrace (Linux)
+✓
+×
+✓
+Table 1: Coverage guided fuzzers evaluation
+3.3
+Testbed
+We decided to analyse only the coverage-guided
+fuzzers present in FuzzBench [Metzman et al., 2021]
+and ProFuzzBench [Natella and Pham, 2021] even
+though the property applies to every coverage-guided
+fuzzer. All fuzzers were executed on an Ubuntu 20.04
+server machine and all our source codes are freely
+available online4 so that our experiments are fully re-
+producible.
+3.4
+Fuzzers evaluation
+We run all selected fuzzers against our three example
+challenges. Table 1 summarises our findings.
+All the fuzzers based on AFL use POSIX signals
+and a bitmap respectively to report bugs and keep
+track of the code coverage.
+As shown in the Table 1, while the bitmaps are
+able to keep track of the child’s code coverage, bugs
+triggered in the child’s processes are not detected
+since AFL catches signals from the main process
+only, as pointed out in the documentation5. The only
+fuzzers able to detect bugs in the child process are
+LibFuzzer6, Entropic [Bohme et al., 2020] and Hong-
+4https://github.com/marcellomaugeri/forks-break-afl
+5https://github.com/google/AFL/blob/master/README.md
+6https://llvm.org/docs/LibFuzzer.html
+fuzz7, as discussed in more detail below:
+• LibFuzzer 8 and Entropic [Bohme et al., 2020]
+employ a set of sanitizers9 to report bugs. These
+mechanisms make the fuzzers able to find the
+bug in Challenge 1 and measure the different
+code paths in Challenge 3, thereby satisfying chal-
+lenges C.1 and C.3, as seen above. Unfortunately,
+challenge C.2 is not satisfied since the fuzzer can-
+not detect hangs in the child process.
+• Honggfuzz supports different software/hardware
+feedback mechanisms and a low-level interface to
+monitor targets.
+When executed on Linux ma-
+chines, Honggfuzz uses the ptrace system call to
+manage processes.
+This mechanism allows the
+fuzzer to capture a wide range of signals.
+As
+shown in Table 1, the use of ptrace (along with
+the SanitizerCoverage) allows the fuzzer to detect
+bugs and to consider coverage also in the child
+process. Unfortunately, neither this mechanism is
+able to detect hangs in the child process.
+In summary, while all selected fuzzers detect the
+code coverage (C3), none detect hangs (C2) and only
+a few detect bugs (C1) in the child process. The eval-
+uation underlines that:
+7https://honggfuzz.dev/
+8https://llvm.org/docs/LibFuzzer.html
+9AddressSanitizer,
+UndefinedBehaviorSanitizer
+and
+MemorySanitizer
+
+• Loops detection challenge is the most difficult be-
+cause fuzzers do not wait for all the child pro-
+cesses but only for the main one;
+• Code coverage challenge is the easiest because
+the instrumentation allows measuring coverage
+from the execution, regardless of the process in-
+volved;
+• Bug detection challenge depends on the technique
+used to observe bugs, as well as the use of sanitis-
+ers.
+We interpret this general outcome as a clear call for
+future research and developments.
+4
+Existing solutions
+Nowadays the only solutions to fuzz programs that
+use forks are manually modifying the code or break-
+ing the multi-process nature of the system (by em-
+ploying tools like defork10) in order to get rid of the
+forks.
+Unfortunately, making modifications to the code,
+as pointed out in the AFLNet documentation 11, to
+remove all the forks is a challenging and error-prone
+task and break the multi-process nature of the system
+often leads to weird system behaviours. The only so-
+lution, therefore, remains to modify the fuzzers.
+5
+Conclusions
+This paper analyses the fork awareness of the
+coverage-guided fuzzers using three different aspects.
+The analysis conducted on 14 well-known fuzzers
+highlights that while is it clear how important is to
+handle multi-process programs, the majority of the
+fuzzers overlook the problem. 11 of 14 fuzzers are
+not able to detect bugs in the child process. The intu-
+ition behind these outcomes is related to the way these
+fuzzers detect bugs. All the AFL-derived fuzzers use
+signals (SIGSEGV, SIGABRT, etc) to detect bugs and
+this mechanism misses bugs in child processes. We
+noticed that dealing with forks is not the only problem
+and other issues may be related to the IPC scheduling.
+For example, the IPC may influence the success of
+the fuzzing process since some bugs may be triggered
+only after a specific process schedule and only after
+access to a particular cell of memory. We believe this
+paper represents a first step towards the devising of
+fuzzers aware of the eventual multiprocess nature of
+10https://github.com/zardus/preeny/blob/master/src/defork.c
+11https://github.com/aflnet/aflnet
+the software. The first step to achieve this goal might
+be the implementation of a loop detector at an early
+stage, e.g. by leveraging a dynamic library to keep
+track of all process identifiers of forked processes. To
+summarise, this work not only provides the first con-
+crete way to evaluate the fuzzers according to their
+fork awareness but sheds light for the first time on a
+class of problems that have been ignored until now,
+showing interesting future directions.
+REFERENCES
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+This figure "orcid.png" is available in "png"� format from:
+http://arxiv.org/ps/2301.05060v1
+
+ScileP
+TeSS
+ScienceandTechnologyPublications

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+page_content='CR]  12 Jan 2023  Evaluating the Fork-Awareness of Coverage-Guided Fuzzers  Marcello Maugeri1  a, Cristian Daniele2  b, Giampaolo Bella1  c, and Erik Poll2  d  1Department of Maths and Computer Science, University of Catania, Catania, Italy  2Department of Digital Security, Radboud University, Nijmegen, The Netherlands  marcello.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+page_content='it, cristian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='daniele@ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='nl, giampaolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='bella@unict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='it, erikpoll@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='nl  Keywords:  Fuzzing, Fork, Security Testing, Software Security  Abstract:  Fuzz testing (or fuzzing) is an effective technique used to find security vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' It consists of feeding a  software under test with malformed inputs, waiting for a weird system behaviour (often a crash of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Over the years, different approaches have been developed, and among the most popular lies the coverage-based  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' It relies on the instrumentation of the system to generate inputs able to cover as much code as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The success of this approach is also due to its usability as fuzzing techniques research approaches that do  not require (or only partial require) human interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Despite the efforts, devising a fully-automated fuzzer  still seems to be a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Target systems may be very complex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' they may integrate cryptographic  primitives, compute and verify check-sums and employ forks to enhance the system security, achieve better  performances or manage different connections at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This paper introduces the fork-awareness  property to express the fuzzer ability to manage systems using forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This property is leveraged to evaluate 14  of the most widely coverage-guided fuzzers and highlight how current fuzzers are ineffective against systems  using forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  1  Introduction  In the last years, plenty of fuzzers have been devel-  oped to deal with sophisticated software and nowa-  days it is extremely common that network systems  employ forks to deal with different connections at the  same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This leads to 1) the need to devise accurate  and ad-hoc fuzzers and 2) the need to evaluate these  fuzzer according to their ability to cope with such ad-  vanced systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Unfortunately,  as  pointed  out  in  [Hazimeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020],  it is not easy to bench-  mark all of them since the fuzzers are very different  from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' faced this problem  by devising FuzzBench [Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2021], an  open-source service for the evaluations of state-  less fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Later, Natella and Pham presented  ProFuzzBench  [Natella and Pham, 2021],  which  similarly to FuzzBench provides a service to evaluate  stateful fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Although FuzzBench includes a sample of real  word programs and ProFuzzBench includes dif-  a  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/0000-0002-6585-5494  b  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/0000-0001-7435-4176  c  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/0000-0002-7615-8643  d  https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/0000-0003-4635-187X  ferent network systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  systems that often  employ forks to deal with multiple connections  [Tanenbaum, 2009]), they do not evaluate the ability  of the fuzzers to cope with programs that use forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Despite forks representing the only way to create  a new process [Tanenbaum, 2009], experimental re-  sults have shown that current fuzzers cannot deal with  forked processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The existing approach merely relies on code mod-  ifications to remove the forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Unfortunately, this ap-  proach goes against the willingness to reduce manual  work and improve automation during a fuzzing cam-  paign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' [Boehme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In this work, we explore and classify the limita-  tions current fuzzers exhibit in front of forking pro-  grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In summary, this paper:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' devises a novel property capturing the ability of  fuzzers to deal with forks appropriately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' evaluates 14 coverage-guided fuzzers based on  this property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' proposes possible improvements to the current  state-of-the-art and future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Section 2 de-  scribes the relevant background, Section 3 presents  our contributions to knowledge, Section 4 shows the  existing approaches that try to cope with the fork  problem and, eventually, Section 5 discuss the results  and propose possible future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2  Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='1  Fuzz testing  Fuzzing is an automated testing technique pioneered  by Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' [Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 1990] in 1990 to test  UNIX utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' As outlined in Figure 1, coverage-  guided fuzzing is composed at least of seed selection,  input generation and system execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  1) Seeds selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The user must provide some  input messages (seeds) representative of some usual  inputs for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2) Input generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The core of every fuzzer is  the generation of slightly malformed input messages  to forward to the software under test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' A fuzzer is as  efficient as the generated inputs are able to break the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' According to the approach used to generate  the messages, the fuzzers may be classified into:  dumb: generate random strings (as the first fuzzer  [Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 1995] did);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  dumb mutational: blindly mutate seed messages  provided by the user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  grammar-based: leverage the grammar of the sys-  tem to craft the input messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  smart mutational (often called evolutionary): re-  quire a sample of inputs and leverage feedback  mechanisms to craft system-tailored messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  An example of feedback mechanisms is the code  coverage feedback, explored in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3) System execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Each execution of the fuzzer  involves three components:  Bugs detector: it reports eventual bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The ma-  jority of the bugs detectors only report crashes,  however for many systems, also a weird deviation  from the happy flow of the protocol may represent  significant security issues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Hangs detector:  it detects program execution  hangs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Code coverage detector: as further explained in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2, the code coverage represents one of  the feedbacks the fuzzer leverages to improve the  quality of the input messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2  Coverage-Guided Fuzzing  Smart mutational fuzzers use feedback mecha-  nisms to steer the generation of the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Different  types  of  feedback  mechanisms  exist  [Shahid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2011], and often different terms are  used to express the same idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' To avoid further noise,  in this work we use the term code coverage to express  the lines of code that are reached by a specific mes-  sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Code coverage fuzzers need to recompile the code  with ad-hoc compilers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' the AFL compiler) to in-  strument the code and obtain run-time information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  AFL [Zalewski, 2017], for example, instruments  the code to fill a bitmap that represents the lines of  the code covered by the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Later, it uses this bitmap to assign a higher score  to messages able to explore previously unseen lines of  code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Start  Seeds selection I  Input  generation  System execution  Hangs  detector  Bugs  detector  Code  coverage  detector  Report O  End  Figure 1: Coverage-guided fuzzing process  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='3  Inter-Process Communication  Operating systems provide system calls to perform  different tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' writing and reading files, access-  ing hardware services, creating and executing new  processes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' On UNIX systems, new processes are cre-  ated by using the fork system call [Tanenbaum, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In short, the first process, called parent process, gen-  erates a clone, called child process, that is an exact  copy of the parent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' After the fork, file de-  scriptors and registers are duplicated, thus a change  in one of the processes does not affect the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Also, the parent and child process will follow sepa-  rate execution paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3  Our contribution  This paper aims to understand how the state-of-the-  art coverage-guided fuzzers deal with software under  tests containing forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  It was not obvious to come up with a way to com-  pare and contrast the various tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  We devised a  novel property, the fork awareness, that must be sat-  isfied when a fuzzer deals with forks effectively and  efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' As we shall see below, fork awareness  rests upon three aspects representing the ability to  deal with child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Also, we evaluate the novel property over the  most widely used fuzzers from two benchmark frame-  works, reaching a total of 14 evaluated tools, 11  drawn from FuzzBench and 3 from ProFuzzBench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='1  Fork-awareness  Abstractly, fork awareness insists that every fuzzer  should address the child process as the parent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  During the system execution, the system monitor  should detect bugs or hangs regardless of their loca-  tion and the coverage should be measured also in the  child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This is formalised through Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' A coverage-guided fuzzer is fork-aware  if it can detect bugs and hangs and measure coverage  in the same way for both the child and the parent’s  branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The three aspects in this definition are called:  [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='1] Child bugs detection: any anomaly is reported  also if it occurs in child processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2] Child hangs detection: any infinite hang is re-  ported also if it occurs in child processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='3] Child code coverage: code coverage is measured  also for child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2  Example challenges  We wrote three simple C programs to use as chal-  lenges for the fuzzers, namely to test whether the  fuzzers satisfy the aspects given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  a) Bugs detection challenge:  1  if(fork()==0){ //Child process  2  raise(SIGSEGV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' //Simulated crash  3  } else { //Parent process  4  wait(NULL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' //Waiting child  5  //termination  6  }  The snippet sends a SIGSEGV signal to simulate  a bug in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' This signal is used to  report a segmentation fault, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' a memory access  violation, which is common in programs written  in low-level languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The fuzzer must detect  this bug also after the parent’s termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  b) Hangs detection challenge:  1  if(fork()==0){ //Child process  2  while(1){ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' } //Simulation of  3  //blocking code  4  }  The snippet simulates an infinite loop in the child  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The fuzzers must report processes still  in execution after the loop and must kill child  processes at the end of the fuzzing campaign,  avoiding pending process executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  c) Code coverage challenge:  1  pid_t pid = fork();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  2  if(pid==0){ //Child process  3  if(data %2 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  4  else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  5  if(data %3 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  6  else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  7  if(data %5 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  8  else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  9  if(data %7 == 0){ do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  10  else { do_something();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' }  11  }  12 else { //Parent process  13  wait(NULL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' //Waiting child  14  //termination  15  }  This snippet simulates a child with several branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  A fuzzer must cover and consider every child’s  branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  We run the 14 fuzzers over these challenges and  organised the results in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' We noticed that none  of the fuzzers succeeded through all three challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Fuzzer  Based on  Monitor technique  Bugs  Detection  (C1)  Hangs  Detection  (C2)  Code  coverage  (C3)  AFL [Zalewski, 2017] POSIX signals  ×  ×  ✓  AFL++ [Fioraldi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020]  AFL  POSIX signals  ×  ×  ✓  AFLFast [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2017]  AFL  POSIX signals  ×  ×  ✓  AFLSmart [Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2021]  AFL  POSIX signals  ×  ×  ✓  Eclipser [Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2019]  AFL  POSIX signals  ×  ×  ✓  FairFuzz [Lemieux and Sen, 2018]  AFL  POSIX signals  ×  ×  ✓  lafintel [Besler and Frederic, 2016]  AFL  POSIX signals  ×  ×  ✓  AFLnwe1  AFL  POSIX signals  ×  ×  ✓  AFLNet [Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020]  AFL  POSIX signals  ×  ×  ✓  MOpt-AFL [Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2019]  AFL  POSIX signals  ×  ×  ✓  StateAFL [Natella, 2022]  AFL  AFLNet  POSIX signals  ×  ×  ✓  LibFuzzer2 UBSAN  ASAN  MSAN  ✓  ×  ✓  Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020]  LibFuzzer  UBSAN  ASAN  MSAN  ✓  ×  ✓  Honggfuzz3 ptrace (Linux)  ✓  ×  ✓  Table 1: Coverage guided fuzzers evaluation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='3  Testbed  We decided to analyse only the coverage-guided  fuzzers present in FuzzBench [Metzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2021]  and ProFuzzBench [Natella and Pham, 2021] even  though the property applies to every coverage-guided  fuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' All fuzzers were executed on an Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='04  server machine and all our source codes are freely  available online4 so that our experiments are fully re-  producible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='4  Fuzzers evaluation  We run all selected fuzzers against our three example  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Table 1 summarises our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  All the fuzzers based on AFL use POSIX signals  and a bitmap respectively to report bugs and keep  track of the code coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  As shown in the Table 1, while the bitmaps are  able to keep track of the child’s code coverage, bugs  triggered in the child’s processes are not detected  since AFL catches signals from the main process  only, as pointed out in the documentation5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The only  fuzzers able to detect bugs in the child process are  LibFuzzer6, Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020] and Hong-  4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com/marcellomaugeri/forks-break-afl  5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com/google/AFL/blob/master/README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='md  6https://llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/docs/LibFuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='html  fuzz7, as discussed in more detail below:  LibFuzzer 8 and Entropic [Bohme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 2020]  employ a set of sanitizers9 to report bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' These  mechanisms make the fuzzers able to find the  bug in Challenge 1 and measure the different  code paths in Challenge 3, thereby satisfying chal-  lenges C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='3, as seen above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Unfortunately,  challenge C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='2 is not satisfied since the fuzzer can-  not detect hangs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Honggfuzz supports different software/hardware  feedback mechanisms and a low-level interface to  monitor targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  When executed on Linux ma-  chines, Honggfuzz uses the ptrace system call to  manage processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  This mechanism allows the  fuzzer to capture a wide range of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  As  shown in Table 1, the use of ptrace (along with  the SanitizerCoverage) allows the fuzzer to detect  bugs and to consider coverage also in the child  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' Unfortunately, neither this mechanism is  able to detect hangs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  In summary, while all selected fuzzers detect the  code coverage (C3), none detect hangs (C2) and only  a few detect bugs (C1) in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The eval-  uation underlines that:  7https://honggfuzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='dev/  8https://llvm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='org/docs/LibFuzzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='html  9AddressSanitizer,  UndefinedBehaviorSanitizer  and  MemorySanitizer  Loops detection challenge is the most difficult be-  cause fuzzers do not wait for all the child pro-  cesses but only for the main one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Code coverage challenge is the easiest because  the instrumentation allows measuring coverage  from the execution, regardless of the process in-  volved;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Bug detection challenge depends on the technique  used to observe bugs, as well as the use of sanitis-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  We interpret this general outcome as a clear call for  future research and developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  4  Existing solutions  Nowadays the only solutions to fuzz programs that  use forks are manually modifying the code or break-  ing the multi-process nature of the system (by em-  ploying tools like defork10) in order to get rid of the  forks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Unfortunately, making modifications to the code,  as pointed out in the AFLNet documentation 11, to  remove all the forks is a challenging and error-prone  task and break the multi-process nature of the system  often leads to weird system behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The only so-  lution, therefore, remains to modify the fuzzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  5  Conclusions  This paper analyses the fork awareness of the  coverage-guided fuzzers using three different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  The analysis conducted on 14 well-known fuzzers  highlights that while is it clear how important is to  handle multi-process programs, the majority of the  fuzzers overlook the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' 11 of 14 fuzzers are  not able to detect bugs in the child process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The intu-  ition behind these outcomes is related to the way these  fuzzers detect bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' All the AFL-derived fuzzers use  signals (SIGSEGV, SIGABRT, etc) to detect bugs and  this mechanism misses bugs in child processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' We  noticed that dealing with forks is not the only problem  and other issues may be related to the IPC scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  For example, the IPC may influence the success of  the fuzzing process since some bugs may be triggered  only after a specific process schedule and only after  access to a particular cell of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' We believe this  paper represents a first step towards the devising of  fuzzers aware of the eventual multiprocess nature of  10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com/zardus/preeny/blob/master/src/defork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='c  11https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com/aflnet/aflnet  the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' The first step to achieve this goal might  be the implementation of a loop detector at an early  stage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' by leveraging a dynamic library to keep  track of all process identifiers of forked processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' To  summarise, this work not only provides the first con-  crete way to evaluate the fuzzers according to their  fork awareness but sheds light for the first time on a  class of problems that have been ignored until now,  showing interesting future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  REFERENCES  Besler  and  Frederic  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Circumventing  fuzzing roadblocks with compiler transforma-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' https://lafintel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='wordpress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Boehme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Cadar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Roychoudhury, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Fuzzing: Challenges and reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  IEEE Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', 38(3):79–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content='  Bohme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', Manes, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=', and Cha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+page_content='  Pearson Education, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
+page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE4T4oBgHgl3EQfaAwh/content/2301.05060v1.pdf'}
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+arXiv:2301.13792v1  [math.FA]  31 Jan 2023
+Linear extension operators for Sobolev spaces on
+uniform trees
+Charles Fefferman1 and Bo’az Klartag2
+Dedicated in friendship to David Jerison
+Abstract
+Let 1 < p < ∞ and suppose that we are given a function f defined on the leaves of a
+weighted tree. We would like to extend f to a function F defined on the entire tree, so as
+to minimize the weighted W 1,p-Sobolev norm of the extension. An easy situation is when
+p = 2, where the harmonic extension operator provides such a function F. In this note
+we record our analysis of the particular case of a uniform binary tree, which is a complete,
+finite, binary tree with weights that depend only on the distance from the root. Neither the
+averaging operator nor the harmonic extension operator work here in general. Nevertheless,
+we prove the existence of a linear extension operator whose norm is bounded by a constant
+depending solely on p. This operator is a variant of the standard harmonic extension operator,
+and in fact it is harmonic extension with respect to a certain Markov kernel determined by p
+and by the weights.
+1
+Introduction
+Consider a full binary tree of height N, whose set of vertices is denoted by
+V =
+N
+�
+k=0
+{0, 1}k,
+i.e., the vertices are strings of zeroes and ones of length at most N. For x ∈ {0, 1}k and ℓ ≤ k
+we write πℓ(x) ∈ {0, 1}ℓ for the prefix of x of length ℓ. Thus for k ≥ 1, the parent of a vertex
+x ∈ {0, 1}k ⊆ V is the vertex πk−1(x). The set {0, 1}0 is a singleton whose unique element
+is denoted by ∅, the empty string, which is the root of the tree. The set of leaves of the tree is
+1Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544,
+USA. Email: [email protected]. Supported by the Air Force Office of Scientific Research, grant number
+FA9950-18-1-0069 and the National Science Foundation (NSF), grant number DMS-1700180.
+2Department of Mathematics,
+Weizmann Institute of Science,
+Rehovot 7610001,
+Israel.
+Email:
+[email protected]. Supported by a grant from the Israel Science Foundation (ISF).
+1
+
+{0, 1}N, and all other vertices in V are internal vertices. A vertex x ∈ {0, 1}k is said to have
+depth
+d(x) = k,
+thus leaves have depth N and the root has depth 0. The collections of bijections from V to
+V that preserve depth and parenthood relations form a group. This group is referred to as the
+symmetry group of the tree. It has 22N−1 elements, and it is a 2-Sylow subgroup of the group of
+all permutations of the leaves. The set
+E =
+N
+�
+k=1
+{0, 1}k = V \ {∅}
+is referred to as the set of edges of the tree, and the depth of an edge e ∈ {0, 1}k is d(e) = k.
+That is, we think of e ∈ E as an undirected edge connecting the vertex whose string corresponds
+to e to its unique parent. Each internal vertex other than the root is connected to three vertices,
+which are its parent and its two children. Assume that we are given edge weights
+W1, W1, . . . , WN > 0,
+where we view Wk as the weight of all edges of depth k. For 1 < p < ∞, the associated
+˙W 1,p-seminorm is defined, for F : V → R, via
+∥F∥ ˙W 1,p(V ) =
+
+
+N
+�
+k=1
+Wk
+�
+x∈{0,1}k
+|F(x) − F(πk−1(x))|p
+
+
+1/p
+.
+(1)
+We write ∂V = {0, 1}N ⊆ V for the set of leaves of the tree. The trace of the ∥·∥ ˙W 1,p-seminorm
+is defined, for f : ∂V → R, via
+∥f∥ ˙W 1,p(∂V ) = inf
+�
+∥F∥ ˙W 1,p(V ) ; F|∂V = f
+�
+,
+(2)
+i.e., the infimum of the ˙W 1,p-seminorm over all extensions of f from the leaves to the entire tree.
+We write RV for the collection of all functions f : V → R, and similarly R∂V is the collection
+of all functions f : ∂V → R. Our main result is the following:
+Theorem 1.1. Let 1 < p < ∞ and let W1, . . . , WN > 0. Then there exists a linear operator
+H : R∂V → RV with the following properties:
+1. It is a linear extension operator, i.e., (Hf)(x) = f(x) for any x ∈ ∂V and any function
+f : ∂V → R.
+2. Its norm is bounded by a constant ¯Cp depending only on p, i.e., for any f : ∂V → R,
+∥Hf∥ ˙W 1,p(V ) ≤ ¯Cp∥f∥ ˙W 1,p(∂V ).
+2
+
+In fact, we have the bound
+¯Cp ≤ 4p1/pq1/q ·
+�
+1 + max{(p − 1)−1/p, (q − 1)−1/q}
+�
+≤ C · max
+�
+1
+p − 1,
+1
+q − 1
+�
+,
+(3)
+where q = p/(p − 1) and where C > 0 is a universal constant.
+The proof of Theorem 1.2 is constructive, and the extension operator H that we construct is
+in fact a harmonic extension operator with respect to a certain random walk defined on the tree.
+At each step the random walk jumps from a vertex to one of its neighbors, where of course the
+neighbors of a vertex are its parent and its children. The Markov kernel corresponding to the
+random walk is invariant under the symmetries of the tree, thus the probability to move from a
+vertex to its neighbor depends only on the weights of the vertex and of its neighbor. The Markov
+kernel of our random walk is determined by the following requirement: For any s = 1, . . . , N,
+the probability that a random walk starting at some vertex of depth s will reach a leaf before
+reaching a vertex of depth s − 1 equals
+qs =
+(2sWs)−1/(p−1)
+�N
+k=s(2kWk)−1/(p−1).
+(4)
+Thus the weights of our random walk typically depend on p ∈ (1, ∞), except for the case where
+Ws is proportional to 2−s. This seems inevitable. Indeed, in some examples such as the 3rd
+example in Section 2 below, the linear extension operator H : R∂V → RV that corresponds to
+the parameter value p = p0, is not uniformly bounded for any p ∈ (1, ∞) \ {p0}. When p = 2,
+our random walk coincides with the usual random walk corresponding to the given weights on
+the edges of the binary tree, and thus in this case H is the standard harmonic extension operator
+(hence ¯Cp = 1 for p = 2).
+There is a certain range of weights where the averaging operator yields a uniformly bounded
+linear extension operator, as proven by Bj¨orn ×2, Gill and Shanmugalingam [1]. The averaging
+operator is the extension operator that assigns to each internal vertex v the average of the function
+values on the leaves of the subtree whose root is v. This averaging operator seems natural also
+from the point of view of Whitney’s extension theory, see the work by Shvartsman [5] on Sobolev
+extension in W 1,p(Rn). However, there are examples of uniform binary trees where the averaging
+operator does not provide a uniformly bounded operator, such as the case where Wk = 2−k for
+all k.
+What about non-uniform trees? Suppose that the edge weights are arbitrary positive numbers
+(We)e∈E that are not necessarily determined by the depth of the edge. When p = 2, there is still
+a harmonic extension operator of norm one from ˙W 1,p(∂V ) to ˙W 1,p(V ). However, for p ̸= 2 the
+situation seems subtle. We conjecture that in the general case of non-uniform tree weights there
+is no linear extension operator whose norm is bounded by a function of p alone. This conjecture
+is closely related to questions about well-complemented subspaces of ℓn
+p that are beyond the
+scope of this note.
+3
+
+In order to prove Theorem 1.1 we reformulate the problem in a way that brings us closer to
+analysis in ℓp-spaces. For 1 < p < ∞, the associated Lp(E)-norm is defined, for f : E → R,
+via
+∥f∥p = ∥f∥Lp(E) =
+
+
+N
+�
+k=1
+Wk
+�
+x∈{0,1}k
+|f(x)|p
+
+
+1/p
+.
+(5)
+We think of a function f : E → R as the gradient of a function ˜f : V → R, uniquely determined
+up to an additive constant. Given f : E → R we thus define a function ˜f : V → R as follows:
+For any x ∈ V other then the root,
+˜f(x) =
+d(x)
+�
+i=1
+f(πix),
+(6)
+while ˜f(x) = 0 if x = ∅ is the root. The only property of ˜f that matters is that for all x ∈ E,
+f(x) = ˜f(x) − ˜f(πd(x)−1(x)).
+We are interested in finding a linear operator T : Lp(E) → Lp(E), with a uniform bound on its
+operator norm, that has the following properties:
+1. The operator T takes the form
+Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)
+(7)
+for some linear operator ˜T : RV → RV . That is, ˜T takes functions on V to functions on
+V , and T is induces from ˜T via formula (7).
+2. The operator ˜T is equivariant with respect to the tree symmetries and it satisfies ˜T(1) ≡ 1,
+i.e., it maps the constant function 1 to itself.
+3. The function ˜Tg coincides with the function g on the leaves of the tree, i.e. ( ˜Tg)|∂V = g|∂V
+for any g : V → R.
+4. The function ˜Tg is determined by the values of the function g on the leaves of the tree.
+The operator norm of ˜T with respect to the ˙W 1,p-seminorm equals to the operator norm
+of T with respect to the Lp(E)-norm. Defining H(f|∂V ) = ˜Tf, Theorem 1.1 may thus be
+reformulated as follows:
+Theorem 1.2. Let 1 < p < ∞ and let W1, . . . , WN > 0. Then there exists a linear operator
+T : Lp(E) → Lp(E) with the above properties, whose operator norm is at most a certain
+constant ¯Cp depending only on p. In fact, we have the bound (3) for the constant ¯Cp.
+4
+
+The proof of Theorem 1.2 occupies the next three sections. In Section 2 we discuss invariant
+random walks on a full binary tree and describe the corresponding harmonic extension operator.
+In Section 3 we deal with the problem of bounding the norm of this operator, and use the sym-
+metries of the problem in order to reduce it to a one-dimensional question. This one-dimensional
+question is then answered in Section 4 using the Muckenhoupt criterion [4].
+When analyzing the binary tree we use the following notation: We write dlca(x, y) for the
+length of the maximal prefix shared by the strings x ∈ {0, 1}k and y ∈ {0, 1}ℓ, while lca(x, y)
+is the maximal prefix itself. Thus for two vertices x, y ∈ V , their least common ancestor is
+lca(x, y) ∈ V and its depth is dlca(x, y) ∈ {0, . . . , N}. Note that for any x ∈ E and ω ∈ ∂V ,
+dlca(πd(x)−1x, ω) = min{d(x) − 1, dlca(x, ω)}.
+(8)
+Acknowledgements. We would like to thank Jacob Carruth, Arie Israel, Anna Skorobogatova
+and Ignacio Uriarte-Tuero for helpful conversations. This research was conducted while BK
+was visiting Princeton University’s Department of Mathematics; he is grateful for their gracious
+hospitality.
+2
+Invariant random walks
+A Markov chain on V is a sequence of random variables R1, R2, . . . ∈ V such that the distribution
+of Ri+1 conditioned on R1, . . . , Ri is the same as its distribution conditioned on Ri. A Markov
+chain is time-homogeneous if for any x, y ∈ V , the probability that Ri+1 = x conditioned on the
+event Ri = y does not depend on i. A random walk on V is a time-homogeneous Markov chain
+R1, R2, . . . ∈ V such that Ri+1 is a neighbor of Ri with probability one.
+We say that the random walk is invariant if the probability to jump from a vertex x to a vertex
+y depends only on the depths d(x) and d(y). Our random walk will be invariant, and it will stop
+when it reaches a leaf, i.e., we have the stopping time
+τ = min{i ≥ 1 ; Ri is a leaf}.
+For s ≥ 1 we define qs to be the probability of the following event: Assuming that R1 is a vertex
+of depth s, the event is that Ri will remain at the subtree whose root is R1 for all 1 ≤ i ≤ τ.
+Equivalently, define
+Xi = d(Ri) ∈ {0, . . . , N}.
+Then X1, X2, . . . ∈ {0, . . . , N} is a random walk, since |Xi+1 − Xi| = 1 for all i. Furthermore,
+qs = P(∀1 ≤ i ≤ τ, Xi ≥ s | X1 = s).
+(9)
+Clearly
+q0 = qN = 1.
+(10)
+For r, s ∈ {0, . . . , N} with r ≤ s we set
+ps,r = P(min{Xi ; i ≤ τ} = r | X1 = s).
+5
+
+That is, the number ps,r is the probability that r is the minimal node that the walker visits when
+starting from node s, before reaching the terminal node N. Clearly �s
+r=0 ps,r = 1.
+Lemma 2.1. For 0 ≤ r ≤ s ≤ N − 1,
+ps,r = qr ·
+s
+�
+k=r+1
+(1 − qk)
+(11)
+where an empty product equals one. Moreover, pN,r = δrN, where δrN is the Kronecker delta.
+Proof. The expression on the right-hand side of (11) is the probability to ever reach s − 1 when
+starting from X1 = s, and from s − 1 to ever reach s − 2, etc. until we finally reach r, yet from
+r we require to never reach r − 1. Alternatively, when 0 ≤ r ≤ s, s ≥ 1 we have the recurrence
+relation
+ps,r = qsδs,r + (1 − qs) · ps−1,r.
+(12)
+This recurrence relation leads to another proof of (11).
+Suppose that our random walk R1, R2, . . . ∈ V begins at a vertex R1 = x with d(x) = s.
+Consider a leaf y ∈ ∂V with dlca(x, y) = r. What is the probability that our random walk will
+reach the leaf y? We claim that this probability is
+bs,r := P(Rτ = y) =
+r
+�
+k=0
+2k−Nps,k.
+(13)
+Indeed, conditioning on the value of k = mini≤τ d(Ri), by symmetry we know that Rτ is dis-
+tributed uniformly among the 2N−k leaf-descendants of the vertex πk(x). When k ≤ r, exactly
+one of these leaf-descendants is the leaf y, since the vertex of minimal depth that (Ri) visits must
+be the vertex πk(x), which is a prefix of y as k ≤ r = dlca(x, y). Hence the probability that
+Rτ = y, conditioning on the value of k, equals to 1/2N−k when k ≤ r and it vanishes other-
+wise. By using the definition of ps,k and the complete probability formula, we obtain (13). The
+harmonic extension operator associated with our invariant random walk is given by
+˜Tg(x) =
+�
+ω∈{0,1}N
+bd(x),dlca(x,ω) · g(ω)
+(x ∈ V ).
+(14)
+The operator T is induced from ˜T via formula (7) above. Requirements 1,...,4 from Section 1 are
+clearly satisfied.
+We stipulate that the collection of descendants of a vertex x ∈ V , denoted by D(x) ⊆ V ,
+includes the vertex x itself. Abbreviate a ∧ b = min{a, b} and a ∨ b = max{a, b}. The operator
+T takes the form
+Tf(x) =
+�
+y∈E
+K(x, y)f(y),
+(15)
+where the kernel K is described next.
+6
+
+Proposition 2.2. Let x, y ∈ E and denote s = d(x), t = d(y), r = dlca(x, y). Then the following
+hold: If x ̸∈ D(y) and y ̸∈ D(x), then r ≤ s ∧ t − 1 and
+K(x, y) = −qs · 2−t ·
+r
+�
+k=0
+2kps−1,k ≤ 0.
+(16)
+Otherwise, i.e., if y ∈ D(x) or if x ∈ D(y) then r = s ∧ t and
+K(x, y) = qs · 2−t ·
+r−1
+�
+k=0
+(2r − 2k)ps−1,k ≥ 0.
+(17)
+Proof. By (7) and (14) we have, for any x ∈ E,
+Tf(x) = ˜T ˜f(x) − ˜T ˜f(πd(x)−1x)
+=
+�
+ω∈{0,1}N
+bd(x),dlca(x,ω) ˜f(ω) −
+�
+ω∈{0,1}N
+bd(x)−1,dlca(πd(x)−1x,ω) ˜f(ω)
+=
+�
+ω∈{0,1}N
+ad(x),dlca(x,w) ˜f(ω)
+(18)
+where for any 0 ≤ r ≤ s, by (8) and (13),
+as,r = bs,r − bs−1,min{s−1,r} =
+r
+�
+k=0
+2k−Nps,k −
+min{s−1,r}
+�
+k=0
+2k−Nps−1,k.
+(19)
+Hence, by (6) and (18),
+Tf(x) =
+�
+ω∈{0,1}N
+ad(x),dlca(x,ω)
+N
+�
+i=1
+f(πiω)
+=
+�
+y∈E
+
+
+�
+ω∈{0,1}N;πd(y)ω=y
+ad(x),dlca(x,ω)
+
+ f(y) =
+�
+y∈E
+K(x, y)f(y),
+where the kernel K of the operator T satisfies, for any x, y ∈ E,
+K(x, y) =
+�
+ω∈{0,1}N ;πd(y)ω=y
+ad(x),dlca(x,ω).
+(20)
+Fix x, y ∈ E with s = d(x), t = d(y) and r = dlca(x, y). Let us consider first the case where x
+is not a descendant of y. This means that the prefix of y that is shared by x, is not the entire string
+y. Hence for any ω ∈ {0, 1}N with πd(y)ω = y we have dlca(x, ω) = dlca(x, y) ≤ d(y) − 1.
+Therefore, from (20) and (19),
+K(x, y) = 2N−d(y)ad(x),dlca(x,y)
+= 2N−t �
+bs,r − bs−1,(s−1)∧r
+�
+=
+r
+�
+k=0
+2k−tps,k −
+(s−1)∧r
+�
+k=0
+2k−tps−1,k,
+7
+
+as bs,r = �r
+k=0 2k−Nps,k. Since ps,s = qs, we have
+K(x, y) = δrs2s−tqs +
+(s−1)∧r
+�
+k=0
+2k−t [ps,k − ps−1,k] = qs ·
+
+δrs2s−t −
+(s−1)∧r
+�
+k=0
+2k−tps−1,k
+
+ ,
+(21)
+where we used the relation (12), which implies that when k ≤ s − 1,
+ps,k − ps−1,k = −qs · ps−1,k.
+(22)
+We may now prove the conclusion of the proposition in the case where x ̸∈ D(y) and y ̸∈ D(x).
+Indeed, in this case r ≤ s ∧ t − 1 and formula (21) applies. Since δrs = 0 in this case, we deduce
+formula (16) from (21).
+The next case we consider is the case where r = s ≤ t − 1, or equivalently, where y ∈
+D(x) \ {x}. Thus x ̸∈ D(y) and formula (21) applies. Recalling that �s−1
+k=0 ps−1,k = 1 we obtain
+from (21) that
+K(x, y) = qs ·
+s−1
+�
+k=0
+(2s−t − 2k−t)ps−1,k = qs · 2−t ·
+s−1
+�
+k=0
+(2r − 2k)ps−1,k,
+proving formula (17) in the case y ∈ D(x) \ {x}.
+We move on to the case where x ∈ D(y) \ {y}, thus dlca(x, y) = t ≤ s − 1. In this case, by
+applying (20), (19), (13) and then (22),
+K(x, y) =
+�
+ω∈{0,1}N ;πd(y)ω=y
+ad(x),dlca(x,ω) = 2N−d(x)ad(x),d(x) +
+d(x)−1
+�
+k=d(y)
+2N−k−1ad(x),k
+= 2N−sas,s +
+s−1
+�
+k=t
+2N−k−1as,k = 2N−s(bs,s − bs−1,s−1) +
+s−1
+�
+k=t
+2N−k−1(bs,k − bs−1,k)
+= 2N−s
+�
+s
+�
+k=0
+2k−Nps,k −
+s−1
+�
+k=0
+2k−Nps−1,k
+�
++
+s−1
+�
+k=t
+2N−k−1
+k
+�
+ℓ=0
+2ℓ−N(ps,ℓ − ps−1,ℓ)
+= qs − qs
+s−1
+�
+k=0
+2k−sps−1,k − qs
+s−1
+�
+ℓ=0
+s−1
+�
+k=ℓ∨t
+2ℓ−k−1ps−1,ℓ
+= qs
+�
+1 −
+s−1
+�
+k=0
+2k−sps−1,k −
+s−1
+�
+ℓ=0
+[2ℓ−ℓ∨t − 2ℓ−s]ps−1,ℓ
+�
+= qs
+�
+1 −
+s−1
+�
+k=0
+2k−k∨tps−1,k
+�
+= qs ·
+�
+1 −
+t−1
+�
+k=0
+2k−tps−1,k −
+s−1
+�
+k=t
+ps−1,k
+�
+= qs ·
+t−1
+�
+k=0
+(1 − 2k−t)ps−1,k = qs · 2−t ·
+t−1
+�
+k=0
+(2t − 2k)ps−1,k.
+8
+
+Since r = t in this case, we have proved formula (17) in the case where y ∈ D(x) \ {x}. Finally,
+the last case that remains is when x = y. In this case r = s = t and
+K(x, y) =
+�
+ω∈{0,1}N;πd(y)ω=y
+ad(x),dlca(x,ω) = 2N−d(x)ad(x),d(x) = 2N−sas,s = 2N−s(bs,s − bs−1,s−1)
+= 2N−s
+�
+s
+�
+k=0
+2k−Nps,k −
+s−1
+�
+k=0
+2k−Nps−1,k
+�
+= qs − qs
+s−1
+�
+k=0
+2k−sps−1,k
+= qs ·
+s−1
+�
+k=0
+(1 − 2k−s)ps−1,k,
+completing the proof of formula (17).
+Some examples.
+1. The simplest example is when qs = 1 for all s ≥ 1. In this case the operator ˜T is the
+familiar averaging operator. That is, the extension operator ˜T is the operator that assigns
+to each vertex the average of the values at the leaves of its subtree. In this case
+ps,r = δs,r.
+2. Consider the case where the invariant random walk is such that Xi := d(Ri) is a symmetric
+random walk on {0, . . . , N}, i.e., the probability to jump from i to i + 1 is exactly 1/2 for
+i = 1, . . . , N − 1. Recall that qs is the probability to never leave the subtree when starting
+at a vertex of depth s. We claim that in this example, for s = 1, . . . , N,
+qs =
+1
+N − s + 1.
+(23)
+Indeed, the function f(i) = i is harmonic on {0, 1, . . . , N} and hence f(Xi) is a mar-
+tingale. Thus for any stopping time ˜τ we have f(X1) = Ef(X˜τ). We pick the stopping
+time
+˜τ = min{ i; Xi ∈ {s − 1, N} }
+and obtain (23) since
+N · qs + (s − 1) · (1 − qs) = s.
+Next we use Lemma 2.1 and find a formula for ps,r. Since formula (23) is valid for any
+s ≥ 1, we conclude that for any r ≥ 0 and s ≥ r + 1,
+s�
+k=r+1
+(1 − qk) =
+s�
+k=r+1
+N − k
+N − k + 1 = N − s
+N − r.
+(24)
+9
+
+Formula (24) is actually valid for any 0 ≤ r ≤ s, since an empty product equals one.
+Recall that q0 = 1. We thus conclude from Lemma 2.1 that for s = 0, . . . , N − 1,
+ps,r = N − s
+N − r ·
+�
+1
+r = 0
+1
+N−r+1
+1 ≤ r ≤ s
+while pN,r = δN,r.
+3. Let 0 < δ < 1, and consider the case where (Xi) is a random walk on {0, . . . , N} such
+that the probability to jump from k to k + 1 equals 1/2 if k < N − 1, and it equals δ if
+k = N − 1. A harmonic function here is
+f(k) =
+�
+k
+k ≤ N − 1
+N − 2 + 1/δ
+k = N
+Therefore, for s = 1, . . . , N − 1,
+(N − 2 + 1/δ) · qs + (s − 1)(1 − qs) = s.
+Thus q0 = qN = 1 while for s = 1, . . . , N − 1,
+qs =
+1
+N − s + 1/δ − 1.
+Hence for any s ≤ N − 1 and r ≤ s − 1,
+s
+�
+k=r+1
+(1 − qk) =
+s
+�
+k=r+1
+N − k + δ−1 − 2
+N − k + δ−1 − 1 = N − s + δ−1 − 2
+N − r + δ−1 − 2.
+We conclude from Lemma 2.1 that for s = 0, . . . , N − 1 and 0 ≤ r ≤ s,
+ps,r = qr ·
+s�
+k=r+1
+(1 − qk) = N − s + δ−1 − 2
+N − r + δ−1 − 2 ·
+�
+1
+r = 0
+1
+N−r+1/δ−1
+1 ≤ r ≤ s
+while pN,r = δN,r.
+We conclude this section with the following:
+Lemma 2.3. For any numbers q1, . . . , qN−1 ∈ (0, 1) there exists a random walk
+X1, X2, . . . ∈ {0, . . . , N}
+satisfying (9) with τ = min{i ≥ 1 ; Xi = N} for s = 1, . . . , N − 1.
+Proof. Write xk for the probability that the random walk jumps from k to k + 1. Then for
+s = 0, . . . , N − 1,
+qs = xs(qs+1 + (1 − qs+1)xs),
+(25)
+where we set q0 = qN = 1. We claim that xs ∈ [0, 1] is determined by equation (25). Indeed, the
+right-hand side of (25) is a continuous, increasing function of xs in the interval [0, 1], that maps
+this interval to itself.
+10
+
+3
+The ancestral and non-ancestral parts of the kernel
+We need to bound the operator norm in Lp(E) of the operator T whose kernel is described in
+Proposition 2.2. Let us consider first the non-ancestral part of the operator, given by the kernel
+K0(x, y) = K(x, y) · 1{x̸∈D(y),y̸∈D(x)} = −1{r≤s∧t−1} · qs · 2−t ·
+r
+�
+k=0
+2kps−1,k.
+(26)
+Here as usual s = d(x), t = d(y) and r = dlca(x, y). Write T0 : Lp(E) → Lp(E) for the
+operator whose kernel is K0. A function f : E → R is invariant under the symmetries of the
+tree, or invariant in short, if it takes the form
+f(x) = F(d(x))
+for some function F : {1, . . . , N} → R. The operator T0 is equivariant under the symmetries
+of the tree. Therefore, if f(x) = F(d(x)) is an invariant function, then so is T0f. In fact, in the
+case where f(x) = F(d(x)) we can write
+T0f(x) =
+N
+�
+t=1
+L0(d(x), t)F(t)
+(27)
+for a certain kernel L0(s, t) defined for s, t = 1, . . . , N.
+Lemma 3.1. For s, t = 1, . . . , N,
+L0(s, t) = −qs ·
+m−1
+�
+k=0
+(1 − 2k−m)ps−1,k ≤ 0.
+Proof. Let r ≤ min{t, s} − 1. A moment of reflection reveals that for x ∈ E with d(x) = s,
+n(t; s, r) := #{y ∈ E ; d(y) = t, dlca(x, y) = r} = 2t−r−1.
+By (26), (27) and the definition of T0, for any x ∈ E with d(x) = s,
+L0(s, t) =
+�
+y∈E;d(y)=t
+K0(x, y) = −
+t∧s−1
+�
+r=0
+n(t; s, r) · qs · 2−t ·
+r
+�
+k=0
+2kps−1,k.
+Denote m = s ∧ t. Then for s, t = 1, . . . , N,
+L0(s, t) = −qs ·
+t∧s−1
+�
+r=0
+r
+�
+k=0
+2k−r−1ps−1,k = −qs ·
+m−1
+�
+k=0
+m−1
+�
+r=k
+2k−r−1ps−1,k
+= −qs ·
+m−1
+�
+k=0
+(1 − 2k−m)ps−1,k.
+11
+
+Write ΩN = {1, . . . , N} and for F : ΩN → R define
+∥F∥p = ∥F∥Lp(ΩN) =
+� N
+�
+k=1
+2k · Wk · |F(k)|p
+�1/p
+.
+(28)
+Observe that ∥f∥Lp(E) = ∥F∥p if f(x) = F(d(x)). Let
+S0F(s) =
+N
+�
+t=1
+L0(s, t)F(t)
+so that by (27),
+T0(F ◦ d) = (S0F) ◦ d.
+(29)
+For f, g : E → R we consider the scalar product
+⟨f, g⟩ =
+�
+x∈E
+Wd(x)f(x)g(x)
+while for F, G : ΩN → R we set
+⟨F, G⟩ := ⟨F ◦ d, G ◦ d⟩ =
+N
+�
+k=1
+2kWkF(k)G(k).
+The adjoint operators T ∗
+0 and S∗
+0 are defined with respect to these scalar products. The follow-
+ing lemma is probably well-known to experts (see, e.g., Howard and Schep [2] for a related
+argument), and its proof is provided for completeness.
+Lemma 3.2. Let 1 < p < ∞. Then the norm of the operator T0 : Lp(E) → Lp(E) is attained at
+an invariant, non-negative function f, and it equals to the norm of the operator S0 : Lp(Ωn) →
+Lp(Ωn).
+Proof. Denote momentarily T = −T0 and S = −S0. By Lemma 3.1 the kernel −L0 of the
+operator S is non-negative, and by (26) the kernel of the operator T is non-negative as well.
+By approximation, we may assume that these two kernels are strictly positive, while keeping
+condition (29), thus
+T (F ◦ d) = (SF) ◦ d.
+(30)
+We deduce that T ∗(F ◦ d) = (S∗F) ◦ d. By compactness,
+sup
+0̸≡F ∈Lp(ΩN )
+∥SF∥p
+∥F∥p
+is attained at some function F. Since the kernel of S is non-negative, we may assume that
+the extremal function F is non-negative. By the Lagrange multipliers theorem, the function F
+satisfies a certain eigenvalue equation, and in fact there exists λ ∈ R such that
+S∗(SF)p−1 = λF p−1.
+(31)
+12
+
+Since the kernel of S is positive and F is non-negative and not identically zero, it follows from
+(31) that F is actually positive. The norm of the operator S : Lp(Ωn) → Lp(Ωn) equals λ1/p > 0,
+since
+∥SF∥p
+p = ⟨(SF)p−1, SF⟩ = ⟨S∗(SF)p−1, F⟩ = λ⟨F p−1, F⟩ = λ∥F∥p
+p.
+Denoting f = F ◦ d, we find from (30) that f is a positive invariant function satisfying
+T ∗(T f)p−1 = λf p−1.
+Since the kernel of T is non-negative, we have the pointwise H¨older inequality
+T (uv) ≤ T (up)1/p · T (vq)1/q,
+valid for any non-negative functions u, v ∈ Lp(E), where q = p/(p − 1). The operator T has
+a non-negative kernel, and hence its norm is attained at a non-negative function u ∈ Lp(E). By
+the pointwise H¨older inequality,
+(T u)p ≤ T (upf −p/q) · (T f)p/q = T (upf 1−p) · (T f)p−1.
+Therefore,
+∥T u∥p
+Lp(E) ≤ ⟨T (upf 1−p), (T f)p−1⟩ = ⟨upf 1−p, T ∗(T f)p−1⟩ = λ⟨upf 1−p, f p−1⟩ = λ∥u∥p
+Lp(E).
+Thus the norm of T : Lp(E) → Lp(E) is at most λ1/p, which is the norm of S : Lp(ΩN) →
+Lp(ΩN). The two norms must therefore be equal, since the operator S is equivalent to the
+restriction of T to the space of invariant functions.
+We move on to the ancestral part of the operator, which according to Proposition 2.2 is given
+by
+K1(x, y) = K(x, y) − K0(x, y) = 1{r=s∧t} · qs · 2−t ·
+r−1
+�
+k=0
+(2r − 2k)ps−1,k ≥ 0.
+(32)
+Write T1 for the operator whose kernel is K1. As before, for an invariant function f(x) =
+F(d(x)) we may write
+T1f(x) =
+N
+�
+t=1
+L1(d(x), t)F(t)
+(33)
+for a certain kernel L1(s, t) defined for s, t = 1, . . . , N. We also write
+S1F(s) =
+N
+�
+t=1
+L1(s, t)F(t).
+It is possible to use formula (7) for the operator T and the definition (14) of the harmonic exten-
+sion operator ˜T and deduce that
+S0 + S1 ≡ 0,
+(34)
+essentially because the only invariant, harmonic function on the vertices of the tree is the constant
+function. An alternative, more direct proof of (34) is provided in the following:
+13
+
+Lemma 3.3. Let 1 < p < ∞. Then the norm of the operator T1 : Lp(E) → Lp(E) is equal to
+the norm of S1 : Lp(Ωn) → Lp(Ωn). Additionally, for s, t = 1, . . . , N with m = s ∧ t we have
+L1(s, t) = qs ·
+m−1
+�
+k=0
+(1 − 2k−m)ps−1,k = −L0(s, t).
+Proof. The first assertion of the lemma follows from fact that the kernel of T1 is non-negative
+and invariant under the symmetries of the tree, as in Lemma 3.2. For the second part, let s, t =
+1, . . . , N and denote m = s ∧ t. We claim that for x ∈ E with d(x) = s,
+n(t; s, m) := #{y ∈ E ; d(y) = t, dlca(x, y) = m} = max{1, 2t−s}.
+(35)
+Indeed, assume first that t ≥ s. How many y’s are there with d(y) = t and dlca(x, y) = m?
+Since m = s, the answer is 2t−s. Next, if s ≥ t, then the number of such y’s is one. This proves
+(35). Therefore,
+L1(s, t) =
+�
+y∈E;d(y)=t
+K1(x, y) = n(t; s, m) · qs · 2−t ·
+m−1
+�
+k=0
+(2m − 2k)ps−1,k
+= max{2−t, 2−s} · qs ·
+m−1
+�
+k=0
+(2m − 2k)ps−1,k.
+= 2−m · qs ·
+m−1
+�
+k=0
+(2m − 2k)ps−1,k.
+Corollary 3.4. We have
+∥T∥Lp(E)→Lp(E) ≤ 2∥S0∥Lp(ΩN)→Lp(ΩN).
+Proof. This follows from the fact that T = T0 + T1 together with the facts that ∥T0∥ = ∥S0∥ and
+∥T1∥ = ∥S1∥ while S1 = −S0.
+In view of Corollary 3.4, we are interested in bounds for the norm of the operator S = −S0 =
+S1 : Lp(ΩN) → Lp(ΩN) whose non-negative kernel is
+L(s, t) = qs ·
+s∧t−1
+�
+k=0
+(1 − 2k−s∧t)ps−1,k ≤ qs ·
+s∧t−1
+�
+k=0
+ps−1,k.
+(36)
+14
+
+From Lemma 2.1 we know that ps,r = qr · �s
+k=r+1(1 − qk) for s ≤ N − 1. A little exercise in
+probability shows that for any 1 ≤ m ≤ s ≤ N,
+m−1
+�
+k=0
+ps−1,k =
+s−1
+�
+k=m
+(1 − qk).
+(37)
+Alternatively, (37) holds true for m = 0 as q0 = 1, and it may be proven by induction on m since
+ps−1,m +
+s−1
+�
+k=m
+(1 − qk) = qm ·
+s−1
+�
+k=m+1
+(1 − qk) +
+s−1
+�
+k=m
+(1 − qk) =
+s�
+k=m+1
+(1 − qk).
+From (36) and (37) we thus obtain
+Corollary 3.5. For s, t = 1, . . . , N, with m = min{s, t},
+0 ≤ L(s, t) ≤ qs
+s−1
+�
+k=m
+(1 − qk),
+where an empty product equals one.
+Some examples (parallel to the ones discussed in Section 2).
+1. For the averaging operator, where qs = 1 and ps,r = δs,r, we have
+L(s, t) = −1
+2 · 1{s≤t},
+i.e., this is the matrix whose entries equal 0 below the diagonal and −1/2 on and above
+the diagonal. This is a rather simple matrix, and it is bounded with respect to the weighted
+Lp-norm for quite a few sequences of weights.
+2. For the symmetric random walk matrix, we have q0 = 1 while for 1 ≤ s ≤ N,
+qs =
+1
+N − s + 1.
+Hence in view of Corollary 3.5, with m = min{s, t},
+0 ≤ L(s, t) ≤
+1
+N − s + 1
+s−1
+�
+k=m
+N − k
+N − k + 1 =
+1
+N − m + 1.
+3. In the case where
+qs =
+1
+N − s + 1/δ − 1
+for some 0 < δ < 1, we have
+0 ≤ L(s, t) ≤
+1
+N − s + 1/δ − 1
+s−1
+�
+k=m
+N − k + 1/δ − 2
+N − k + 1/δ − 1 =
+1
+N − m + 1/δ − 1.
+15
+
+All that remains is to bound the Lp(ΩN)-norm of the operator whose kernel is discussed
+in Corollary 3.5. Recall from Lemma 2.3 that we have the freedom to choose the parameters
+q1, . . . , qN−1 ∈ (0, 1) as we please.
+How should we choose these parameters? Since L(N, t) ≤ �N−1
+k=t (1−qk) and we are looking
+for upper bounds for the norm, the qs should not be too tiny. On the other hand, L(s, t) ≤ qs for
+s ≤ N − 1 and hence it is beneficial to choose qs rather small. We would therefore need some
+balance for the qs, which is the subject of the next section.
+4
+One-dimensional analysis
+Let 1 < p < ∞. It will be slightly more convenient to denote
+K(s, t) = L(N + 1 − s, N + 1 − t)
+and
+Qs = qN+1−s.
+Recalling from (10) that qN = 1, we see that
+Q1 = 1.
+From Corollary 3.5 we know that for s, t = 1, . . . , N,
+K(s, t) = L(N + 1 − s, N + 1 − t) ≤ qN+1−s
+N−s
+�
+k=N+1−max{s,t}
+(1 − qk) = Qs
+max{s,t}
+�
+k=s+1
+(1 − Qk).
+From Corollary 3.5 we know that L ≥ 0. Consequently, for s, t = 1, . . . , N,
+0 ≤ K(s, t) ≤
+�
+Qs
+t ≤ s
+Qs · �t
+k=s+1(1 − Qk)
+t ≥ s + 1
+(38)
+Recall that we are given edge weights W1, . . . , WN > 0, and that the associated Lp(E)-norm is
+given by (5). Denote
+ws := 2N+1−s · WN+1−s > 0.
+Consider the weighted ℓp-norm
+∥f∥p,w =
+� N
+�
+k=1
+wk|f(k)|p
+�1/p
+(39)
+and the operator T whose kernel is K(s, t). We are allowed to choose the weights q1, . . . , qN−1 ∈
+(0, 1) as we please, or equivalently, we have the freedom to determine Q2, . . . , QN ∈ (0, 1). We
+must keep Q1 = 1. Based on considerations related to the Muckenhoupt criterion discussed
+below, we set
+Qs =
+w−1/(p−1)
+s
+�s
+k=1 w−1/(p−1)
+k
+.
+(40)
+It is clear that Q1 = 1 and that Qs ∈ (0, 1) for all s ≥ 2. Recall that q = p/(p − 1).
+16
+
+Lemma 4.1. In order to prove Theorem 1.2, it suffices to show that the operator norm of T with
+respect to the ∥ · ∥p,w-norm is bounded by a constant ˆCp > 0 depending only on p, where in fact
+ˆCp ≤ 2p1/pq1/q ·
+�
+1 + max{(p − 1)−1/p, (q − 1)−1/q}
+�
+.
+(41)
+Proof. In view of Corollary 3.4, it suffices to bound the operator norm of −S0, whose kernel is
+L, with respect to the Lp(ΩN)-norm defined in (28). Under the transformation
+s �→ N + 1 − s
+the operator −S0 whose kernel is L transforms to the operator T whose kernel is K. The Lp(ΩN)-
+norm from (28) transforms to the ∥·∥p,w-norm defined in (39). Hence Theorem 1.2 would follow
+once we obtain the bound (41), where ¯Cp ≤ 2 ˆCp by Corollary 3.4.
+The remainder of this section is devoted to the proof of the following:
+Proposition 4.2. The operator norm of T with respect to the norm (39) is bounded by a number
+ˆCp depending only on p ∈ (1, ∞). In fact, we have the bound (41) for the constant ˆCp.
+Our main tool in the proof of Proposition 4.2 is the Muckenhoupt criterion [4], which is an
+indispensable tool for proving one-dimensional inequalities of Poincar´e-Sobolev type. For the
+reader’s convenience, we include here a statement and a proof of a straightforward modification
+of the Muckenhoupt criterion, with sums in place of integrals:
+Theorem 4.3. (Muckenhoupt) Let 1 < p < ∞ and write ΩN = {1, . . . , N}. Let U, V : ΩN →
+(0, ∞) and let A > 0 be such that for all r = 1, . . . , N,
+� N
+�
+k=r
+|U(k)|p
+�1/p
+≤ A
+�
+r
+�
+k=1
+|V (k)|−q
+�−1/q
+.
+(42)
+Then for any function f : ΩN → R,
+� N
+�
+k=1
+�����U(k)
+k
+�
+ℓ=1
+f(ℓ)
+�����
+p�1/p
+≤ CpA
+� N
+�
+k=1
+|V (k)f(k)|p
+�1/p
+,
+(43)
+with Cp = p1/pq1/q.
+By continuity, the analog of Theorem 4.3 for p = 1, ∞ holds true with C1 = C∞ = 1. We
+remark that as in [4], this criterion is tight, in the sense that the infimum over all A > 0 satisfying
+(42) is equivalent to the best constant in inequality (43). For the proof of Theorem 4.3 we require
+the following:
+17
+
+Lemma 4.4. For α1, . . . , αN > 0 and r = 1, . . . , N,
+r
+�
+k=1
+αk
+�
+k
+�
+ℓ=1
+αℓ
+�−1/p
+≤ q ·
+�
+r
+�
+k=1
+αk
+�1/q
+.
+(44)
+Proof. We will use the simple inequality
+(a + b)1/q − a1/q ≥ b ·
+min
+ξ∈(a,a+b)
+ξ1/q−1
+q
+= 1
+q(a + b)1/q−1 · b,
+(45)
+valid for any a, b ≥ 0 with a + b > 0. From (45), for k = 1, . . . , N,
+� k
+�
+ℓ=1
+αℓ
+�1/q
+−
+�k−1
+�
+ℓ=1
+αℓ
+�1/q
+≥ 1
+q · αk ·
+�
+k
+�
+ℓ=1
+αℓ
+�1/q−1
+,
+where an empty sum equals zero. By summing this for k = 1, . . . , r we obtain (44).
+Proof of Theorem 4.3 (Muckenhoupt). By the H¨older inequality, for any function f : ΩN → R
+and weights h : ΩN → (0, ∞),
+N
+�
+k=1
+�����U(k)
+k
+�
+ℓ=1
+f(ℓ)
+�����
+p
+≤
+N
+�
+k=1
+Up(k) ·
+k
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p ·
+�
+k
+�
+j=1
+|V (j)h(j)|−q
+�p/q
+=
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p
+N
+�
+k=ℓ
+Up(k)
+� k
+�
+j=1
+|V (j)h(j)|−q
+�p/q
+.
+(46)
+Set h(k) =
+��k
+ℓ=1 V (ℓ)−q�1/(pq)
+. By applying Lemma 4.4 with αk = V (k)−q we obtain
+r
+�
+k=1
+|V (k)h(k)|−q =
+r
+�
+k=1
+αk
+� k
+�
+ℓ=1
+αℓ
+�−1/p
+≤ q ·
+�
+r
+�
+k=1
+αk
+�1/q
+= q ·
+�
+r
+�
+k=1
+V (k)−q
+�1/q
+.
+Hence for any f : ΩN → R, the expression in (46) is at most
+qp/q ·
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p
+N
+�
+k=ℓ
+Up(k)
+�
+k
+�
+j=1
+|V (j)|−q
+�p/q2
+.
+(47)
+By applying (42) and then Lemma 4.4 with αk = Up(N + 1 − k) and with p ∈ (1, ∞) playing
+the rˆole of q ∈ (1, ∞), we see that
+N
+�
+k=ℓ
+|U(k)|p
+�
+k
+�
+j=1
+|V (j)|−q
+�p/q2
+≤ Ap/q
+N
+�
+k=ℓ
+|U(k)|p
+� N
+�
+j=k
+|U(j)|p
+�−1/q
+= Ap/q
+N+1−ℓ
+�
+k=1
+αk
+� k
+�
+j=1
+αj
+�−1/q
+≤ p · Ap/q
+�N+1−ℓ
+�
+k=1
+αk
+�1/p
+= p · Ap/q
+� N
+�
+k=ℓ
+Up(k)
+�1/p
+.
+18
+
+Hence the expression in (47) is at most
+p · (qA)p/q ·
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p ·
+� N
+�
+k=ℓ
+Up(k)
+�1/p
+.
+Applying (42) again we bound the last expression from above by
+p · (qA)p/q · A ·
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)h(ℓ)|p ·
+�
+ℓ
+�
+k=1
+V −q(k)
+�−1/q
+= pqp/qAp
+N
+�
+ℓ=1
+|f(ℓ)V (ℓ)|p,
+completing the proof.
+Corollary 4.5. Let 1 < p < ∞ and ΩN = {1, . . . , N}. Let U, V : ΩN → (0, ∞) and let A > 0
+be such that for all r = 1, . . . , N,
+�
+r
+�
+k=1
+|U(k)|p
+�1/p
+≤ A
+� N
+�
+k=r
+|V (k)|−q
+�−1/q
+.
+(48)
+Then for any f : ΩN → R,
+� N
+�
+k=1
+�����U(k)
+N
+�
+ℓ=k
+f(ℓ)
+�����
+p�1/p
+≤ CpA
+� N
+�
+k=1
+|V (k)f(k)|p
+�1/p
+,
+(49)
+with Cp = p1/pq1/q.
+Proof. Denote ˜U(k) = U(N + 1 − k) and ˜V (k) = V (N + 1 − k). Then from (48), for all
+r = 1, . . . , N,
+� N
+�
+k=r
+| ˜U(k)|p
+�1/p
+≤ A
+�
+r
+�
+k=1
+| ˜V (k)|−q
+�−1/q
+.
+By Theorem 4.3, this implies that for any g : ΩN → R, denoting f(r) = g(N + 1 − r),
+� N
+�
+k=1
+�����
+˜U(k)
+k
+�
+ℓ=1
+f(N + 1 − ℓ)
+�����
+p�1/p
+≤ CpA
+� N
+�
+k=1
+| ˜V (k)f(N + 1 − k)|p
+�1/p
+,
+or equivalently,
+� N
+�
+k=1
+�����
+˜U(N + 1 − k)
+N
+�
+ℓ=k
+f(ℓ)
+�����
+p�1/p
+≤ CpA
+� N
+�
+k=1
+| ˜V (N + 1 − k)f(k)|p
+�1/p
+.
+This implies (49).
+19
+
+Proposition 4.6. For f : Ωn → R and s = 1, . . . , N denote
+T0f(s) = Qs
+s
+�
+t=1
+f(t).
+Then the operator norm of T0 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by a
+number ˜Cp depending only on p ∈ (1, ∞). In fact, ˜Cp ≤ 21/pp1/pq1/q · max{1, (p − 1)−1/p}.
+The proof of Proposition 4.6 requires the following:
+Lemma 4.7. For any α1, . . . , αN > 0 and r = 1, . . . , N,
+N
+�
+k=r
+αk
+� k
+�
+ℓ=1
+αℓ
+�−p
+≤
+2
+min{1, p − 1} ·
+�
+r
+�
+k=1
+αk
+�−p+1
+.
+(50)
+Proof. We use the inequality
+(p − 1) · b(a + b)−p ≤ a−p+1 − (a + b)−p+1,
+which is valid for any a, b > 0. Then for k = 2, . . . , N,
+(p − 1) · αk
+� k
+�
+ℓ=1
+αℓ
+�−p
+≤
+�k−1
+�
+ℓ=1
+αℓ
+�−p+1
+−
+�
+k
+�
+ℓ=1
+αℓ
+�−p+1
+.
+By summing this for k = r + 1, . . . , N we obtain
+(p − 1) ·
+N
+�
+k=r+1
+αk
+� k
+�
+ℓ=1
+αℓ
+�−p
+≤
+�
+r
+�
+k=1
+αk
+�−p+1
+−
+� N
+�
+ℓ=1
+αℓ
+�−p+1
+≤
+�
+r
+�
+k=1
+αk
+�−p+1
+,
+where an empty sum equals zero. We conclude (50) by summing this with the trivial inequality
+min{1, p − 1} · αr
+�
+r
+�
+ℓ=1
+αℓ
+�−p
+≤
+�
+r
+�
+k=1
+αk
+�−p+1
+.
+Proof of Proposition 4.6. Define
+U(k) = w1/p
+k
+· Qk
+and
+V (k) = w1/p
+k .
+Let us verify the condition of the Muckenhoupt criterion. We need to find A > 0 such that for
+all r = 1, . . . , N inequality (42) holds true, that is,
+N
+�
+k=r
+wkQp
+k ≤ Ap
+�
+r
+�
+k=1
+w−q/p
+k
+�−p/q
+.
+(51)
+20
+
+Recall that p/q = p − 1. From the definition (40) of Qk, we need
+N
+�
+k=r
+w−1/(p−1)
+k
+�
+k
+�
+ℓ=1
+w−1/(p−1)
+ℓ
+�−p
+≤ Ap
+�
+r
+�
+k=1
+w−1/(p−1)
+k
+�−(p−1)
+.
+Setting αk = w−1/(p−1)
+k
+and using Lemma 4.7, we see that (51) holds true with
+A = 21/p · max{1, (p − 1)−1/p}.
+From Theorem 4.3 we thus conclude that for any f : ΩN → R,
+� N
+�
+k=1
+wk
+�����Qk
+k
+�
+ℓ=1
+f(ℓ)
+�����
+p�1/p
+≤ p1/pq1/qA ·
+� N
+�
+k=1
+wk|f(k)|p
+�1/p
+.
+This implies the required bound for the operator norm of f.
+Proposition 4.8. For f : Ωn → R and s = 1, . . . , N denote
+T1f(s) =
+N
+�
+t=s+1
+�
+Qs
+t�
+k=s+1
+(1 − Qk)
+�
+f(t).
+Then the operator norm of T1 with respect to the norm ∥ · ∥p,w defined in (39) is bounded by
+21/qp1/pq1/q · max{1, (q − 1)−1/q}.
+Proof. Denote αk = w−1/(p−1)
+k
+and recall from (40) that Qk = αk/ �k
+ℓ=1 αℓ. We claim that for
+any t ≥ s + 1,
+Qs
+t�
+k=s+1
+(1 − Qk) =
+αs
+�t
+k=1 αk
+.
+(52)
+Indeed, (52) holds true for t = s, since an empty product equals one, and for t ≥ s + 1 it is
+proven by an easy induction on t. Consequently,
+T1f(s) = αs
+N
+�
+t=s+1
+1
+�t
+j=1 αj
+f(t).
+Since p, q ∈ (1, ∞), the elementary inequality of Lemma 4.7 is valid also when p is replaced by
+q. It implies that for r = 1, . . . , N,
+�
+r
+�
+k=1
+αk
+�−q+1
+≥ min{1, q − 1}
+2
+·
+N
+�
+k=r
+αk
+�
+k
+�
+j=1
+αj
+�−q
+.
+(53)
+21
+
+Set A = 21/q min{1, q − 1}−1/q. Since αk = w−1/(p−1)
+k
+and q/p = q − 1 = 1/(p − 1), it follows
+from (53) that for r = 1, . . . , N,
+�
+r
+�
+k=1
+wkαp
+k
+�1/p
+≤ A
+
+
+N
+�
+k=r
+w−q/p
+k
+� k
+�
+j=1
+αj
+�−q
+
+−1/q
+.
+This is precisely the Muckenhoupt criterion from Corollary 4.5, with
+U(k) = w1/p
+k αk
+and
+V (k) = w1/p
+k
+k
+�
+j=1
+αj.
+Thus, by Corollary 4.5, for any g : ΩN → R,
+N
+�
+k=1
+wk
+�����αk
+N
+�
+ℓ=k
+g(ℓ)
+�����
+p
+≤ (CpA)p
+N
+�
+k=1
+wk
+�������
+�
+k
+�
+j=1
+αj
+�
+g(k)
+�����
+p
+,
+(54)
+with Cp = p1/pq1/q. By restricting attention to non-negative functions g in (54), we may alter
+(54) and replace �N
+ℓ=k by the shorter sum �N
+ℓ=k+1. Inequality (54) remains correct, for non-
+negative g, also after this modification. Denoting g(k) = f(k)/ �k
+j=1 αj, we conclude that for
+any non-negative function f : ΩN → R,
+N
+�
+k=1
+wk
+�����αk
+N
+�
+ℓ=k+1
+1
+�ℓ
+j=1 αj
+f(ℓ)
+�����
+p
+≤ (CpA)p
+N
+�
+k=1
+wk |f(k)|p .
+(55)
+Since the kernel of T1 is non-negative, its operator norm is attained at a non-negative function
+f : ΩN → R. Therefore (55) implies the required bound for the operator norm of T1.
+Proof of Proposition 4.2. The kernel of the operator T is given in (38). It is a non-negative
+kernel, and therefore the operator norm of T is at most the operator norm of the operator whose
+kernel is the expression on the right-hand side of (38). The latter operator equals
+T0 + T1
+with T0 from Proposition 4.6 and T1 from Proposition 4.8. From these two propositions it follows
+that the operator norm of T is at most
+21/pp1/pq1/q· max{1, (p − 1)−1/p} + 21/qp1/pq1/q · max{1, (q − 1)−1/q}.
+Theorem 1.2 follows from Lemma 4.1 and Proposition 4.2.
+22
+
+Remarks.
+1. In this paper we have left open several natural questions, including the existence of linear
+extension operators for ˙W 1,p(T) for weighted trees T in the extreme cases p = 1, p = ∞,
+as well as the analog of our result for the inhomogeneous Sobolev space W 1,p(T) in place
+of ˙W 1,p(T).
+2. The problem of existence of linear Sobolev extension operators for weighted trees arose in
+connection with an extension problem for W 1,p(R2). More precisely, given E ⊆ R2, let
+˙W 1,p(E) denote the space of restrictions to E of functions in ˙W 1,p(R2), endowed with the
+natural seminorm. Does there exist a linear extension operator from ˙W 1,p(E) to ˙W 1,p(R2)?
+The answer is affirmative for p ≥ 2; see A. Israel [3]. For 1 < p < 2, the answer
+is unknown. For a particular class of examples E, the problem reduces to the question
+answered by Theorem 1.1.
+References
+[1] Bj¨orn, A., Bj¨orn, J., Gill, J. T., Shanmugalingam, N., Geometric analysis on Cantor sets
+and trees. J. Reine Angew. Math., Vol. 725, (2017), 63–114.
+[2] Howard, R., Schep, A. R., Norms of positive operators on Lp-spaces. Proc. Amer. Math.
+Soc., Vol. 109, no. 1, (1990), 135–146.
+[3] Israel, A., A bounded linear extension operator for L2,p(R2). Ann. of Math. (2), Vol. 178,
+no. 1, (2013), 183–230.
+[4] Muckenhoupt, B., Hardy’s inequality with weights. Studia Math., Vol. 44, (1972), 31–38.
+[5] Shvartsman, P., Sobolev W 1
+p -spaces on closed subsets of Rn. Adv. Math., Vol. 220, no. 6,
+(2009), 1842–1922.
+23
+

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+arXiv:2301.08639v1  [math.AC]  20 Jan 2023
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+ALESSANDRO LINZI ID
+Abstract. The main aim of this article is to study and develop valuation theory for Krasner
+hyperfields. In analogy with classical valuation theory for fields, we generalise the formalism of
+valuation rings to describe equivalence of valuations on hyperfields. After proving basic results
+and discussing several examples, we focus on the valued hyperfields that Krasner originally defined
+in 1957. We find that these must have a particular additive structure which in turns implies the
+existence of a valuation a’la Krasner. We note that given such a valued hyperfield (F, v), the
+valuation induced by its additive structure does not have to be equivalent to v. We discuss the
+cases in which it does.
+1. Introduction
+In 1957, M. Krasner in [24] (the article is included in Krasner’s collected works [4, pages 413–
+490]) formulated for the first time an axiomatisation of structures that generalise fields by allowing
+the additive operation to be multivalued (see [4, pages 407–490] for more details). He called these
+hyperfields (in french hypercorps).
+On the one hand, he found his inspiration in the work of Marty [35, 36, 37] which is considered
+as the starting point of hypergroup theory. In fact, the additive part of a hyperfield is a hypergroup
+(see also [48, 38] for a more detailed historical overview).
+On the other hand, Krasner was motivated by his interest in valuation theory (in connection with
+p-adic numbers) and he sensed the importance of some hyperfields which are canonically associated
+to any valued field (cf. Example 4.11). Let us briefly recall some basic notions of classical valuation
+theory.
+Let K be a field and Γ a linearly ordered abelian group (written additively). A surjective map
+v : K → Γ ∪ {∞}
+is called a (Krull) valuation on K if it satisfies for all x, y ∈ K:
+– v(x) = ∞ if and only if x = 0,
+– v(xy) = v(x) + v(y),
+– v(x + y) ≥ min{v(x), v(y)}.
+Here, ∞ is a symbol such that γ + ∞ = ∞ + γ = ∞ > γ for all γ ∈ Γ. If a valuation v on a field
+K is given, then (K, v) is called a valued field. One usually denotes Γ by vK and call it the value
+group of (K, v). The value v(x) of x ∈ K will often be written as vx, if there is no risk of confusion.
+2020 Mathematics Subject Classification. Primary: 12J20, 20N20 Secondary: 13A18.
+Key words and phrases. Hyperfield, multifield, hyperring, valuation, ordered abelian group, tropical hyperfield.
+The author would like to spend a few words to thank H. Stojałowska, P. Touchard, Franz-Viktor and Katarzyna
+Kuhlmann, I. Cristea as well as Ch. Massouros, who all, in one way or another, contributed to the realisation of the
+final version of this manuscript.
+1
+
+2
+LINZI, A.
+If (K, v) is a valued field, then
+Ov := {x ∈ K | vx ≥ 0}
+is a subring of K, called the valuation ring of (K, v). It determines the valuation map v up to
+equivalence, i.e., up to composition with an order preserving isomorphism of the value group. Any
+valuation ring has a unique maximal ideal
+Mv := {x ∈ K | vx > 0}
+and the field Kv := Ov/Mv is called the residue field of (K, v). For further details, let us mention
+[10, 44] as general references on classical valuation theory.
+While it is quite natural to generalise the concept of valuation to hyperfields (see Definition 3.1),
+Krasner noticed that the structures which attracted his attention come equipped with a map similar
+to a valuation which satisfies two additional properties (see Definition 4.7). These properties would
+be vacuous if postulated for valued fields. Thus, Krasner included them in his axiomatisation of
+valued hyperfields (hypercorps valué).
+Krasner’s motivation was not model theoretical. Nevertheless, it turns out that the structures he
+studied play an important role in the model theory of valued fields; specifically for the problem of
+quantifier elimination for henselian valued fields (of characteristic 0). In this setting, these objects
+are known as RV-structures or leading-term structures and have been considered, independently of
+Krasner, by Flenner in [11] (see also [3, 27, 33, 34] for more details). In addition, an interesting
+application of hyperfields in the model theory of valued fields recently appeared in [32].
+The interest for valued hyperfields may also be motivated by the following observations. Clas-
+sically, real algebra, which studies real fields (i.e., linearly ordered fields, with an order compatible
+with the operations) and was developed by E. Artin in his solution of Hilbert’s seventeenth problem,
+is in relation with valuation theory. After the works of Marshall and Gładki [15, 16] on real hyper-
+fields, it is to expect that a development of valuation theory for hyperfields would be beneficial for
+this line of research. Significant developments have already been achieved with the generalisation
+of classical results, such as the Baer–Krull Theorem, to the multivalued setting (see [26, 31]). The
+reader interested in these aspects may look also at [12, 13, 14]. Furthermore, the unpublished work
+of Viro [47], followed up by Jun and Jell et al. [20, 18], indicate applications in the realm of trop-
+icalization maps and analytification, topics that are as well, classically, in relation with valuation
+theory.
+We believe that developing valuation theory for hyperfields will eventually lead to further appli-
+cations back to the classical theory of valuations for fields.
+The switch from singlevalued operations to multivalued ones may be not difficult to describe
+and understand. Nevertheless, the consequences of this switch have to be handled very carefully.
+For instance, valuation theorists are accostumed to the fact that v(x − y) always defines a metric
+(in fact, an ultrametric, cf. Section 4.1) on a valued field (K, v), but this clearly ceases to be true
+(in general) if the operation of addition (and hence of difference) is multivalued. Indeed, in the
+latter case, the distance map may depend on the choice of some element of the set x − y and this
+choice is not canonical, in general. This obstacle is overtaken using one of Krasner’s postulates
+which forces all the elements of x − y, for x ̸= y, to have the same value. However, one may expect
+the latter being a quite strong requirement. In fact, the restrictions due to Krasner’s axioms for
+valued hyperfields are so strong that, e.g., the trivial valuation on a hyperfield F (Example 3.2)
+satisfies them only if F is actually a field (Remark 4.10). For this and other reasons (which we
+will discuss later in the paper), it makes sense to consider also valuations on hyperfields which do
+not satisfy the two additional properties imposed by Krasner (cf. Example 4.17) and it turns out
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+3
+that this choice is more beneficial than harmful. As a consequence of these observations, the term
+“valued hyperfield” will be used in this paper in a broader sense than the original one; the valued
+hyperfields satisfying in addition Krasner’s axioms will be called “Krasner valued hyperfields”and
+their valuations “Krasner valuations”.
+The manuscript is organised as follows. In Section 2 we introduce the necessary concepts and
+terminology from hypercompositional algebra, taking the opportunity to discuss several examples.
+In Section 3, we show that the formalism of valuation rings to describe valuations up to equivalence
+generalises without major modifications to the multivalued framework. We use this formalism to
+state and prove one of our main results Theorem 4.22 on Krasner valued hyperfields in Section 4.
+Our main theorem states that the existence of a Krasner valuation on a hyperfield F implies that
+the additive structure of F satisfies certain additional axioms (cf. Proposition 4.20). Conversely, if
+the additive structure of a hyperfield F satisfies those axioms, then F admits a Krasner valuation
+(cf. Proposition 4.21). Let v be a Krasner valuation on a hyperfield F. Then the additive structure
+of F induces a Krasner valuation w on F. We observe that, in general, w is not equivalent to
+v (Example 4.25). After Section 5, where we put into our context the notion of coarsening of a
+valuation, which turns out to be necessary for our discussion, we describe situations in which v and
+w are equivalent in Section 6. The final Section 7 of this manuscript contains possible further lines
+of investigation.
+Besides presenting some new results, the article is also thought to serve as a unified reference for
+basic hyperring and hyperfield theory and related terminology. On the one hand, these hypercom-
+positional structures have recently attracted increasing interest from the mathematical community
+(including top mathematicians such as A. Connes [7, 8]). On the other hand, we found the relevant
+literature on hyperrings and hyperfields to be quite fragmented. After the time and effort spent
+during the PhD studies and thanks to the fruitful connections with some of the mathematicians who
+are part of the early history of hypercompositional algebra, we felt that we could give a contribution
+from this point of view as well.
+2. Preliminaries
+Let H be a non-empty set and P(H) its power set. A multivalued operation + on H is a function
+which associates to every pair (x, y) ∈ H × H an element of P(H), denoted by x + y. If + is a
+multivalued operation on H ̸= ∅, then for x ∈ H and A, B ⊆ H we set
+A + B :=
+�
+a∈A,b∈B
+a + b,
+A + x := A + {x} and x + A := {x} + A. If A or B is empty, then so is A + B.
+A hypergroup can be defined as a non-empty set H with a multivalued operation + which is
+associative (see Definition 2.3 (CH1) below) and reproductive on H (i.e., x+H = H +x = H for all
+x ∈ H). This notion was first considered by F. Marty in [35, 36, 37]. The theory of hypergroups,
+with a detailed historical overview, is presented in [38], where an extensive bibliography is also
+provided.
+Definition 2.1. A hyperoperation + on H is a multivalued operation such that x + y ̸= ∅ for all
+x, y ∈ H.
+Lemma 2.2 (Theorem 12 in [38]). If (H, +) is a hypergroup, then + is a hyperoperation on H.
+Proof. Aiming for a contradiction, suppose that x + y = ∅ for some x, y ∈ H. Then
+H = x + H = x + (y + H) = (x + y) + H = ∅ + H = ∅,
+
+4
+LINZI, A.
+which is excluded.
+□
+The following special class of hypergroups (cf. Remark 2.4 below) will be of interest for us.
+Definition 2.3. A canonical hypergroup is a triple (H, +, 0), where H ̸= ∅, + is a multivalued
+operation on H and 0 is an element of H such that the following axioms hold:
+(CH1) + is associative, i.e., (x + y) + z = x + (y + z) for all x, y, z ∈ H,
+(CH2) x + y = y + x for all x, y ∈ H,
+(CH3) for every x ∈ H there exists a unique x′ ∈ H such that 0 ∈ x + x′ (the element x′ will be
+denoted by −x),
+(CH4) z ∈ x + y implies y ∈ z − x := z + (−x) for all x, y, z ∈ H.
+Lemma 2.4. Let (H, +, 0) be a canonical hypergroup. Then the multivalued operation + is repro-
+ductive on H. In particular, (H, +) is a hypergroup and + is a hyperoperation.
+Proof. Fix a ∈ H. For all x ∈ H + a there exists y ∈ H such that x ∈ y + a ⊆ H. Therefore,
+H + a ⊆ H. For the other inclusion, observe that for all x ∈ H we have that
+x ∈ x + 0 ⊆ x + (a − a) = (x − a) + a,
+so there exists y ∈ x − a ⊆ H such that x ∈ y + a ⊆ H + a. We have proved that H = H + a
+for an arbitrary a ∈ H. Now the conclusions of the lemma follow from (CH2), the definition of
+hypergroups and Lemma 2.2.
+□
+Let (H, +, 0) be a canonical hypergroup. Then 0 is called the neutral element for + in H because
+of the following observation.
+Lemma 2.5 (Section III, (b) in [39]). Let (H, +, 0) be a canonical hypergroup. Then x + 0 = {x}
+for all x ∈ H.
+Proof. Take x ∈ H. If y ∈ x + 0, then 0 ∈ y − x follows by (CH4). Now y = x follows from the
+uniqueness required in (CH3).
+□
+Remark 2.6. Note that an abelian group (G, ∗, 0) is not a priori a canonical hypergroup, because the
+operation on G is not a multivalued operation, as it takes values in G and not in P(G). However,
+it can be turned into a canonical hypergroup (G, +, 0) by setting x + y := {x ∗ y} for all x, y ∈ G.
+Conversely, if a canonical hypergroup (H, +, 0) satisfies that x + y is a singleton for all x, y ∈ H,
+then one can define on H a standard operation which makes it an abelian group with identity 0. In
+these cases, we sometimes abusively say that (G, ∗, 0) is a canonical hypergroup or that (H, +, 0)
+is a group.
+2.1. Hyperrings and hyperfields. Let us now define the structures of main interest in this paper.
+Definition 2.7. A (commutative) hyperring is a tuple (R, +, ·, 0) which satisfies the following
+axioms:
+(HR1) (R, +, 0) is a canonical hypergroup,
+(HR2) (R, ·) is a (commutative) semigroup and 0 is an absorbing element, i.e., 0 · x = x · 0 = 0, for
+all x ∈ R,
+(HR3) the operation · is distributive with respect to +. That is, for all x, y, z ∈ R,
+x(y + z) = xy + xz,
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+5
+where, for x ∈ R and A ⊆ R, we have set
+xA := {xa | a ∈ A}.
+If the operation · has a neutral element 1 ̸= 0, then we say that (R, +, ·, 0, 1) is a hyperring with unity.
+If (R, +, ·, 0, 1) is a hyperring with unity and (R \ {0}, ·, 1) is an abelian group, then (R, +, ·, 0, 1) is
+called a hyperfield. If R is a hyperring with unity, then we denote by R× the multiplicative group
+of the units of R, i.e.,
+R× := {x ∈ R | ∃y ∈ R : xy = 1}.
+In particular, if R is a hyperfield, then R× = R \ {0}.
+Since we will only consider the commutative case, in the following sections we will call commu-
+tative hyperrings simply hyperrings.
+Remark 2.8. In the literature, the name “hyperring” can be found for structures (R, +, ·) where +
+is an operation and · is a multivalued operation or where both + and · are multivalued operations.
+Some authors refer to the structures (R, +, ·, 0) where only the addition is multivalued as Krasner
+hyperrings (some more historical remarks on this are given in in [17, Section 4]).
+Structures (R, +, ·, 0) where + is an operation and · is a multivalued operation (satisfying similar
+axioms) were introduced in [45] and called multiplicative hyperrings (in italian iperanelli moltiplica-
+tivi).
+Structures (R, +, ·, 0) with both + and · multivalued operations (satisfying similar axioms) are
+instead called general hyperrings (see e.g. [9, Section 2]).
+In this paper, the name “hyperring” will be used for Krasner hyperrings exclusively, as indicated
+in the above definition.
+In the literature, one may also find the term multiring referring to a structure (R, +, ·, 0), where
++ is a multivalued operation and · is an operation, satisfying (HR1), (HR2) and the following
+weaker version of (HR3):
+(MR) x · (y + z) ⊆ x · y + x · z, for all x, y, z ∈ M.
+Thus, in this case, the different name reflects a difference in the axioms rather than in the structure.
+Similarly, multifields are defined as hyperfields satisfying (MR) instead of (HR3). Multirings and
+multifields have been considered for instance in [12], where, among other facts, it is observed that
+all multifields are hyperfields, while there are several meaningful examples of multirings that are
+not hyperrings.
+In this paper, we preferred to use the name “hyperfield” solely.
+We say that a hyperring (resp. hyperfield) R is a ring (resp. field) if the additive canonical
+hypergroup of R is a group (cf. Remark 2.6). The next observation gives a necessary condition for
+a hyperring with unity to be a ring. This fact is an immediate corollary of a result already noted
+in [42, page 369] (see also Example 2.45 below). We wish to state it for later reference and we take
+the opportunity to write a quick proof .
+Lemma 2.9 ([42]). A hyperring with unity R is a ring if and only if 1 − 1 = {0}.
+Proof. If R is a field, then 1 − 1 = {0} holds trivially. Conversely, by axiom (HR3) 1 − 1 = {0}
+implies x − x = {0} for all x ∈ R. Take a, b ∈ R and x, y ∈ a + b. We have that
+x − y ⊆ (a + b) − (a + b) = (a − a) + (b − b) = {0}
+In particular, 0 ∈ x − y and x = y follows fro (CH3). We have proved that a + b is a singleton for
+all a, b ∈ R.
+□
+
+6
+LINZI, A.
+Example 2.10. The K-hyperfield K is the set {0, 1} with the hyperoperation + which has 0 as its
+neutral element and satisfies 1 + 1 = {0, 1}. The multiplication is the obvious one.
+Remark 2.11. It is customary among some authors to call the above hyperfield the Krasner hy-
+perfield. We prefer to avoid that terminology which, in view of Remark 2.8 above, might lead to
+unnecessary confusion.
+Example 2.12. The sign hyperfield S is the set {−1, 0, 1} with the hyperoperation + which has
+0 as its neutral element and satisfies x + x = {x} for x ∈ {−1, 1} and 1 − 1 = {−1, 0, 1}. The
+multiplication is the obvious one.
+Example 2.13. The weak sign hyperfield W is the set {−1, 0, 1} with the hyperoperation + which
+has 0 as its neutral element and satisfies x + x = {x, −x} for x ∈ {−1, 1} and 1 − 1 = {−1, 0, 1}.
+The multiplication is the obvious one.
+Other examples of finite hyperfields are described e.g. in [1]. Let us now consider some infinite
+examples.
+Example 2.14. Let (Γ, +, <, 0) be an ordered abelian group and ∞ be a symbol such that γ+∞ =
+∞+γ = ∞ > γ for all γ ∈ Γ. For x, y ∈ Γ∪{∞} such that x ≤ y (i.e., x < y or x = y), let us denote
+by [x, y] the set consisting of all z ∈ Γ ∪ {∞} such that x ≤ z ≤ y. Consider the hyperoperation ⊞
+defined on T (Γ) := Γ ∪ {∞} as x ⊞ ∞ = ∞ ⊞ x = {x} for all x ∈ T (Γ) and:
+x ⊞ y :=
+�
+{min{x, y}}
+if x ̸= y,
+[x, ∞]
+if x = y.
+(x, y ∈ T (Γ))
+It is not difficult to check that (T (Γ), ⊞, +, ∞, 0) is a hyperfield. The hyperfield T (R, +, 0, >),
+where > denotes the standard order of the real numbers, is known as the tropical hyperfield (see
+[47, Section 5.3]). Therefore, we call the hyperfields of the form T (Γ) generalised tropical hyperfields.
+Example 2.15. Let (Γ, +, <, 0) be an ordered abelian group and ∞ be a symbol such that γ+∞ =
+∞ + γ = ∞ > γ for all γ ∈ Γ. For x, y ∈ Γ ∪ {∞} such that x < y, let us denote by (x, y] the set
+consisting of all z ∈ Γ∪{∞} such that x < z ≤ y. Consider the hyperoperation ⊞′ defined on T (Γ)
+as x + ∞ = ∞ + x = {x} for all x ∈ T (Γ) and:
+x ⊞′ y :=
+�
+{min{x, y}}
+if x ̸= y,
+(x, ∞]
+if x = y.
+(x, y ∈ T (Γ))
+It is not difficult to check that (T (Γ), ⊞′, +, ∞, 0) is a hyperfield. We will denote it by T ′(Γ) and
+call it the strict generalised tropical hyperfield.
+The above examples are (up to isomorphism) all instances of a general construction which yields
+a hyperring from a ring and a subgroup of its multiplicative semigroup (see Proposition 2.17 below).
+This construction was first described by Krasner in [25]. We briefly recall it in the following example.
+Example 2.16. Let A be a commutative ring and T a subgroup of the commutative semigroup
+(A, ·). Let AT denote the set of all multiplicative cosets xT for x ∈ A, in particular 0T = {0}.
+Krasner showed that by setting
+xT + yT := {(x + yt)T | t ∈ T }
+and
+xT · yT := xyT
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+7
+we have that (AT , +, ·, 0T ) is a hyperring where −(xT ) = (−x)T for all x ∈ A. In addition, if A
+is a field, then this construction always yields a hyperfield. This is called the factor (or quotient)
+hyperring (resp. hyperfield) of A modulo T .
+For the next proposition we employ the notion of isomorphism of hyperfields (see Definition 2.33
+below).
+Proposition 2.17. The following assertions hold:
+(i) For any field K with more than two elements, we have that K is isomorphic to KK×.
+(ii) For any ordered field K with positive cone P, we have that S is isomorphic to KP .
+(iii) If p > 3 is a prime such that p ≡ 3 mod 4, then W is isomorphic to (Fp)(F×
+p )2.
+(iv) Let Γ be an ordered abelian group. For any field k with more than two elements, let K :=
+k((tΓ)) denote the Hahn series field and let v be its canonical t-adic valuation. Then KO×
+v
+is isomorphic to T (Γ).
+(v) Let Γ be an ordered abelian group. Let K := F2((tΓ)) denote the Hahn series field and let
+v be its canonical t-adic valuation. Then KO×
+v is isomorphic to T ′(Γ).
+Proof. Let K be a field with more than two elements. Then KK× contains precisely two elements,
+the coset 0K× and the coset 1K×, the latter containing all non-zero elements of K. If x ∈ K×, then
+−x ∈ K× as well and thus 0K× belongs to 1K× + 1K×. Since by assumption there are x, y ∈ K×
+with x ̸= y, also (x − y)K× ̸= 0K× is an element of 1K× + 1K×. This shows (i).
+Let K be an ordered field with positive cone P. Here, this means that P is a subset of K× such
+that P + P, P · P ⊆ P and K× is the disjoint union of P and −P. It follows that KP contains
+precisely three1 elements: 0P, 1P (containing all the elements of P) and −1P (containing all the
+elements of −P). Since P + P ⊆ P, we have that 1P + 1P only contains 1P. By definition, if
+x ∈ P, then −x ∈ −P. Thus, 1P − 1P contains 0P. It does also contain 1P and −1P since e.g.
+1 + 1 ∈ P (because 1 ∈ P) and (1 + 1) − 1 = 1 ∈ P while 1 − (1 + 1) = −1 ∈ −P. This shows (ii).
+Let K be Fp (the finite field with p elements) for some prime number p > 3 such that p ≡ 3
+mod 4. The prime number p is certainly odd and thus the cardinality of (K×)2 is p−1
+2 . If −1 is
+a square in K, then K× contains an element of order 4 but this is excluded by the assumption
+p ≡ 3 mod 4. We have that x ∈ K× is a square if and only if −x is not a square. It follows that
+K(K×)2 contains precisely three elements: 0(K×)2, 1(K×)2 (containing all the non-zero squares)
+and −1(K×)2 (containing all the non-squares). Moreover, 1(K×)2 − 1(K×)2 contains 0(K×)2.
+This shows that −1(K×)2 is the hyperinverse of 1(K×)2 in K(K×)2. Since 1 + 1 ̸= 0 in K and
+1 = (1 + 1) − 1, we have that either 1(K×)2 or −1(K×)2 (depending on which among 1 + 1 and
+−(1 + 1) is a square) belongs to 1(K×)2 − 1(K×)2. By the symmetry of 1(K×)2 − 1(K×)2 under
+the application of − (i.e., multiplication by −1(K×)2), we obtain that both 1(K×)2 and −1(K×)2
+must be elements of 1(K×)2 −1(K×)2. In a finite field, any non-square is the sum of two (non-zero)
+squares. Thus, −1(K×)2 is an element of 1(K×)2 + 1(K×)2. In order to show that 1(K×)2 is an
+element of 1(K×)2+1(K×)2 as well, we distinguish two cases. If 1+1 is a square in K, then 1(K×)2
+is in 1(K×)2 + 1(K×)2 trivially. Otherwise, 1 + 1 is not a square. In this case, either 1 + 1 + 1 is a
+square, in which case 1 = (1+1+1)−(1+1) shows that 1(K×)2 is an element of 1(K×)2 +1(K×)2,
+or −(1 + 1 + 1) = −((1 + 1) + 1)) is a square, in which case −(1 + 1) = −(1 + 1 + 1) + 1 implies
+that 1(K×)2 is an element of 1(K×)2 + 1(K×)2. This proves (iii).
+1In the literature, sometimes positive cones are defined as “non-negative cones”, i.e., containing 0. Notice that for
+our statement to hold it is necessary to exclude 0 from positive cones.
+
+8
+LINZI, A.
+Let k be a field with more thatn 2 elements and consider the Hahn series field K = k((tΓ))
+equipped with the t-adic valuation v. A non-zero element x of K× is represented as a formal series
+x =
+�
+γ∈Γ
+aγtγ
+with well-ordered support, i.e., {γ ∈ Γ | aγ ̸= 0} is a well-ordered set with respect to the order of
+Γ. The t-adic value of x under v is defined to be the minimum of the support of x (cf. [10, Exercise
+3.5.6]).
+The value group of v is thus Γ and it follows from general valuation theory that K×/O×
+v is
+isomorphic to Γ as an ordered abelian group (cf. [10, Section 2.1] or Lemma 3.20 below).
+For
+x ∈ K×, we have that xO×
+v contains all the elements of K with the same value of x under v. If
+x, y ∈ K are such that vx ̸= vy (i.e., xO×
+v ̸= yO×
+v ), then
+v(x + yt) = v(x + y) = min{vx, vy}
+for any t ∈ O×
+v (i.e., any t with vt = 0). This follows from [10, Section 1.3 (1.3.4)] (or Corollary 3.5
+(iv) below). We now note that if x ∈ K has any positive value under v, then v(x − 1) = 0 and thus
+vx = v(1 + (x − 1)) is the value of some element of K generating a coset in KO×
+v which belongs to
+1O×
+v +1O×
+v . Moreover, by the assumption on the cardinality of k, there exists a ∈ k \{1}, so by the
+definition of the t-adic valuation, we have that va = 0 and v(1 − a) = 0. Since v(1) = v(−1) = 0,
+we conclude that the image of 1O×
+v − aO×
+v = 1O×
+v + 1O×
+v under v is [0, ∞]. By distributivity,
+this suffices to show that T (Γ) ≃ KO×
+v (see also Lemma 3.3 below). The proof of (iv) is now also
+complete.
+The proof for (iv) can easily be adapted to prove (v).
+□
+Remark 2.18. Let us mention at this point that not all hyperfields are factor hyperfields. This fact
+was proved by Massouros in [39] who then further improved his results in [40]. According to [21,
+Theorem 6] finite hyperfields such that 1 /∈ 1 + 1 and with the property that 0 does not belong to
+any finite sum of 1 with itself are also not quotient hyperfields.
+2.2. Subhyperrings. To choose a good notion of subhyperring is not as straightforward as it might
+seem. Two options are presented in the next definition.
+Definition 2.19. Let (R, +, ·, 0) be a hyperring. A subset S of R is a relational subhyperring of R
+if it is multiplicatively closed and with the induced multivalued operation:
+x +S y := (x + y) ∩ S
+(x, y ∈ S)
+we have that (S, +S, ·, 0) is a hyperring as well.
+A subset S of R is a (traditional) subhyperring of R if 0 ∈ S and for all x, y ∈ S we have that
+x − y ⊆ S and xy ∈ S.
+If R is a hyperring with unity 1, then we say that a relational subhyperring S of R is a relational
+subhyperfield of R if 1 ∈ S and (S, +S, ·, 0, 1) is a hyperfield. In addition, a subhyperring S of R is
+called a (traditional) subhyperfield if 1 ∈ S and (S \ {0}, ·, 1) is an abelian group.
+Remark 2.20. The adjective “relational” has been chosen for the following reason. A multivalued
+operation can be encoded in a first-order language via the ternary relation z ∈ x + y (as noticed
+e.g. in [32]). If we consider a first-order language L with a constant symbol for 0, a binary function
+symbol for · and a ternary relation symbol for +, then a hyperring naturally becomes a structure on
+L, which is a model of all the axioms of Definitions 2.7 and 2.3 (written as sentences in L). Under
+this interpretation, the relational subhyperrings of a hyperring R are precisely the submodels of R,
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+9
+i.e., the substructures of R (see [43, Section 2.3]) which satisfy the axioms. For more details on the
+model theoretical point of view let us refer the reader to [33].
+Our choices regarding terminology are also motivated by historical reasons. In fact, subhyper-
+groups (and consequently subhyperfields) have been defined long time ago (see e.g. Definition 2 and
+the subsequent remark in [22]2).
+Remark 2.21. It is clear that a subhyperring S of a hyperring (R, +, ·, 0) is a relational subhyperring
+of R and that +S = + in this case.
+On the other hand, not all relational subhyperrings are
+subhyperrings, as the following example shows.
+Example 2.22. Consider an ordered abelian group (Γ, +, 0, <). Let R := T (Γ) be the hyperfield
+obtained as in Example 2.14 and consider the subset S := {∞, 0} of R. It is straightforward to check
+that S is a relational subhyperfield of R. However, S is not a subhyperfield of R since 0⊞0 = [0, ∞]
+is not a subset of S.
+Remark 2.23. Let Γ be an ordered abelian group. It is not difficult to see that the only subhyperfield
+of T (Γ) is T (Γ) itself. On the other hand, if ∆ is a subgroup of Γ, then T (∆) is a relational
+subhyperfield of T (Γ) (Example 2.22 above corresponds to the case ∆ = {0}).
+2.3. Homomorphisms. Next, let us consider a notion of homomorphism for hyperrings and hy-
+perfields.
+Definition 2.24. Let (R, +R, ·R, 0R) and (S, +S, ·S, 0S) be hyperrings. We say that a map σ :
+R → S is a homomorphism of hyperrings if
+(HH1) σ(0R) = 0S,
+(HH2) σ(x ·R y) = σ(x) ·S σ(y), for all x, y ∈ R,
+(HH3) σ(x +R y) ⊆ σ(x) +S σ(y), for all x, y ∈ R.
+If (R, +R, ·R, 0R, 1R) and (S, +S, ·S, 0S, 1S) are hyperfields, then we say that a homomorphism of
+hyperrings σ : R → S is a homomorphism of hyperfields when the following properties
+(HH4) σ(1R) = 1S,
+(HH5) σ(x−1) = σ(x)−1, for all x ∈ R \ {0},
+hold as well.
+Example 2.25. Let R be a hyperring. The map defined by σ(0) := 0 and σ(x) := 1 for x ∈ R\{0}
+is a homomorphism of hyperrings R → K.
+Example 2.26. Let (Γ, <, +, 0Γ) be an ordered abelian group. The map
+ι : K → T (Γ)
+0K �→ ∞
+1K �→ 0Γ
+is a homomorphism of hyperfields.
+Example 2.27. Let K be a field and T a subgroup of K×. The function K → KT mapping x ∈ K
+to [x]T ∈ KT is a homomorphism of hyperfields.
+Remark 2.28. Let σ : R → S be a homomorphism of hyperrings. Then the kernel of σ
+ker σ := {x ∈ R | σ(x) = 0S}
+2This article is included in Krasner’s collected works [4, pages 280–406]
+
+10
+LINZI, A.
+is a subhyperring of R. Indeed, for x, y ∈ ker σ, we have that
+σ(x − y) ⊆ σ(x) − σ(y) = 0S − 0S = {0S}.
+Thus, σ(z) = 0S for all z ∈ x − y, i.e., x − y ⊆ ker σ.
+Remark 2.29. Let (R, +, ·, 0, 1) be a hyperring and S ⊆ R. Consider the inclusion map ι : S ֒→ R.
+It is straightforward to verify that if S is a relational subhyperring of R, then ι is a homomoprhism
+of hyperrings.
+On the other hand, if (R, +R, ·R, 0R) and (S, +S, ·S, 0S) are hyperrings with S ⊆ R and the
+inclusion ι : S ֒→ R is a homomorphism of hyperrings, then one cannot conclude that S is a
+relational subhyperring of R, as the following example shows.
+Example 2.30. The identity S → W is a homomorphism of hyperrings. However, S is not a
+subhyperring of W as the hypersum of 1 with itself in S is {1}, while the hypersum of 1 with itself
+in W intersected with S is {−1, 1}.
+The above example motivates the following definition.
+Definition 2.31. Let (R, +R, ·R, 0R) and (S, +S, ·S, 0S) be hyperrings (resp. hyperfields with uni-
+ties 1R and 1S). An injective homomormphism of hyperrings (resp. hyperfields) σ : R → S is called
+an embedding of hyperrings (resp. hyperfields) if
+(EM1) σ(x +R y) = (σ(x) +S σ(y)) ∩ Im σ, for all x, y ∈ R.
+We leave the straightforward proof of following lemma to the reader.
+Lemma 2.32. If S is a relational subhyperring of a hyperring R, then the inclusion map S ֒→ R is
+an embedding of hyperrings. Conversely, If R and S are hyperrings with S ⊆ R and the inclusion
+map S ֒→ R is an embedding of hyperrings, then S is a relational subhyperring of R.
+Definition 2.33. A homomorphism of hyperrings (resp. hyperfields) σ : R → S is called an
+isomorphism of hyperrings (resp. hyperfields) if it is a bijective map and σ−1 : S → R is also a
+homomorphism of hyperrings (resp. hyperfields). We say that two hyperrings (resp. hyperfields) R
+and S are isomorphic and we write R ≃ S if there exists an isomorphism σ : R → S.
+Lemma 2.34. Let R and S be hyperrings (resp. hyperfields) and denote by +R and +S their
+hyperoperations, respectively. A map σ : R → S is an isomorphism of hyperrings (resp. hyperfields)
+if and only if σ is a surjective embedding of hyperrings (resp. hyperfields).
+Proof. If σ is an isomorphism, then it is bijective by definition, in particular Im σ = S. Since σ is
+a homomorphism, we have that σ(x +R y) ⊆ σ(x) +S σ(y) holds. On the other hand, since σ−1
+satisfies (HH3) as well, we obtain that
+σ−1(σ(x) +S σ(y)) ⊆ x +R y.
+Applying σ to both sides then yields σ(x +R y) ⊇ σ(x) +S σ(y).
+Assume now that σ is a surjective embedding and let us show that σ−1 is a homomorphism.
+Clearly, (HH1) and (HH2) hold for σ−1.
+Since by (EM1) and the surjectivity we have that
+σ(x +R y) = σ(x) +S σ(y), property (HH3) follows applying σ−1 to both sides and using again
+the surjectivity of σ.
+□
+Remark 2.35. Note that a bijective homomorphism is not necessarily an isomorphism as it can be
+deduced from Example 2.30.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+11
+Example 2.36. Let (Γ, <, +, 0Γ) be an ordered abelian group. Consider the relational subhyper-
+field S := {∞, 0} of T (Γ) as in Example 2.22. Then (S, +S, ·, ∞, 0Γ) is isomorphic to K.
+As usual we identify isomorphic structures: Examples 2.22 and 2.36 can be expressed by saying
+that K is a subhyperfield of T (Γ) for any ordered abelian group Γ.
+2.4. Hyperideals. We now briefly discuss how the classical ring theory notion of ideal generalises
+in the multivalued setting.
+Definition 2.37 (Definition 2.1 in [5] and Definition 2.9 in [31]). Let R be a hyperring. A subhy-
+perring I of R is a hyperideal of R if it satisfies
+(HID1) xy ∈ I, for all x ∈ R and y ∈ I.
+Lemma 2.38. Let R be a hyperring with no zero divisors, i.e., xy = 0R implies x = 0R or y = 0R.
+A relational subhyperring I of R satisfying (HID1) is a hyperideal of R.
+Proof. Pick x, y ∈ I and let z ∈ x − y. It suffices to prove that z ∈ I. If x = 0R or y = 0R,
+then there is nothing to show. Otherwise, by distributivity in R we have that zy ∈ xy − y2 and by
+(HID1) we obtain that zy ∈ I, since y ∈ I. Therefore, zy ∈ (xy − y2) ∩ I. Now, distributivity in I
+(with respect to the induced multivalued operation +I) yields zy ∈
+�
+(x − y) ∩ I
+�
+y, so there exists
+z′ ∈ (x − y) ∩ I such that zy = z′y. Hence, 0R ∈ zy − z′y = (z − z′)y. Since R has no zero divisors,
+this implies that 0R ∈ z − z′ and hence z = z′ ∈ I.
+□
+Remark 2.39. We wish to justify the choice of requiring hyperideals to be traditional subhyperrings.
+In [19, Section 3.1], Jun provides a generalisation of the classical quotient construction of a ring
+modulo an ideal in the multivalued setting. One property that is certainly desirable to preserve is for
+the canonical epimorphism from a hyperring R to the quotient hyperring R/I modulo a hyperideal
+I to be a homomorphism of hyperrings R → R/I having I as its kernel. We have no examples
+of hyperrings with a relational subhyperring satisfying (HID1), but at the same time we did not
+succeed in proving Lemma 2.38 without the assumption on zero divisors. If such an example would
+exist, then that relational subhyperring could not be the kernel of a homomorphism of hyperrings
+by Remark 2.28 and so we should not call it a hyperideal.
+Remark 2.40. Let σ : R → S be a homomorphism of hyperrings. Then it is easy to show that ker σ
+is a hyperideal of R. Conversely, for all hyperideals I of a hyperring R the map σ : R → K defined
+as
+σ(x) :=
+�
+0
+if x ∈ I,
+1
+otherwise.
+is a homomorphism of hyperrings such that ker σ = I.
+The following statements can be shown to hold as in the classical theory of rings.
+Lemma 2.41 (Lemma 2.12 in [31]). If a hyperideal I of a hyperring R contains a unit, then I = R.
+Corollary 2.42 (Corollary 2.13 in [31]). The only hyperideals of a hyperfield F are {0} and F.
+Definition 2.43 (Definition 2.14 (ii) in [31]). Let I be a hyperideal of a hyperring R. Then I is
+called maximal if I ⊊ R and for all hyperideals J of R we have that I ⊊ J implies J = R.
+We recall without proof the following result.
+Proposition 2.44 (Proposition 2.16 (ii) in [31]). Assume that R is a hyperring with unity. Let I
+be a hyperideal R. Then I is maximal if and only if R/I is a hyperfield.
+
+12
+LINZI, A.
+Example 2.45 ([42]). Let R be a hyperring with unity. An element s of R is called a scalar if
+x + s is a singleton for all x ∈ R. As in the proof of Lemma 2.9, it is not difficult to show that
+s ∈ R is a scalar if and only if s − s = {0}. Interestingly, the set of all scalar elements of R forms a
+hyperideal of R.
+Assume now that R is a hyperring with unity. If 1 is a scalar, then by Lemma 2.41 the hyperideal
+of scalars is R and thus R is a ring (as all of its elements are scalars). Moreover, we deduce from
+Corollary 2.42 that if F is a hyperfield, but not a field, then the only scalar of F is 0, i.e., x − x is
+not a singleton for all x ∈ F ×.
+3. Valued hyperfields
+Some of the results presented in this section can be found in [31].
+Some proofs have been
+improved, others we will repeat for the convenience of the reader.
+3.1. Valuations. The next definition is a straightforward generalisation of the definition of valua-
+tion for fields.
+Definition 3.1 (Definition 4.1 in [31]). Take a hyperfield F and an ordered abelian group Γ (written
+additively). A surjective map v : F → Γ ∪ {∞} is called a valuation on F if it has the following
+properties:
+(V1) vx = ∞ ⇐⇒ x = 0, for all x ∈ F,
+(V2) v(xy) = vx + vy, for all x, y ∈ F,
+(V3) z ∈ x + y =⇒ vz ≥ min{vx, vy}, for all x, y, z ∈ F.
+If v is a valuation on a hyperfield F we call (F, v) a valued hyperfield and we denote by vF the
+ordered abelian group v(F ×), i.e., the value group of (F, v).
+Example 3.2. Let F be a hyperfield. The trivial valuation vx = 0 for all x ∈ F × is always a
+valuation on F with value group {0}.
+The next result provides further examples of valued hyperfields.
+Lemma 3.3 (Corollary 4.2 in [31]). Let (K, v) be a valued field and take a subgroup T of K×.
+Denote by O×
+v the group of units of the valuation ring of K, i.e., O×
+v := {x ∈ K | vx = 0}. If
+T ⊆ O×
+v , then
+vT : KT → vK ∪ {∞}
+xT �→ vx
+is a valuation on KT.
+Proof. The assumption T ⊆ O×
+v guarantees that the map vT is well-defined. Moreover, surjectivity,
+(V1) and (V2) follow immediately from the corresponding properties of v and the definition of the
+factor hyperfield KT . It remains to verify (V3). Take x, y ∈ K and t ∈ T . We obtain that
+v(x + yt) ≥ min{vx, v(yt)} = min{vx, vy + vt} = min{vx, vy}.
+From this it easily follows that (V3) holds for (KT , vT ).
+□
+The next lemma gives an alternative definition of valuation on (hyper)fields (cf. [2, Example
+1.8]).
+Lemma 3.4. Let F be a hyperfield and Γ an ordered abelian group. Then (F, v) is a valued hyperfield
+with vF = Γ if and only if the map v : F → T (Γ) is a surjective homomorphism of hyperfields.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+13
+Proof. Assume that (F, v) is a valued hyperfield. Then v : F → T (vF) is a surjective map, (HH1)
+follows from (V1), (HH2) follows from (V2), (HH3) follows from (V3). Property (HH4) follows
+because vx = vx + v(1) for all x ∈ F and (HH5) follows since, for x ∈ F ×, we have that
+0 = v(1) = v(xx−1) = vx + v(x−1).
+Thus, v is a homomorphism of hyperfields.
+Conversely, let F be a hyperfield and v : F → T (Γ) a surjective homomoprhism of hyperfields.
+Property (V2) for v follows from (HH2) and (V3) follows from (HH3) together with the definition
+of the hyperoperation of T (Γ). Property (V1) follows from Corollary 2.42 since v(1) = 0 ̸= ∞ and
+thus ker v = {x ∈ F | vx = ∞} is a proper hyperideal of F. We proved that v is a valuation on F
+with vF = Γ.
+□
+On the basis of the previous lemma, the following result is almost immediate to verify. Thus,
+we omit its proof.
+Corollary 3.5 (Lemma 4.5 in [31]). Let v : F → Γ ∪ {∞} be a valuation on a hyperfield F. Then:
+(i) v(1) = v(−1) = 0,
+(ii) v(−x) = vx for all x ∈ F,
+(iii) vx−1 = −vx for all x ∈ F ×,
+(iv) if vx ̸= vy, then vz = min{vx, vy}, for every x, y ∈ F and z ∈ x + y.
+Lemma 3.6. Let (KT , w) be a valued hyperfield which is a factor hyperfield. Then there exists a
+valuation v on K such that T ⊆ O×
+v and w = vT .
+Proof. Let Γ be the value group of (KT , w). Since K → KT , x �→ [x]T and w : KT → T (Γ) are
+surjective homomorphisms of hyperfields, their composition v : K → T (Γ) is a valuation on K
+satisfying the conditions of the statement.
+□
+Example 3.7. By Proposition 2.17 (iv) we have that T (Γ) is isomorphic to all factor hyperfields of
+the form k((tΓ))O×
+vt for some field k. By the above lemmas, the isomorphism σ : k((tΓ))O×
+vt → T (Γ)
+is a valuation on k((tΓ))O×
+vt and the identity map id = σ ◦ σ−1 : T (Γ) → T (Γ) is a valuation on
+T (Γ).
+3.2. Valuation hyperrings. The next definition is inspired by classical valuation theory.
+Definition 3.8 (Definition 4.6 in [31]). Let F be a hyperfield. A relational subhyperring O of F
+is called a valuation hyperring in F if for all x ∈ F × we have that either x ∈ O or x−1 ∈ O.
+Observe that, by definition, it follows that 1 ∈ O for any valuation hyperring O in F. Let us
+now prove some more basic properties of valuation hyperrings.
+Lemma 3.9 (Proposition 4.7 in [31]). A valuation hyperring O in a hyperfield F is a subhyperring
+of F.
+Proof. It suffices to show that a − b ⊆ O for all a, b ∈ O. Take a, b ∈ O and x ∈ a − b. If x ∈ O,
+then there is nothing to show (note that this case also includes x = 0). Otherwise, we have that
+x−1 ∈ O and thus ax−1, bx−1 ∈ O. Since x ∈ a − b we obtain from (CH4) that a ∈ x + b, so, using
+axiom (HR3),
+ax−1 ∈ (x + b)x−1 = 1 + bx−1.
+
+14
+LINZI, A.
+We have obtained that ax−1 ∈ (1 + bx−1) ∩ O = 1 +O bx−1. By (CH4) and (HR3) applied to the
+hyperring (O, +O, ·, 0), it follows that
+xx−1 = 1 ∈ ax−1 +O (−bx−1) = (a +O (−b))x−1.
+Therefore, x ∈ a +O (−b) ⊆ O. This shows that a − b ⊆ O.
+□
+Lemma 3.10 (Lemma 4.8 in [31]). Let O be a valuation hyperring in a hyperfield F.
+Then
+M := O \ O× is the unique maximal hyperideal of O.
+Proof. Take a ∈ M and c ∈ O. If ca is invertible in O, then there exists x ∈ O such that x(ca) = 1.
+Hence (xc)a = 1 and a−1 = xc ∈ O contradicting a ∈ M. This proves that ca ∈ M and shows that
+M satisfies (HID1).
+Take a, b ∈ M. We may assume that ab−1 ∈ O (otherwise ba−1 ∈ O and we can interchange the
+roles of a and b). Since O is a subhyperring of F (cf. Lemma 3.9), we obtain that 1 − ab−1 ⊆ O
+and therefore, using what we have just proved, we conclude that
+b − a = b(1 − ab−1) ⊆ M.
+We have shown that M is a hyperideal of O.
+Since, by the definition of M, we have that O\M = O×, by Lemma 2.41, every proper hyperideal
+of O must be contained in M, showing that M is the unique maximal hyperideal of O.
+□
+3.3. Residue hyperfield. Any valuation on a hyperfield F induces a valuation hyperring in F.
+Proposition 3.11 (Proposition 4.11 in [31]). Let v : F → Γ ∪ {∞} be a valuation on a hyperfield
+F. Then
+Ov := {x ∈ F | vx ≥ 0}
+is a valuation hyperring in F and
+Mv := {x ∈ F | vx > 0}
+is its unique maximal hyperideal.
+Proof. We first prove that Ov is a subhyperring of F. Take a, b ∈ Ov. By (V3), for all c ∈ a − b we
+have vc ≥ min{va, v(−b)} = min{va, vb} ≥ 0, so a − b ⊆ Ov. Further, we have ab ∈ Ov by (V2).
+By Corollary 3.5 (iii) we conclude that if x /∈ Ov, then x−1 ∈ Ov so Ov is a valuation hyperring in
+F.
+Next we show that Mv is the unique maximal hyperideal of Ov. Observe that, by virtue of
+Corollary 3.5 (iii),
+O×
+v = {x ∈ Ov | vx = 0}.
+Hence, Mv = Ov \ O×
+v and then Mv is the unique maximal hyperideal of Ov by Lemma 3.10.
+□
+Remark 3.12. Let (F, v) be a valued hyperfield. Note that Ov can be described in terms of the
+hyperoperation ⊞ of T (vF) as the preimage in F of v(1) ⊞ v(1) under v:
+Ov = v−1(v(1) ⊞ v(1)).
+It follows from Proposition 2.44 that, for a valuation hyperring O with maximal hyperideal M,
+the quotient hyperring
+O/M = {x + M | x ∈ O},
+with the multivalued operation ⊕ defined as
+(x + M) ⊕ (y + M) := {z + M | z ∈ x + y},
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+15
+is a hyperfield (see [19, Section 3] and Remark 2.39 above).
+Definition 3.13. If (F, v) is a valued hyperfield, then we call the hyperfield Ov/Mv the residue
+hyperfield of (F, v) and we denote it by Fv. For an element x ∈ Ov, we denote by xv its natural
+image in Fv.
+Proposition 3.14. Let (K, v) be a valued field and T ⊆ O×
+v a subgroup of K×. Then KT vT ≃
+(Kv)(T v), where T v := {tv | t ∈ T }.
+Proof. By definition vT [x]T = vx for all x ∈ K, thus OvT = (Ov)T and MvT = (Mv)T . It follows
+that
+KT vT = OvT /MvT = (Ov)T /(Mv)T .
+Let us define
+σ : (Ov)T /(Mv)T → (Kv)(T v)
+[x]T + (Mv)T �→ [xv]T v
+and show that σ is an isomorphism of hyperfields. By [19, Lemma 3.3] we have that [x]T +(Mv)T =
+[y]T + (Mv)T if and only if v(x − yt) > 0 for some t ∈ T . This implies that
+0 = (x − yt)v = xv − yv · tv ∈ [xv]T v − [yv]T v ,
+where we have used the assumption T ⊆ O×
+v . This shows that σ is well-defined and injective. The
+surjectivity of σ is clear. Moreover, σ is easily seen to be a homomorphism of the corresponding
+multiplicative groups. Finally, we observe that
+σ
+�
+[x]T + (Mv)T ⊕ ([y]T + (Mv)T )
+�
+= {σ
+�
+[z]T + (Mv)T
+�
+| [z]T ∈ [x]T + [y]T }
+= {[zv]T v | z = x + yt for some t ∈ T }
+= {[xv + yv · tv]T v | t ∈ T }
+= [xv]T v + [yv]T v
+hence σ is an isomorphism of hyperfields by Lemma 2.34.
+□
+Example 3.15. By Proposition 2.17, T (Γ) ≃ KO×
+v , where K = k((tΓ)) for some field k with more
+than two elements and v is its canonical t-adic valuation. Since Kv = k and O×
+v v = k×, by the
+above proposition and Proposition 2.17 (i) we have that the residue field of T (Γ) with respect to
+its valuation given by the identity map is isomorphic to kk× ≃ K which, moreover, is a relational
+subhyperfield of T (Γ) (cf. Examples 2.22 and 2.36).
+For a valued field (K, v) we denote by 1 + Mv the set of 1-units. That is, those x ∈ K such that
+v(x − 1) > 0 or equivalently, xv = 1v.
+Proposition 3.16. Let (K, v) be a valued field and T ⊆ O×
+v a subgroup of K×. Then the map
+ι : KT vT → KT
+[xv]T v �→ [x]T
+is an embedding of hyperfields if and only if 1 + Mv ⊆ T .
+Proof. If 1 + Mv ⊆ T and x, y ∈ O×
+v , then [xv]T v = [yv]T v if and only if xv = yv · tv = (yt)v for
+some t ∈ T if and only if v(1 − ytx−1) = v(x − yt) > 0 if and only if ytx−1 ∈ 1 + Mv ⊆ T . It
+follows that [y]T [x]−1
+T
+= [ytx−1]T = [1]T and ι is well-defined. On the other hand if [x]T = [y]T for
+
+16
+LINZI, A.
+some x, y ∈ O×
+v , then for some t ∈ T we have that x = yt and thus xv = (yt)v = yv · tv. It follows
+that ι is injective. Moreover, we have that
+ι
+�
+[xv]T v · [yv]−1
+T v
+�
+= ι
+�
+[xv · y−1v]T v
+�
+= [xy−1]T = [x]T [y]−1
+T
+and hence ι satisfies (HH2) and (HH5). It clearly satisfies (HH1) and (HH4). It remains to show
+that (EM1) holds. First observe that Im ι consists of classes [x]T where x = 0 or x ∈ R ⊆ O×
+v ,
+where R is a set of some selected representatives for Kv. Take again x, y ∈ O×
+v . We have that
+ι ([xv]T v + [yv]T v) = {[x + yt]T | x, y ∈ R, t ∈ T ∩ R} = ([x]T + [y]T ) ∩ Im ι.
+This completes the proof of one implication.
+If a ∈ 1 + Mv is not in T , then av = 1v and thus [av]T v = [1v]T v holds. On the other hand, we
+have that
+ι[av] = [a]T ̸= [1]T = ι[1v]T
+and thus ι is not well-defined and in particular not an embedding of hyperfields.
+□
+3.4. Equivalence of valuations. In analogy with classical valuation theory, our aim is now to
+show that valuation hyperrings can be used to describe valuations up to composition with an order
+preserving isomorphism of the value group.
+Proposition 3.17 (Proposition 4.12 in [31]). Let F be a hyperfield and O a valuation hyperring
+in F. Consider the multiplicative group Γ := F ×/O× and define a relation ≤ on Γ as follows:
+aO× ≤ bO× ⇐⇒ ba−1 ∈ O.
+Then (Γ, ·, ≤) is an ordered abelian group and the canonical projection
+π : F → Γ ∪ {∞},
+extended so that π(0F ) = ∞, is a valuation on F. Furthermore, Oπ = O.
+Proof. First we show that ≤ is an ordering for (Γ, ·). Since aa−1 = 1F ∈ O, reflexivity is clear. If
+ab−1, ba−1 ∈ O, then ab−1 ∈ O× so aO× = bO×. Hence ≤ is antisymmetric. If ab−1, bc−1 ∈ O,
+then ac−1 = ab−1bc−1 ∈ O, showing that ≤ is transitive. Take now a, b ∈ F × such that aO× ≤ bO×
+and c ∈ F ×. We have that bc(ac)−1 = bcc−1a−1 = ba−1 ∈ O, whence acO× ≤ bcO×. This shows
+that ≤ is compatible with the operation of Γ. Finally, ≤ is a total order since O is a valuation
+hyperring, so that ab−1 ∈ O or ba−1 ∈ O for all a, b ∈ F ×.
+We now show that π is a valuation on F. Clearly, π is a surjective map, onto the ordered abelian
+group Γ with ∞ and (V1) holds. Since π is a homomorphism of groups we obtain (V2). It remains
+to show that (V3) holds for π. Take x, y ∈ F. If one of them is 0F , then (V3) is straightforward.
+We may then assume that x, y ∈ F × and that xO× ≤ yO×. Take z ∈ x + y. We wish to show that
+zx−1 ∈ O. By assumption we have that yx−1 ∈ O, thus
+zx−1 ∈ (x + y)x−1 = 1 + yx−1 ⊆ O,
+where we used Lemma 3.9.
+Finally, we observe that, by definition
+Oπ = {x ∈ F | πx ≥ 1} = {x ∈ F | xO× ≥ 1O×} = {x ∈ F | x ∈ O} = O.
+□
+Definition 3.18. Let (Γi, <i) be (partially) ordered sets for i = 1, 2. A map σ : Γ1 → Γ2 is said
+to be order preserving if γ1 ≤1 γ2 implies that σ(γ1) ≤2 σ(γ2) for all γ1, γ2 ∈ Γ1.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+17
+Definition 3.19. For i = 1, 2 let vi : F → Γi ∪ {∞} be valuations on a hyperfield F. We say
+that v1 and v2 are equivalent if there exists an isomorphism of groups σ : Γ1 → Γ2 which is order
+preserving and such that v2 = σ ◦ v1.
+Lemma 3.20. Let v : F → Γ ∪ {∞} be a valuation on a hyperfield F. Then Γ ≃ F ×/O×
+v with an
+isomorphism of groups which is order preserving.
+Proof. We consider F ×/O×
+v as an ordered abelian group with the ordering defined in Proposition
+3.17. Using the surjectivity of v, we define a map
+σ : Γ → F ×/O×
+v
+by σ(va) = aO×
+v for all a ∈ F ×. This is well-defined since if va = vb, then va − vb = v(ab−1) = 0
+so that ab−1 ∈ O×
+v and then aO×
+v = bO×
+v . Using property (V2) of valuations, we obtain that σ is
+a homomorphism of groups. Further, if va ≤ vb, then ba−1 ∈ Ov which means that σ(va) ≤ σ(vb).
+Thus, σ is order preserving. It is clear that σ is surjective. It therefore remains to show that σ is
+injective. To this end, assume that aO×
+v = bO×
+v for some a, b ∈ F ×. Then there exists c ∈ O×
+v such
+that a = bc. Since vc = 0, by (V2) we obtain that va = vb. This completes the proof.
+□
+Remark 3.21. Observe that by construction of σ in the above proof, we have that σ ◦ v = π where
+π is the canonical epimorphism F × → F ×/O×
+v .
+Corollary 3.22. For i = 1, 2 let vi : F → Γi ∪ {∞} be valuations on a hyperfield F. Then v1 and
+v2 are equivalent if and only if Ov1 = Ov2.
+Proof. By the previous lemma we obtain for i = 1, 2 that Γi ≃ F ×/O×
+vi as ordered abelian groups
+with isomorphisms σi such that σi ◦ vi = πi where πi : F × → F ×/O×
+vi is the canonical projection
+for i = 1, 2. Thus, if Ov1 = Ov2, then π1 = π2 and σ := σ−1
+2
+◦ σ1 is an isomorphism of ordered
+abelian groups Γ1 → Γ2. Further we have that
+σ ◦ v1 = σ−1
+2
+◦ (σ1 ◦ v1) = σ−1
+2
+◦ π2 = v2.
+Hence, v1 and v2 are equivalent.
+On the other hand, if v1 and v2 are equivalent, then we obtain that F ×/O×
+v1 ≃ F ×/O×
+v2 as ordered
+abelian groups. In particular, for a ∈ F × we have that 1O×
+v1 ≤ aO×
+v1 if and only if 1O×
+v2 ≤ aO×
+v2.
+Using the definition of the ordering in F ×/O×
+vi we see that this means that a ∈ Ov1 if and only if
+a ∈ Ov2. Since 0 ∈ Ovi for i = 1, 2, we conclude that Ov1 = Ov2 as claimed.
+□
+In the following sections we will always consider valuations on hyperfields up to equivalence.
+4. Krasner valued hyperfields
+In this section, we focus our attention on those valued hyperfields which satisfy the original more
+restrictive axioms of Krasner and were called by him hypercorps valué. These valued hyperfields
+still attract the attention of mathematicians. For instance, they have recently been considered by
+Tolliver and Lee (see [46, 32]) who called them simply “valued hyperfields”.
+4.1. Ultrametric spaces. We begin by presenting some basic theory of ultrametric spaces, a no-
+tion as well studied by Krasner (cf. [23]). The axioms for an ultrametric distance can be formulated
+using just the linear order of non-negative real numbers where 0 is a bottom element. Since nothing
+but the order is used from the structure of real numbers, we will use the term ultrametric space in a
+broader sense allowing ultrametric distances to take values in any linearly ordered set. In addition,
+since the value group of a valuation has a top element, our definition of ultrametric space below
+
+18
+LINZI, A.
+is a modification which better fits into our context (the same approach can be found e.g. in [28]).
+In the case of real-valued distances, this modification corresponds to a replacement of the linearly
+ordered set (R≥0, <) with (R ∪ {∞}, >).
+Definition 4.1. An ultrametric distance (or simply an ultrametric) on a set X is a function
+d : X × X → Γ ∪ {∞}, where (Γ, <) is a linearly ordered set and ∞ satisfies γ < ∞ for all γ ∈ Γ,
+such that for all x, y, z ∈ X
+(U1) d(x, y) = ∞ if and only if x = y,
+(U2) d(x, y) = d(y, x),
+(U3) d(x, z) ≥ min{d(x, y), d(y, z)}.
+We call (X, d) an ultrametric space whenever d is an ultrametric on X. We call the set dX :=
+{d(x, y) | x, y ∈ X, x ̸= y} ⊆ Γ the value set of d.
+Example 4.2. Let (K, v) be a valued field. Then the function
+K × K → Γ ∪ {∞}
+(x, y) �→ v(x − y)
+is an ultrametric on K.
+Definition 4.3. Let (X, d) be an ultrametric space. A subset B ⊆ X is called a ball if for all
+y, z ∈ B and all x ∈ X we have that the following implication
+d(x, y) ≥ d(y, z)
+=⇒
+x ∈ B
+holds for all x, y, z ∈ X.
+Definition 4.4. A subset ρ of a linearly ordered set (Γ, <) is called an initial segment (resp. final
+segment) if for all δ ∈ ρ and all γ ∈ Γ if γ < δ (resp. γ > δ), then γ ∈ ρ.
+Lemma 4.5. For every element x of an ultrametric space (X, d) and every final segment ρ of
+dX ∪ {∞}, we have that
+Bρ(x) := {y ∈ X | d(x, y) ∈ ρ}.
+is a ball in X. Conversely, if B is a ball in X and ρ is the smallest (with respect to inclusion) final
+segment of dX ∪ {∞} containing d(y, z) for all y, z ∈ B, then for every x ∈ B,
+B = Bρ(x).
+In particular, Bρ(x) = Bρ(y) for every y ∈ Bρ(x).
+Proof. Assume that y, z ∈ Bρ(x), that is, d(x, y) ∈ ρ and d(x, z) ∈ ρ.
+If t ∈ X is such that
+d(y, t) ≥ d(y, z), then the inequalities
+d(x, t) ≥ min{d(x, y), d(y, t)} ≥ min{d(x, y), d(y, z)} ≥ min{d(x, y), d(y, x), d(x, z)} ∈ ρ
+follow from (U2) and (U3). Since ρ is a final segment of dX ∪ {∞}, we conclude that d(x, t) ∈ ρ.
+Hence, t ∈ Bρ(x)and Bρ(x) is a ball.
+For the converse, assume that B is a ball and let ρ be as in the assertion. Further, let x be any
+element in B. If y ∈ B, then d(x, y) ∈ ρ and thus y ∈ Bρ(x). On the other hand, if y ∈ Bρ(x),
+then d(x, y) ∈ ρ. So by definition of ρ, there is some z ∈ B such that d(x, z) ≤ d(x, y). Since B is
+a ball, it follows that y ∈ B. We have proved that B = Bρ(x).
+□
+Corollary 4.6. Let (X, d) be an ultrametric space. Every two balls with non-empty intersection
+are comparable by inclusion.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+19
+Proof. Take two balls B and B′ and suppose that z ∈ B ∩ B′. By Lemma 4.5 there are final
+segments ρ, ς of dX ∪ {∞} such that B = Bρ(z) and B′ = Bς(z). Since ρ and ς are final segments,
+we must have ρ ⊆ ς or ς ⊆ ρ. Hence, B ⊆ B′ or B′ ⊆ B.
+□
+4.2. Krasner valuations. Let (Γ, <, +, 0) be an ordered abelian group, ρ be an initial segment of
+Γ and γ an element of Γ. We will sometimes write γ > ρ to indicate that γ /∈ ρ.
+The following class of valued hyperfields is of special interest.
+Definition 4.7 (Section 3 of [24], Definition 1.4 in [46] and Definition 2.4 in [32]). We call a valued
+hyperfield (F, v) a Krasner valued hyperfield if
+(KVH1) For all x, y ∈ F, v(x + y) is a singleton unless 0 ∈ x + y.
+(KVH2) There exists an initial segment ρv of vF such that 0 ∈ ρv and for all x, y, z, t ∈ F we have
+that z ∈ x + y implies that t ∈ x + y if and only if vs > ρv + min{vx, vy} for all s ∈ z − t.
+The initial segment ρv is called the norm of v. We will also say that v is a Krasner valuation on F
+when (F, v) is a Krasner valued hyperfield.
+Example 4.8. Let (K, v) be a valued field and consider K as a hyperfield. Then v is a Krasner
+valuation on K with norm vK.
+Proposition 4.9. Let (F, v) be a Krasner valued hyperfield. For all x, y ∈ F such that x ̸= y, by
+axiom (KVH1), v(x−y) contains a unique element γx,y ∈ vF. Define a map dv : F ×F → Γ∪{∞}
+as
+dv(x, y) :=
+�
+γx,y
+if x ̸= y,
+∞
+otherwise.
+Then dv is an ultrametric on F. Moreover, for all x, y ∈ F and any z ∈ x + y, in the ultrametric
+space (F, dv) we have that
+x + y = Bρv+min{vx,vy}(z).
+Proof. Axiom (U1) follows from axiom (V1), axiom (U2) follows from Corollary 3.5 (ii) and axiom
+(U3) is a direct consequence of axiom (V3). The last statement is just a reformulation of axiom
+(KVH2).
+□
+We call the ultrametric dv on a Krasner valued hyperfield (F, v) the ultrametric induced by v on
+F.
+Remark 4.10. Let F be a hyperfield and let v be the trivial valuation on F. If v is a Krasner
+valuation of norm ρv, then by (KVH2) for all x, y ∈ F × we have that x ∈ 1 − 1 if and only if
+0 = vx > ρv. Since 0 ∈ ρv, it follows that x − x = {0} for all x ∈ F and then F is a field by Lemma
+2.9.
+Conversely, if K is a field, then the map v(0) := ∞ and vx := 0 for all x ∈ K× is a Krasner
+valuation on K with value group vK = {0} and norm vK.
+Krasner’s main motivation probably came from the following example.
+Example 4.11. For a valued field (K, v) and an initial segment ρ ⊆ vK containing 0, let us
+consider the (multiplicative) group of 1-units of level ρ:
+1 + Mρ
+v = {x ∈ K | v(x − 1) > ρ} ⊆ O×
+v .
+Then vρ := v1+Mρ
+v is a Krasner valuation on Kρ := K1+Mρ
+v and the norm of vρ is ρ.
+
+20
+LINZI, A.
+Actually, any Krasner valued hyperfield which is a factor hyperfield is of this form, as we show
+below.
+Proposition 4.12. Let F be a factor hyperfield admitting a Krasner valuation w. Then F = Kρ
+and w is vρ for some valued field (K, v) and some initial segment ρ of vK = wF containing 0.
+Proof. By Lemma 3.6 there is a valued field (K, v) and a subgroup T ⊆ O×
+w of K× such that
+vT = w. Suppose that t ∈ T is not a 1-unit in K. Then vt = 0 and v(t − 1) = 0 must hold.
+now, on the one hand, since w = vT is a Krasner valuation, for all [x]T ∈ [1]T − [1]T we have that
+vx = vT [x]T > 0 by (KVH2). On the other hand, since t ∈ T we have that [t − 1]T ∈ [1]T − [1]T .
+This contradiction proves that T ⊆ 1 + Mv must hold and the result follows.
+□
+Remark 4.13. We do not know if all Krasner valued hyperfields are factor hyperfields. Some results
+connected to this problem are provided in [34].
+Example 4.14. Consider the Hahn series field K := F2((tΓ)) for some non-trivial ordered abelian
+group Γ and let v denote its canonical t-adic valuation. In this case, since Kv ≃ F2 we have that
+O×
+v = 1 + Mv. We conclude that
+T ′(Γ) ≃ Kρ,
+where ρ = {γ ∈ Γ | γ ≤ 0}. We have that the identity map vρ is the identity map on T (Γ) = T ′(Γ)
+and it is a Krasner valuation on T ′(Γ) with norm ρ. Note that, the same map is not a Krasner
+valuation on T (Γ) as 0 ∈ [0, ∞] = 0 ⊞ 0 violates (KVH2).
+We now study the residue hyperfield of a Krasner valued hyperfield.
+Proposition 4.15. The residue hyperfield Fv of a Krasner valued hyperfield (F, v) is a field.
+Proof. Take xv ∈ 1v − 1v. This means that x ∈ 1 − 1 and by axiom (KVH2), we deduce that
+vx > ρv. Since 0 ∈ ρv, it follows that xv = 0v and therefore Fv is a field by Lemma 2.9.
+□
+Example 4.16. Let (K, v) be a valued field. It follows from Proposition 3.14 that, for any initial
+segment ρ of vK containing 0, the residue hyperfield of Kρ is a field isomorphic to Kv.
+Let us now consider a valued hyperfield which is not a Krasner valued hyperfield.
+Example 4.17. Consider the rational function field K := Q(X) over the rationals, and its mul-
+tiplicative subgroup T := Q×. Under the canonical X-adic valuation v, we have that T ⊆ O×
+v .
+Hence, Lemma 3.3 yields a valued hyperfield (KT , vT ). By Proposition 3.14 and Proposition 2.17
+(i), we have that
+KT vT ≃ QQ× ≃ K.
+Since K is not a field, the Proposition 4.15 implies that (KT , vT ) is not a Krasner valued hyperfield.
+4.3. Mittas and superiorly canonical hypergroups. J. Mittas was a student of Krasner. He
+introduced superiorly canonical hypergroups (defined below) and published (sometimes without
+proofs) some results which connect them to Krasner valuations (see e.g. [41]). Some of the contents
+of this subsection were inspired by his work.
+Definition 4.18 ([41] and page 81 of [38]). A canonical hypergroup H is called superiorly canonical
+if
+(SCH1) For all x, y ∈ H, if x ∈ x + y, then x + y = {x}.
+(SCH2) For all x, y, z, t ∈ H, if (x + y) ∩ (z + t) ̸= ∅, then x + y ⊆ z + t or z + t ⊆ x + y.
+(SCH3) For all x, y ∈ H such that x ̸= y we have that z, t ∈ x − y implies z − z = t − t.
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+21
+(SCH4) For all x, y, z ∈ H, if x ∈ z − z and y /∈ z − z, then x − x ⊆ y − y.
+Example 4.19. Any abelian group is a superiorly canonical hypergroup.
+Other examples of superiorly canonical hypergroups are provided by the following result.
+Proposition 4.20. Let (F, v) be a Krasner valued hyperfield. Then the additive canonical hyper-
+group of F is superiorly canonical.
+Proof. We verify the axioms one by one. Assume that x ∈ x+ y for some x, y ∈ F. By reversibility,
+y ∈ x − x, so the inequalities vy > ρv + vx ≥ vx follow from axiom (KVH2) and 0 ∈ ρv. Therefore,
+if z ∈ x + y, then vx = vz by Corollary 3.5 (iv) and since by reversibility y ∈ z − x and dv(y, 0) =
+vy > ρv + min{vx, vz}, axiom (KVH2) implies that 0 ∈ z − x and so x = z must hold. This shows
+(SCH1).
+Axiom (SCH2) follows from Proposition 4.9 and Corollary 4.6.
+For (SCH3), take x, y ∈ F and assume that x ̸= y, i.e., 0 /∈ x − y. Take z, t ∈ x − y. We
+claim that vz = vt. Suppose that vz < vt, then by Corollary 3.5 (iv) we have that va = vz for all
+a ∈ z − t, thus
+dv(z, 0) = vz = dv(z, t) > ρv + min{vx, vy}.
+Axiom (KVH2) now implies that 0 ∈ x − y, a contradiction. Therefore, vz ≥ vt. Symmetrically,
+vt ≥ vz and so vz = vt as claimed.
+Now, by (KVH2) we have that a ∈ z − z if and only if
+va > ρv + vz and a ∈ t − t if and only if va > ρv + vt. Since vz = vt we conclude that z − z = t − t.
+This shows that (SCH3) holds.
+For (SCH4), take x ∈ z − z and y /∈ z − z. By axiom (KVH2) it follows that vx > ρv + vz and
+vy ∈ ρv +vz. In particular, vx > vy. Now, again by axiom (KVH2), if a ∈ x−x, then va > ρv +vx
+which implies va > ρv + vy and so a ∈ y − y. Hence, x − x ⊆ y − y. This completes the proof.
+□
+The theory of superiorly canonical hypergroups and the theory of Krasner valued hyperfields are
+even more deeply related.
+Proposition 4.21. Let F be a hyperfield with a superiorly canonical additive hypergroup. Then
+O := {x ∈ F | x − x ⊆ 1 − 1}
+is a valuation hyperring in F. Moreover, if v is the valuation such that Ov = O, then (F, v) is a
+Krasner valued hyperfield.
+Proof. If F is a field, then O = F and thus v is the trivial valuation on F. Since F is a field, (F, v)
+is a Krasner valued hyperfield. We can then assume that F is not a field.
+If x − x ⊆ 1 − 1 and y − y ⊆ 1 − 1, then
+xy − xy = (x − x)y ⊆ (1 − 1)y = y − y ⊆ 1 − 1.
+Hence, O is multiplicatively closed.
+If x − x ⊈ 1 − 1, then 1 − 1 ⊊ x − x by axiom (SCH2), since 0 ∈ (x − x) ∩ (1 − 1). Multiplying
+by x−1 we obtain that x−1 − x−1 ⊊ 1 − 1. Therefore, x−1 ∈ O.
+Assume that x − x ⊆ 1 − 1 and y − y ⊆ 1 − 1. We claim that if z ∈ x + y, then z − z ⊆ 1 − 1.
+Pick a ∈ z − z. Since z ∈ x + y we have that
+a ∈ z − z ⊆ (x − x) + (y − y) ⊆ (1 − 1) + (1 − 1),
+so there exists x′ ∈ 1 − 1 and y′ ∈ 1 − 1 such that a ∈ x′ + y′. Hence, 1 ∈ x′ + 1 by reversibility, and
+a ∈ x′ + y′ ⊆ (x′ + 1) − 1 = 1 − 1,
+
+22
+LINZI, A.
+where we have used axiom (SCH1). We proved that O is a valuation hyperring in F.
+Clearly, O× = {x ∈ F | x−x = 1−1}. Set vF := F ×/O× and let v : F × → vF be the canonical
+epimorphism. We have to show that (F, v) is a Krasner valued hyperfield. It is a valued hyperfield
+by Proposition 3.17. Let us now verify the validity of axioms (KVH1) and (KVH2).
+Take x, y ∈ F × and assume that 0 /∈ x + y. Pick z, t ∈ x + y and suppose that vz < vt. This
+means that tz−1 ∈ O \ O×, so
+tz−1 − tz−1 ⊊ 1 − 1.
+But then t − t ⊊ z − z which contradicts axiom (SCH3). We have proved that (KVH1) holds for
+(F, v).
+We now have to verify (KVH2). Our first claim is that v(1 − 1) is a final segment of vF which
+does not contain 0. Pick a ∈ 1 − 1 and γ > va. Let b ∈ F × be such that vb = γ. Since vb > va, we
+have that b − b ⊊ a − a. Now, if b /∈ 1 − 1, then axiom (SCH4) implies a − a ⊆ b − b. Therefore,
+b ∈ 1 − 1 must hold and thus γ = vb ∈ v(1 − 1) and v(1 − 1) is a final segment.
+For x ∈ F ×, if x ∈ x − x, then x − x = {x} by axiom (SCH1). However, since 0 ∈ x − x, this
+cannot be. An element x ∈ F × has value 0 if and only if x − x = 1 − 1. Hence it cannot belong to
+1 − 1 as x /∈ x − x for all x ∈ F ×. This shows that v(1 − 1) does not contain 0.
+We let ρv be the complement in vF of v(1 − 1). Then ρv is an initial segment of vF which
+contains 0.
+Observe that for all x ∈ F we have that a ∈ x − x = (1 − 1)x if and only if there exists y ∈ 1 − 1
+such that a = yx. This implies that va = vy + vx > ρv + vx. Conversely, if va > ρv + vx, then
+v(ax−1) ∈ v(1 − 1), so there exists b ∈ 1 − 1 such that vb = v(ax−1). Therefore,
+b ∈ 1 − 1 = bxa−1 − bxa−1 = b(xa−1 − xa−1),
+so 1 ∈ xa−1 − xa−1 implying that a ∈ x − x.
+Take now x, y ∈ F such that x ̸= y and vx ≤ vy. Fix z ∈ x − y. We have to show that t ∈ x − y
+if and only if dv(z, t) > ρv + vx.
+Assume first that t ∈ x − y. If t = z there is nothing to show. Otherwise, it suffices to show that
+z − t ⊆ x − x by what we have already shown above. Take a ∈ z − t. Since z ∈ x − y, we have that
+a ∈ z − t ⊆ x − (y + t).
+Hence, there exists b ∈ y+t such that a ∈ x−b. We obtain that b ∈ (x−a)∩(y+t), so y+t ⊆ x−a
+or x − a ⊆ y + t by axiom (SCH2). In the first case, since x ∈ y + t we have that x ∈ x − a and
+a ∈ x − x follows. It remains to deal with the case x − a ⊊ y + t. In this case, since t ∈ x − y and
+y − y ⊆ x − x, we obtain that
+x − a ⊊ y + t ⊆ y + x − y ⊆ x + (x − x).
+Take c ∈ x − a. There exists x′ ∈ x − x such that c ∈ x + x′. This implies that
+a ∈ x − c ⊆ x − (x + x′) = x − x
+where we have used axiom (CH4) and axiom (SCH1) to conclude that x + x′ = {x}.
+Now, assume that dv(z, t) > ρv + vx. We have to prove that t ∈ x − y. If z = t, then there is
+nothing to show. Hence, we can assume that z ̸= t and so z − t ⊊ x − x must hold. We have that
+t ∈ t + 0 ⊆ t + z − z ⊊ x − x + z.
+Hence, there is b ∈ z − x such that t ∈ x + b. We conclude that b ∈ (z − x) ∩ (t − x). By axiom
+(SCH2) we then obtain that z − x ⊆ t − x or t − x ⊆ z − x. In the first case, −y ∈ z − x ⊆ t − x
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+23
+and t ∈ x − y follows. It remains to deal with the case t − x ⊊ z − x. In this case, since z ∈ x − y,
+we have that
+t − x ⊊ z − x ⊆ x − y − x = x − x − y.
+Take a ∈ t − x. There exists x′ ∈ x − x such that a ∈ x′ − y. Now, by reversibility and since
+a ∈ t − x, we have that
+−y ∈ a − x′ ⊆ t − (x + x′) = t − x,
+where for the last equality we have used axiom (SCH1). Now, t ∈ x−y follows by axiom (CH4).
+□
+Directly from Proposition 4.20 and Proposition 4.21 above, we deduce the following characteri-
+zation theorem for the hyperfields which admit a Krasner valuation.
+Theorem 4.22. Let F be a hyperfield.
+Then F admits a Krasner valuation if and only if the
+additive hypergroup of F is superiorly canonical.
+Example 4.23. The additive hypergroup of the hyperfield that we have considered in Example
+4.17 above is not superiorly canonical. Indeed,
+1T + 1T = {0T, 1T }
+and thus (SCH1) fails. It follows from Theorem 4.22 that this hyperfield does not admit Krasner
+valuations at all.
+Example 4.24. Consider a generalised tropical hyperfield T (Γ), where Γ is some non-trivial ordered
+abelian group (see Example 2.14). By Proposition 2.17 (iv) and Lemma 3.3 we conclude that T (Γ)
+is a valued hyperfield (see also the discussion after Proposition 5.9). Nevertheless, by Theorem 4.22,
+T (Γ) does not admit Krasner valuations as its additive hypergroup does not satisfy (SCH1).
+Let us now analyse another example.
+Example 4.25. Take K = Q(X) with the valuation v := vp ◦ vX where vX denotes the X-adic
+valuation on Q(X) and vp denotes the p-adic valuation on Q, for some prime number p. More
+explicitly, for a polynomial f(X) = �n
+i=0 aiXi in Q[X], we have
+vf(X) =
+�
+vXf(X), vp(avXf(X))
+�
+∈ Z × Z.
+The order relation in vK is the lexicographic order of Z × Z, that is, (n1, n2) < (m1, m2) if and
+only if n1 < m1 ∨ (n1 = m1 ∧ n2 < m2).
+Consider the Krasner valued hyperfield F := Kρ where ρ is the smallest initial segment of Z × Z
+which contains {0} × Z (cf. Example 4.11). Let us denote its Krasner valuation vρ by w.
+Note that, if (m1, m2) ∈ ρ and m1 > 0, then, since ρ is an initial segment, we have that
+{0} × Z ⊆ {(k1, k2) ∈ Z × Z | (k1, k2) ≤ (m1 − 1, m2)} ⊊ ρ,
+in contradiction with the minimality of ρ. Therefore, m1 ≤ 0 for all (m1, m2) ∈ ρ. Conversely, if
+m1 ≤ 0, then (m1, m2) ≤ (0, m2) for all m2 ∈ Z and therefore (m1, m2) ∈ ρ since ρ is an initial
+segment containing {0} × Z. It follows that
+(1)
+ρ = {(m1, m2) ∈ Z × Z | m1 ≤ 0} = {(m1, m2 + n) ∈ Z × Z | m1 ≤ 0} = ρ + (0, n)
+for any (0, n) ∈ {0} × Z.
+By Proposition 4.20, the additive hypergroup of F is superiorly canonical. Therefore, by Propo-
+sition 4.21, we have the valuation hyperring
+Ou = {x ∈ F | x − x ⊆ 1 − 1}
+
+24
+LINZI, A.
+for some Krasner valuation u on F. Let us now show that w and u are not equivalent valuations
+on F.
+In fact, we will show that Ow ⊊ Ou. Pick x ∈ Ow, i.e., wx ≥ (0, 0). By axiom (KVH2) we have
+that y ∈ x − x if and only if wy > ρ + wx ≥ ρ. Another application of axiom (KVH2) shows that
+x − x ⊆ 1 − 1 so that x ∈ Ou. Now, consider e.g. the element x of F corresponding to the rational
+number p−1 in K. Since vp(p−1) = −1 we have that wx = (0, −1) < (0, 0) so that x /∈ Ow. On
+the other hand, since wx = (0, −1) ∈ {0} × Z, we have that ρ + wx = ρ by (1) and thus using
+(KVH2), we obtain that y ∈ x − x if and only if wy > ρ if and only if y ∈ 1 − 1. This implies that
+x − x ⊆ 1 − 1 and thus x ∈ Ou.
+The above example constitutes the starting point for the discussion that we will have in Section
+6.
+5. More on ordered abelian groups
+In this section, we characterise generalised tropical hyperfields and discuss the concept of coars-
+ening of a valuation in the multivalued setting. We derive some conclusions on the relation between
+ordered abelian groups and generalised tropical hyperfields.
+5.1. Characterisation of generalised tropical hyperfields. To characterise generalised tropi-
+cal hyperfields, we will apply another remarkable characterisation result. A hyperfield is stringent
+if x + y is a singleton whenever 0 /∈ x + y. Bowler and Su in [6] characterised stringent hyperfields
+and we will now briefly recall their result.
+In [6, Section 4] a natural construction of a hyperfield arising from a short exact sequence
+(2)
+{1}
+F ×
+H×
+Γ
+{0}
+ϕ
+ψ
+where Γ is an ordered abelian group and F is any hyperfield is described.
+The thus obtained
+hyperfield has H× as multiplicative group and is called the Γ-layering of F along the short exact
+sequence (2). After that the following theorem is proved.
+Theorem 5.1 (Theorem 4.10 in [6]). A hyperfield H is stringent if and only if it is the Γ-layering
+of F along the short exact sequence (2) with F being (isomorphic to) either K, S or a field.
+From the details of the construction (which we omit for brevity) it is not difficult to verify that
+the extension of the map ϕ sending 0F to 0H is in any case an embedding of hyperfields.
+We say that a hyperfield has characteristic 2 if 0 ∈ 1 + 1 and that it has C-characteristic 1 if
+1 ∈ 1 + 1. For example, S satisfies the latter but not the former while K satisfies both. For more
+information on characteristic and C-characteristic of hyperfields the reader can see [21].
+We are now ready to state and prove our characterisation theorem.
+Theorem 5.2 (Characterisation of generalised tropical hyperfields). Generalised tropical hyper-
+fields are precisely stringent hyperfields of characteristic 2 and C-characteristic 1.
+Proof. The reader can easily verify from the relevant definitions that a generalised tropical hyperfield
+T (Γ) is a stringent hyperfield of characteristic 2 and C-characteristic 1.
+For the converse, let H be a stringent hyperfield of characteristic 2 and C-characteristic 1. By
+Theorem 5.1 and our observations on the extension of the map ϕ above, we have that either K, S or
+a field embed into H. Since H has characteristic 2 we have that 1 = −1, thus S cannot embed into
+H. On the other hand, any field cannot embed into H neither. Indeed, our assumption 0, 1 ∈ 1 + 1
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+25
+implies that {0, 1} = (1 + 1) ∩ {0, 1} ⊆ (1 + 1) ∩ F = 1 +F 1 is not a singleton. It follows that H is
+the Γ-layering of K along a short exact sequence
+{1}
+K×
+H×
+Γ
+{0}
+ϕ
+ψ
+for some ordered abelian group Γ. Now, since K× = {1}, we obtain that the multiplicative group
+of H is isomorphic to Γ and the details of the construction given in [6, Section 4] show that the
+hyperoperation of H is the one of T (Γ). We conclude that H is a generalised tropical hyperfield.
+□
+An analogous reasoning yields to the following (but maybe less interesting) characterisation of
+strict generalised tropical hyperfields.
+Theorem 5.3. Generalised tropical hyperfields are precisely the Γ-layerings of F along the short
+exact sequence (2), with F = F2.
+5.2. Coarsenings.
+Definition 5.4. Let (Γ, <, +, 0) be an ordered abelian group. A subgroup ∆ of Γ is convex if for
+all γ ∈ Γ we have that if there exist δ1, δ2 ∈ ∆ such that δ1 < γ < δ2, then γ ∈ ∆.
+It is not difficult to see that the intersection of a family of convex subgroups of an ordered
+abelian group is again a convex subgroup and that the collection of all convex subgroups of an
+ordered abelian group is linearly ordered by inclusion. Let us recall another basic fact which makes
+convex subgroups important.
+Fact 5.5. Let (Γ, <, +, 0) be an ordered abelian group and ∆ a convex subgroup of Γ. Then (Γ/∆, ≺
+, +, 0) is an ordered abelian group, where
+x + ∆ ≺ y + ∆ ⇐⇒ x < y and y − x /∈ ∆.
+In particular, the canonical epimorphism Γ → Γ/∆ is order preserving.
+The following notion will play a fundamental role later.
+Definition 5.6. Let (Γ, <, +, 0) be an ordered abelian group and ρ an initial segment of Γ. We
+call the set
+ig(ρ) := {γ ∈ Γ | ρ + γ = ρ}
+the invariance group of ρ. If ig(ρ) = {0}, then we say that ρ has trivial invariance group.
+Remark 5.7. In [29, 30] F.-V. Kuhlmann proves several results about invariance groups associated
+to initial segments in ordered abelian groups. In particular, it follows from [30, Lemma 2.3] that
+the invariance group of an initial segment ρ of an ordered abelian group Γ such that 0 ∈ ρ is a
+convex subgroup of Γ contained in ρ.
+Definition 5.8. Let v, w be two valuations on a hyperfield F. We say that w is a coarsening of v
+if Ov ⊆ Ow.
+In classical valuation theory for fields, it is well-known that to any convex subgroup of the value
+group one can associate a coarsening. We note that this result generalises to valued hyperfields,
+actually with a much more conceptual proof.
+Proposition 5.9. Let (F, v) be a valued hyperfield and let ∆ be a convex subgroup of vF. Then
+v∆ : F → (vF/∆) ∪ {∞}
+x �→ vx + ∆
+is a valuation on F which is a coarsening of v.
+
+26
+LINZI, A.
+Proof. By Lemma 3.4, v : F → T (vF) is a surjective homomorphism of hyperfields. In addition,
+the canonical epimorphism vF → vF/∆ extends to a surjective map T (vF) → T (vF/∆). From the
+fact that the latter is order preserving, it easily follows that it is a homomorphism of hyperfields.
+We conclude that the composition v∆ : F → T (vF/∆) of v with the latter map is a surjective
+homomorphism of hyperfields. This proofs the first part of the statement. Now, since
+Ov∆ = {x ∈ F | v∆x ⪰ 0vF/∆}
+= {x ∈ F | vx + ∆ ⪰ ∆}
+= {x ∈ F | vx ≥ 0 or vx ∈ ∆}
+⊇ {x ∈ F | vx ≥ 0} = Ov,
+we conclude that v∆ is a coarsening of v.
+□
+In the above proof, we have seen that if ∆ is a convex subgroup of an ordered abelian group Γ,
+then the natural map
+π∆ : T (Γ) → T (Γ/∆)
+is a surjective homomorphism of hyperfields, which, by Lemma 3.4, is a valuation on T (Γ) with
+value group Γ/∆. The valuation hyperring of this valuation is
+O∆ = {γ ∈ Γ | γ + ∆ ⪰ 0 + ∆} = {γ ∈ Γ | γ ∈ ∆ or γ > ∆}
+and its unique maximal hypeideal is
+M∆ = {γ ∈ Γ | γ > ∆}
+It follows that the residue hyperfield T (Γ)π∆ = O∆/M∆ of T (Γ) with respect to the valuation π∆
+is isomorphic to T (∆). An isomorphism is given by the map
+T (∆) → T (Γ)π∆
+∞ �→ ∞
+δ �→ δπ∆
+In the next result we show that the valuations of T (Γ) are (up to equivalence) all of this form.
+Proposition 5.10. Let Γ be an ordered abelian group and assume that v : T (Γ) → T (Γ′) is a
+valuation. Then there exists a convex subgroup ∆ of Γ such that Γ/∆ ≃ Γ′ via an order preserving
+isomorphism.
+Proof. Let 0′ ∈ Γ′ denote the neutral element of the group Γ′, ⊞ the hyperoperation of T (Γ) and ⊞′
+the hyperoperation of T (Γ′). Set ∆ := v−1(0′). Since v is a homomorphism of groups Γ → Γ′, we
+have that ∆ is a subgroup of Γ. Moreover, since v is surjective, we have that Γ/∆ ≃ Γ′ as groups
+by the first homomorphism theorem for groups. Assume that δ1, δ2 ∈ ∆ and that γ ∈ Γ is such
+that δ1 ≤ γ ≤ δ2. Hence, δ2 ∈ γ ⊞ γ and so 0′ = v(δ2) ∈ v(γ) ⊞′ v(γ) since v is a homomorphism
+of hyperfields. On the other hand, γ ∈ δ1 ⊞ δ1 so that v(γ) ∈ 0′ ⊞′ 0′ = [0′, ∞]. If v(γ) > 0′, then
+0′ /∈ [v(γ), ∞] = v(γ) ⊞′ v(γ), a contradiction. We conclude that v(γ) = 0′ must hold and thus
+γ ∈ ∆. We have shown that ∆ is a convex subgroup of Γ. It remains to show that the isomorphism
+of groups σ : γ + ∆ �→ v(γ) is order preserving. Assume that γ1 + ∆ ≺ γ2 + ∆. This means that
+γ1 < γ2 and γ2 −γ1 /∈ ∆ hold in Γ. Thus, {γ1} = γ1 ⊞γ2 and then v(γ1) ∈ v(γ1)⊞′ v(γ2). Since σ is
+bijective and γ1 ̸= γ2, we have that v(γ1) ̸= v(γ2), so v(γ1) ∈ v(γ1) ⊞′ v(γ2) =
+�
+min{v(γ1), v(γ2)}
+�
+.
+This shows that v(γ1) < v(γ2) must hold in Γ′ and thus σ is order preserving and the proof is
+complete.
+□
+
+NOTES ON VALUATION THEORY FOR KRASNER HYPERFIELDS
+27
+The next result now follows from Corollary 3.22.
+Corollary 5.11. Let Γ be an ordered abelian group. There is a bijective correspondence between
+convex subgroups of Γ and valuation hyperrings of T (Γ).
+6. Krasner valuations induced by the additive structure
+In this section we apply some of the results obtained in the previous section to Krasner valued
+hyperfields. We begin by observing that coarsenings of Krasner valuations might not be Krasner
+valuations.
+Example 6.1. Let Γ be a non-trivial ordered abelian group with a non-trivial convex subgroup ∆
+and consider the identity map as a Krasner valuation v : T ′(Γ) → T (Γ). As a map, the coarsening
+v∆ : T ′(Γ) → T (Γ/∆)
+is just the canonical epimorphism Γ → Γ/∆ extended to send ∞ to ∞. Suppose that v∆ is a
+Krasner valuation of norm ρ for some initial segment ρ of Γ/∆ containing 0Γ/∆. Then by (KVH2)
+for all γ ∈ Γ we would have that
+γ > 0 ⇐⇒ γ ∈ 0 ⊞′ 0 ⇐⇒ γ + ∆ > ρ + (0 + ∆).
+On the other hand, since 0Γ/∆ ∈ ρ , any positive γ ∈ ∆ would violate this equivalence.
+Nevertheless, we have the following result.
+Theorem 6.2. Let (F, v) be a Krasner valued hyperfield and let w be the Krasner valuation on F
+determined by
+Ow = {x ∈ F | x − x ⊆ 1 − 1}.
+Then w is equivalent to the coarsening of v corresponding to ig(ρv). In particular, the coarsening
+of a Krasner valuation v corresponding to ig(ρv) is a Krasner valuation.
+Proof. We show that these two valuations have the same valuation hyperring. Indeed, for all x ∈ F
+we have that vx + ig(ρv) ⪰ 0vF/ ig(ρv) if and only if vx ∈ ig(ρv) or vx > ig(ρv). In both cases by
+(KVH2) applied to v we have that
+y ∈ x − x ⇐⇒ vy > ρv + vx ⊇ ρv =⇒ y ∈ 1 − 1,
+i.e., y ∈ Ow. Conversely, vx + ig(ρv) ≺ 0vF/ ig(ρv) if and only if vx /∈ ig(ρv) and vx < 0. Therefore,
+ρv + vx ⊊ ρv and there exists y ∈ F such that vy ∈ ρv and vy /∈ ρv + vx. By (KVH2) applied to v,
+the former implies y /∈ 1 − 1 while the latter is equivalent to y ∈ x − x. We conclude that x /∈ Ow.
+We have proved that vig(ρv) and w have the same valuation hyperring in F. The theorem follows
+from Corollary 3.22.
+□
+Corollary 6.3. Let F be a hyperfield admitting two Krasner valuations v1 and v2 of norm ρ1 and
+ρ2, respectively. Then the coarsening of v1 corresponding to ig(ρ1) is equivalent to the coarsening
+of v2 corresponding to ig(ρ2).
+Corollary 6.4. Let (F, v) be a Krasner valued hyperfield. If ρv has trivial invariance group, then
+Ov = {x ∈ F | x − x ⊆ 1 − 1}.
+Corollary 6.5. Let F be a hyperfield with a superiorly canonical additive hypergroup. Then the
+norm of the Krasner valuation v determined on F by the valuation hyperring
+Ov = {x ∈ F | x − x ⊆ 1 − 1}
+has trivial invariance group.
+
+28
+LINZI, A.
+Corollary 6.6. Let F be a hyperfield with a superiorly canonical additive hypergroup. There is a
+unique (up to equivalence) Krasner valuation v on F such that ig(ρv) = {0}.
+Remark 6.7. In the model theory of valued fields, the RV-structure (mentioned in the introduction)
+of level γ of a valued field (K, v), where γ is a non-negative element of vK, is essentially the Krasner
+valued hyperfield3 Kγ := Kργ, where ργ := {δ ∈ vK | δ ≤ γ}. Since ig(ργ) = {0}, it follows, as a
+consequence of Corollary 6.4, that the valuation hyperring of the Krasner valuation on Kγ induced
+by v is always definable in the language of hyperfields, e.g., using the ternary relation z ∈ x + y to
+encode the multivalued operation +.
+7. Further research
+There are at least three interesting points that have been touched but not developed further in
+this manuscript. Below we briefly describe them.
+(1) In the multivalued setting, (F, v), vF and Fv can all be described as (valued) hyperfields
+and they are not distinct structures. This applies in particular when F is a field.
+(2) In the literature there are not many examples of infinite hyperfields that are not factor
+hyperfields. Essentially, only the work of Massouros [39, 40] provides such examples. In
+all those examples M we have by definition x + x = M \ {x} for all x ∈ M ×. If v is a
+valuation on M and x ̸= 1, then x, x−1 ∈ 1 + 1, so by (V3) and Corollary 3.5 (iii) we have
+that vx = 0 must hold for all x ∈ M ×. It follows that M only admits the trivial valuation.
+Is it true that any hyperfield admitting a non-trivial valuation is a factor hyperfield?
+(3) In view of Theorem 5.2, a non-trivial valuation on a hyperfield F is a surjective homomor-
+phism onto a hyperfield H satisfying the following three conditions:
+• H has characteristic 2;
+• H has C-characteristic 1;
+• H is stringent.
+From this point of view, the image of a valuation is a quite special hyperfield. For example,
+one could think that responsible for the fact that finite (hyper)fields only admit the trivial
+valuation is the third property of H, since there are many examples of finite hyperfields
+satisfying 0, 1 ∈ 1+1. We speculate that investigating weakenings of the stringency property
+for H may lead to a notion of generalised valuation which have the potential to be applied
+also in the classical singlevalued setting e.g. to use (generalised) valuation-theoretic methods
+over finite fields.
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+Email address: [email protected]
+

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+Fault-tolerant error correction for a universal non-Abelian topological quantum
+computer at finite temperature
+Alexis Schotte,1, ∗ Lander Burgelman,2 and Guanyu Zhu1, 3, †
+1IBM Quantum, IBM Almaden Research Center, San Jose, CA 95120, USA
+2Department of Physics and Astronomy, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
+3IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA
+We study fault-tolerant error correction in a quantum memory constructed as a two-dimensional model of Fi-
+bonacci anyons on a torus, in the presence of thermal noise represented by pair-creation processes and measure-
+ment errors. The correction procedure is based on the cellular automaton decoders originating in the works of
+G´acs [1] and Harrington [2]. Through numerical simulations, we observe that this code behaves fault-tolerantly
+and that threshold behavior is likely present. Hence, we provide strong evidence for the existence of a fault-
+tolerant universal non-Abelian topological quantum computer.
+I.
+INTRODUCTION
+Anyons are emergent quasi-particles that exist in
+two-dimensional condensed matter systems and whose
+exchange statistics generalize that of Bosons and
+Fermions. These particles have spurred much interest
+due to their potential applications for quantum compu-
+tation. In particular, it was found that with certain types
+of non-Abelian anyons, a universal quantum compu-
+tation can be performed by braiding and fusing these
+particles [3–5]. An intriguing benefit of this paradigm
+is that, due to their topological nature, computations
+are intrinsically robust to perturbations at zero tem-
+perature. At non-zero temperature, however, thermal
+anyonic excitations can corrupt the computation by
+performing non-trivial braids with the computational
+anyons. Since systems exhibiting anyonic excitations
+have a spectral gap ∆, this source of errors can be sup-
+pressed to some extent at temperatures T ≪ ∆/kB
+as the density of thermal anyons scales as e−∆/kBT .
+Alas, this passive protection does not suffice, because
+the presence of thermal anyons is unavoidable at non-
+zero temperatures when scaling up the size of com-
+putations. Therefore, proficient active error correction
+schemes for non-Abelian models are paramount for the
+realization of topological quantum computers.
+Besides their envisaged use for topological quantum
+computation, topologically ordered systems (i.e., those
+that support anyonic excitations on top of their ground
+space) are also of much interest for quantum error cor-
+rection. In particular, one of the characteristics of such
+systems is a robust ground space degeneracy, which al-
+∗ [email protected]
+† [email protected]
+lows one to use their ground space as the code space of
+an error correcting code. This realization led to the dis-
+covery of topological quantum error correcting codes,
+which encode logical quantum states in topologically
+ordered states of a system of qudits (typically arranged
+on a two-dimensional lattice). Since their discovery
+in the 90s, most research has focused exclusively on
+Abelian topological codes such as the surface code and
+the color code [5–14], which admit an elegant charac-
+terization in terms of the stabilizer formalism [15]. Due
+to their geometrical locality and high error thresholds,
+these codes are considered to be promising candidates
+for protecting quantum information from noise in error-
+corrected quantum computers. One of the drawbacks
+of Abelian topological codes, however, is that they do
+not allow one to execute a universal set of logical gates
+in a protected fashion in two dimensions. Hence, they
+must be supplemented with additional protocols such
+as magic state distillation [16] or code switching to
+higher-dimensional codes [17, 18] , which introduce a
+large space-time overhead [19]. Alternatively, there ex-
+ist non-Abelian topological codes which do not suffer
+from this inherent limitation, and are able to perform
+a universal gate set natively within their code space in
+two dimensions [3]. The trade-off is that such codes go
+beyond the stabilizer formalism and are therefore very
+hard to simulate classically.
+While active error correction in Abelian anyon mod-
+els and Abelian topological codes has been studied
+extensively, quantum error correction based on non-
+Abelian anyon models has not enjoyed the same fo-
+cus. Nevertheless, important progress has been made
+over the last decade, including both analytical proofs
+and numerical demonstrations of threshold behavior
+for various non-Abelian topological error correcting
+codes [20–24]. Moreover, syndrome extraction circuits
+for such non-Abelian string-net codes have been devel-
+arXiv:2301.00054v1  [quant-ph]  30 Dec 2022
+
+2
+oped in recent years [24, 25]. In addition, state prepa-
+ration for non-Abelian codes based on the Kitaev quan-
+tum double models via measurements has also been
+proposed recently for the experimental implementa-
+tion on qubit lattices [26, 27], although further devel-
+opment is still needed in the context of fault-tolerant
+state preparation. Notably, previous studies in this field
+already include codes based on the Fibonacci anyon
+model, which is universal for quantum computation
+[23, 24]. In particular, a quantum memory of qubits
+supporting doubled Fibonacci anyonic excitations was
+found to have a threshold that lies remarkably close to
+that of the surface code under similar assumptions [24].
+These results, however, all assume perfect syndrome
+measurements, which are topological charge measure-
+ments in this context. As we aim to model more re-
+alistic scenarios, we must take faulty measurements
+into consideration. Again, much is known in the case
+of Abelian topological codes [2, 7, 28–31]. For their
+non-Abelian counterparts, one key result stands out: in
+Ref. [32] a proof was formulated that topological codes
+based on non-cyclic anyon models admit a error cor-
+rection thresholds with faulty topological charge mea-
+surements. While this result is remarkable, non-cyclic
+anyon models are not universal for quantum computa-
+tion, and it remains an open question whether similar
+claims can be made for universal models.
+In this work, we take a step towards demonstrat-
+ing that fault-tolerance is indeed possible for universal
+non-Abelian topological codes. To this end, we define
+a quantum memory constructed as a two-dimensional
+model of Fibonacci anyons on a torus. We study ac-
+tive continuous quantum error correction on this model
+in the presence of thermal noise represented by pair-
+creation processes, and with faulty syndrome measure-
+ments. The correction procedure is based on the cel-
+lular automaton decoders originating in the works of
+G´acs [1] and Harrington [2], and further studied in the
+context of non-Abelian models in Ref. [32]. Through
+numerical simulations, we study how the average mem-
+ory lifetime changes with the error rate. The results in-
+dicate that this code is indeed fault-tolerant, which is
+strong evidence for the existence of fault-tolerant uni-
+versal non-Abelian codes.
+The structure of this work is as follows. In Sec. II
+we introduce the topological Fibonacci code. We then
+describe the details of the noise model in Sec. III and
+introduce the cellular automaton decoder in Sec. IV.
+We proceed by giving an outline of the numerical sim-
+ulations performed in this work in Sec. V. Finally, we
+present the corresponding numerical results in Sec. VI
+and conclude with a discussion in Sec. VII.
+II.
+THE FIBONACCI CODE
+We consider a two-dimensional model comprised of
+hexagonal tiles laid out on the surface of a torus. The
+resulting geometry can be represented as an L × L
+hexagonal lattice with periodic boundary conditions in
+both directions (Fig. 2). Each of these hexagonal tiles
+can contain an excitation known as a Fibonacci anyon.
+Anyons are point-like quasi-particle excitations
+which can be characterized algebraically in terms of a
+unitary modular tensor category (UMTC). A thorough
+description of anyon models using UMTCs goes be-
+yond the scope of this work, however, some details are
+given in Sec. A. For now, it is sufficient to state that
+an anyon model specifies a set of anyon labels, also
+referred to as particle types, which can fuse according
+to a specific set of fusion rules. The Fibonacci anyon
+model considered in this work contains two labels, 1
+and τ, which obey the fusion rules
+1×1 = 1 ,
+1×τ = τ×1 = τ ,
+τ×τ = 1+τ . (1)
+In general, one can associate a vector space to a
+given set of anyons, where the basis vectors are la-
+beled by the different ways in which the anyons can
+fuse. This fusion space has a topological degeneracy,
+and can therefore be used to robustly encode quan-
+tum information. In particular, for the Fibonacci anyon
+model the anyonic vacuum on a two-dimensional torus
+has a twofold degeneracy [33]. Starting from our two-
+dimensional model, we can therefore define an error
+correcting code whose code space is identified with the
+anyonic vacuum on the torus and which encodes a sin-
+gle logical qubit. A basis for this code space can be
+defined using Wilson line operators along the homo-
+logically non-trivial cycles x and y shown in Fig. 1:
+W a
+x |1⟩x = Sa1
+S11 |1⟩x ,
+W a
+x |τ⟩x = Saτ
+S1τ |τ⟩x ,
+a ∈ {1, τ} ,
+(2)
+We note that a different basis, {|0⟩y , |1⟩y}, can be
+defined analogously by swapping the x and y labels
+above, where the two bases are related through the
+modular S matrix Sab =
+y⟨a|b⟩x. For the Fibonacci
+anyon model its numerical values are
+S =
+1
+�
+1 + φ2
+�
+1
+φ
+φ −1
+�
+,
+(3)
+where φ = 1 +
+√
+5
+2
+.
+The action of the mapping class group on the any-
+onic vacuum then corresponds to unitary operations on
+
+3
+x
+y
+Figure 1: The two homologically non-trivial cycles on
+a torus.
+the code space. For the Fibonacci category, any logical
+unitary operator can be realized in this way, up to arbi-
+trary precision [3]. Therefore, the quantum error cor-
+recting code defined above natively supports universal
+quantum computation.
+Errors in this code appear as spurious anyonic ex-
+citations, which can corrupt the encoded informa-
+tion if their world lines between creation and re-
+annihilation are topologically non-trivial, i.e., form a
+non-contractible cycle 1. The objective of error cor-
+rection is then to systematically remove these spurious
+excitations, without corrupting the quantum memory in
+the process. This correction is performed in an active
+and continuous manner, and can be broken down into a
+series of discrete steps. At each step, a suitable recov-
+ery operation is performed based on a measured list of
+positions and types of the excitations, called the error
+syndrome.
+We conclude this section by noting that the numer-
+ical simulation of the error-correction process requires
+the introduction of some additional manipulations on
+fusion states of multiple Fibonacci anyons. As these
+are technical details that do not contribute to the in-
+tuition of the procedure, we defer their definition to
+Sec. A.
+III.
+NOISE MODEL AND CORRECTABILITY
+Having defined our model and code space, we now
+turn to the description of the noise model used in our
+1 Note that a pair of Fibonacci anyons can also fuse to a single non-
+trivial anyon when one member of the pair has been transported
+along a non-trivial cycle. Since the resulting state is no longer in
+the code space, this does not constitute a logical operation on the
+encoded information. However, one can show that the encoded
+information is irrevocably lost in case of such an event [21].
+simulations.
+We model continuous active error cor-
+rection in our Fibonacci code as a sequence of time
+steps, where each time step itself consists of three parts:
+the application of pair-creation noise, faulty syndrome
+measurement, and error correction respectively.
+At each time step, first, for each edge of the hexago-
+nal lattice graph a pair of anyons is created across this
+edge with a probability p. Immediately after each pair
+creation event, the resulting charge in the two affected
+tiles is sampled, effectively collapsing all superposi-
+tions of anyonic charge within each tile to either 1 or
+τ. After the pair creation noise has been applied, faulty
+syndrome extraction is simulated by generating a list of
+the anyon charge in all tiles, and flipping each outcome
+individually with a probability q. In addition to the
+charges that are correctly detected, the resulting faulty
+syndrome can contain both “ghost defects” (indicating
+a non-trivial charge when none is truly present) and
+“missing defects” (failing to report a true non-trivial
+charge). Finally, this faulty syndrome is passed to a de-
+coder, introduced in the following section, which then
+performs a set of local operations based on the current
+(and past) syndrome information in an attempt to move
+the system back towards the initial state.
+After each time step, the current state of the sys-
+tem is copied and it is checked whether it is still cor-
+rectable. This is done by passing the copy to a clus-
+tering decoder [22–24] and simulating a decoding pro-
+cedure with perfect syndrome measurements starting
+from this given initial state. If this perfect decoding
+is successful, the memory is considered intact and the
+simulation is continued. If perfect decoding is unsuc-
+cessful, the memory is considered corrupted and the
+simulation is aborted. The memory lifetime is then de-
+fined as the number of time steps after which a perfect
+clustering decoder can no longer successfully restore
+the initial state.
+For a given pair of tiles which share an edge, the
+process of pair creation across this edge corresponds to
+the matrix elements
+⟨a′, b′; c′| Upc |a, b; c⟩ = δc,c′F aa1
+ττa′F a′τa
+b c b′ ,
+(4)
+where we have used the F-symbols of the Fibonacci
+category, given in (A5). Here, |a, b; c⟩ represents the
+state where the affected tiles have anyon charges a
+and b, respectively, with total charge c. This then de-
+fines the probability distribution according to which
+outcomes a′ and b′ are sampled.
+Since our noise
+model does not allow any superposition in the any-
+onic charge of individual tiles, it should be considered
+semi-classical rather than fully quantum-mechanical.
+
+4
+(a)
+(b)
+Figure 2: (a) Pair-creation events creating anyonic
+excitations in neighboring tiles. The dotted ellipse
+represents a collapse to the total charge of the anyons
+it contains. A ghost defect is shown in blue, a missing
+defect is highlighted in orange with a cross. (b) The
+outcome of this noise process represented on the
+decoding graph. Note that the missing defect
+(highlighted in orange) will (by definition) not be
+visible in the syndrome.
+Note, however, that this does not render our model
+completely classical. Indeed, superpositions in the total
+charge c of the affected tiles are an inherent part of the
+state evolution that cannot be captured faithfully by any
+classical probabilistic process. Furthermore, while the
+extreme decoherence assumption for the anyon charge
+in individual tiles greatly simplifies the numerical sim-
+ulation outlined in this work, it was argued in Ref. [22]
+that this decoherence is unlikely to have any tangible
+influence on the observed memory lifetimes, as the es-
+sential topological nature of the noise processes is still
+captured correctly.
+We note that this type of noise can be understood as
+originating from the connection to a thermal bath with
+inverse temperature β = 1/(kBT) determined by the
+error rate p through the relation
+p
+1 − p = e−β∆ .
+(5)
+Here, ∆ represents the energy required to create a pair
+of anyonic excitations and place them in neighboring
+tiles.
+To conclude this section, we emphasize that in the
+case of non-Abelian error correction, even decoding
+with perfect syndrome measurements is still an inher-
+ently stochastic procedure due to the indeterminacy of
+anyonic charge measurements. This means that perfect
+decoding can sometimes either be successful or unsuc-
+cessful even starting from the same initial state. Our
+definition of the memory lifetime therefore simply cor-
+responds to a statistical estimate of the actual memory
+lifetime. Furthermore, it is not known which decoder
+is optimal for the Fibonacci code. Hence, the choice
+for the clustering decoder to verify the correctability of
+states is, in a way, an arbitrary one. This choice, how-
+ever, is motivated by the recent discovery that the clus-
+tering decoder yields high thresholds for a related er-
+ror correcting code exhibiting doubled Fibonacci any-
+onic excitations, and performed significantly better in
+this context than decoders based on a perfect matching
+strategy [24]. In any case, one should keep in mind that
+the memory lifetime as defined above, does not repre-
+sent the true memory lifetime. Instead the sub-optimal
+verification process entails that it merely provides us
+with a lower bound on the true value.
+IV.
+HARRINGTON’S CELLULAR AUTOMATON
+DECODER
+The model described above is paired with a decoder
+which is a straight-forward adaptation of the cellu-
+lar automaton decoder introduced in Ref. [2]. Previ-
+ously, this decoder has also been used for a similar
+phenomenological model of Ising anyons in Ref. [32],
+where the existence of an error correction threshold
+was proven analytically. At each time step during the
+error correction simulation, based on the reported mea-
+surement outcomes in the faulty syndrome, the decod-
+ing algorithm will apply local transition rules to fuse
+neighboring anyons or to move anyons to neighboring
+tiles.
+Intuitively these transition rules work as follows.
+The lattice is divided into square colonies of size Q×Q.
+At each time step, the transition rules will attempt to
+fuse neighboring non-trivial anyons, as observed in
+the faulty syndrome.
+If a non-trivial anyon has no
+neighbors, the transition rules will move it to the cen-
+ter of its colony.
+At larger timescales, higher-level
+transition rules are applied on a renormalized lattice
+where anyons located at colony centers will be fused
+with anyons at neighboring colony centers, or moved
+toward the center of their respective super-colonies,
+which consist of Q × Q colonies. This renormalization
+scheme is then continued at higher levels until eventu-
+ally the Qn × Qn super-colony covers the entire lattice
+for some integer n. To ensure the latter is possible, we
+will always assume that the linear lattice size satisfies
+L = Qn for some integer n. An example of these pro-
+cesses is shown in Fig. 4.
+To describe the action of the decoding algorithm
+more precisely, we will define its action at different
+renormalization levels k. The level-0 transition rules
+
+5
+are those already discussed above and are applied at
+every time step based on the reported faulty syndrome
+obtained from the most recent round of faulty measure-
+ments. The transition rules are applied to one location
+at a time and take into consideration only the anyon
+content of that site and of its eight neighbors. A de-
+tailed definition of these rules is given in App. C. When
+an anyon is moved from a site l to a neighboring site
+l′, the (true) anyon content of site l is fused with that
+of site l′ and the resulting charge is placed on site l′
+while the charge of site l is restored to the vacuum.
+This happens irrespective of whether or not the syn-
+drome for both sites was correct. Hence, when the de-
+coder attempts to move a ghost defect (a trivial charge
+misidentified as a non-trivial one) to a neighboring site,
+this process does not create additional excitations. This
+does not, however, mean that mistaking a trivial charge
+for a non-trivial one has no negative consequences. In-
+deed, these wrong syndromes may cause the decoder
+to stretch out existing errors.
+The level-1 transition rules are not applied in every
+time step, but only when t is a multiple of a param-
+eter called the work period, which we will denote by
+U. We require that U = b2 for some positive integer
+b. One should think of U as the time scale at which a
+coarse-graining is performed. Level-1 transition rules
+act at on a coarse-grained lattice where the sites corre-
+spond to the centers of the level-0 colonies, and these
+are grouped into level-1 colonies of size Q2 × Q2.
+Hence, the actions determined by the level-1 transition
+rules involve pairs of level-0 colony centers separated
+by a distance Q. An example of such a move is pro-
+vided in Fig. 4(d). The transition rules themselves are
+nearly identical to the level-0 rules, but are based on
+two sets of level-1 syndromes s1,c and s1,n (defined
+below) rather than one. For a site l (which is a level-
+0 colony center), the transition rules use s1,c(l) as the
+anyon content of site, while the anyon content of its
+neighbors (that is, the neighboring level-0 colony cen-
+ters) is taken to be s1,n(l′).
+The definitions of the level-1 syndromes s1,c and
+s1,n require a pair of variables fc, fn ∈ [0, 1].
+In-
+tuitively these variables serve as detection thresholds
+for the level-1 syndromes by determining the fraction
+of measurements that must return a non-trivial out-
+come at a site before it qualifies as a non-trivial level-1
+syndrome. The proper definition, however, is slightly
+more complicated and uses a coarse-grained counting
+method. Below, we give the precise definition of s1,c,
+the definition of s1,n is entirely analogous (using fn
+instead of fc). We start by dividing the work period
+U = b2, into b intervals of b time steps each. For
+each of these intervals, we say a non-trivial syndrome
+is present at a colony center l if a non-trivial charge
+was reported there for at least fcb of the b time steps in
+the interval. When at least fcb of the b intervals have
+a non-trivial syndrome, s1,c(l) is set to one. A visual
+example of this coarse-grained counting procedure is
+shown in Fig. 3.
+≥ fc · b
+< fc · b
+≥ fc · b
+Figure 3: A visual example of the coarse-grained
+counting procedure to determine the level-1 syndrome
+for a level-0 colony center. The row of dots represent
+U time steps, divided in b intervals of size b. The time
+steps during which a non-trivial measurement
+outcome was reported are indicated by the colored
+dots. The crosses in the second row indicate in which
+intervals the fraction of non-trivial measurement
+outcomes is equal to or higher than fc.
+The motivation for using two types of syndromes for
+k > 0 is as follows. Suppose that an error spans across
+two neighboring colonies, which we will label ρ and ρ′.
+The level-0 transition rules will transport all resulting
+anyons to the respective colony centers, where they can
+now be acted upon by level-1 transition rules at the end
+of the work period. Imagine that a non-trivial anyon
+is now present at both colony centers. When consid-
+ering the level-1 transition rules acting on ρ, there are
+four possible scenarios for the syndromes s1,c(ρ) and
+s1,n(ρ′). In case s1,c(ρ) = 0, the transition rules act
+trivially on ρ. If both s1,c(ρ) = 1 and s1,n(ρ′) = 1
+then the transition rules will be applied correctly and
+the anyons will be fused. However, if s1,c(ρ) = 1 but
+s1,n(ρ′) = 0, the transition rules may move the anyon
+in ρ away from ρ′, thereby increasing the weight of
+the error. Hence, it is desirable to set fc > fn to de-
+crease the odds that when a level-k syndrome reports
+an non-trivial anyon at a colony center, the level-k syn-
+drome for its neighbors are false negatives. We must
+be careful not to set fc too high or fn too low, how-
+ever. If we choose fc to high, low-weight errors could
+cause s1,c to never report any non-trivial charges, de-
+laying any necessary corrections. Similarly, setting fn
+too low will result in low-weight error triggering many
+false positives for s1,n, which can cause the decoder to
+make wrong decisions.
+
+6
+Level-k
+transition
+rules
+are
+applied
+when
+t
+mod U k = 0. They operate on a renormalized lattice
+that uses the centers of level-(k − 1) colonies as sites,
+and groups these into level-k colonies of size Qk ×Qk.
+The level-k syndromes sk,c and sk,n are determined by
+the coarse-grained counting method described above,
+using bk intervals of bk time steps each. For linear sys-
+tem size L, k ranges from 0 to kmax = logQ(L).
+It is important to note that non-Abelian anyons do
+not allow for instantaneous moves. Indeed, while one
+can construct a unitary string operator for Abelian
+anyons, no such operator can be constructed for the
+non-Abelian case. This discrepancy can be traced back
+to the fact that fusion outcomes are non-deterministic
+for non-Abelian anyons, implying it is not possible to
+move an non-Abelian anyon by annihilating it with one
+member of a particle-antiparticle pair (as is done in
+e.g., the surface code).
+Therefore, the actions determined by level-k transi-
+tion rules, for k > 0 cannot be applied withing a sin-
+gle time step. Instead, they will be broken up into a
+sequence of moves involving only pairs of neighbor-
+ing sites which will be applied in Qk consecutive time
+steps. We further limit the model by requiring that the
+number recovery operations affecting a single tile in the
+lattice (or site in the decoding graph), is no greater than
+one in each time step. This allows all recovery opera-
+tions applied in one time step to be performed in par-
+allel. Hence, we must define a hierarchy determining
+which actions (moves or fusions between neighboring
+tiles) get prioritized based on the renormalization level
+from which they originated. In our case, it was opted to
+always prioritize correction processes from the highest
+renormalization level 2
+It was argued in [32] that the prohibition of instanta-
+neous corrections would likely not influence the thresh-
+old behavior other than slightly lowering the memory
+lifetimes relative to a hypothetical case where this re-
+striction is dropped. We explicitly verify this claim for
+our Fibonacci model below in Sec. VI.
+V.
+OUTLINE OF THE SIMULATION
+The goal of this work is to numerically determine
+a fault-tolerant error threshold for the error correcting
+2 Note that if one were to prioritize the level-0 corrections, higher-
+level correction could never be completed, as they would be un-
+done immediately after their first action is applied.
+(a)
+(b)
+(c)
+(d)
+Figure 4: Illustration of the transition rules on the
+decoding graph. The gray disks represent non-trivial
+syndromes, and the blue arrows represent the actions
+suggested by the decoder. The blue dotted lines
+represent the 3 × 3 colonies. (a-c) show a sequence of
+level-0 transition rules and possible outcomes of those
+actions. In (d) non-trivial anyons have been
+transported to two neighboring colony centers, the
+blue arrow represent a level-1 transition which could
+be applied at the end of the work period.
+code defined in Sec. II with pair-creation noise and
+measurement noise as outlined in Sec. III, and with the
+cellular automaton decoder introduced in Sec. IV. This
+is achieved by performing Monte-Carlo simulations to
+determine the average memory lifetime for a range of
+system sizes and error rates. These results then allow
+one to estimate the value of the error threshold.
+A single Monte-Carlo sample (with some fixed val-
+ues for the noise strength p and the measurement er-
+ror rate q) is obtained as follows. First, the state of
+the system is initialized as a ground state (i.e.: con-
+taining no anyons). Then a sequence of time steps is
+performed consisting of the application of pair-creation
+noise with rate p, a round of faulty syndrome measure-
+ments with error probability q, and finally a sequence
+of recovery operations. At the end of each time step,
+it is verified whether or not the state is considered cor-
+rectable, according to the criteria specified in Sec. III.
+
+7
+(a)
+(b)
+(c)
+Figure 5: (a) Level-0 colonies of size Q × Q. (b)
+Level-1 colonies defined as Q × Q level-0 colonies.
+(c) Renormalized lattice used for the level-1 transition
+rules.
+This sequence of time steps is continued until one of
+the following three outcomes occurs: (1) The largest
+connected group of anyons grows too large, rendering
+its classical simulation intractable 3; (2) A noise pro-
+cess or recovery operation induces a logical error by
+fusing a pair of anyons along a path that forms a non-
+contractible loop when combined with their fusion tree;
+(3) The verification procedure at the end of a time step
+fails. The memory lifetime is then set as the number of
+time steps that were completed. The course of a single
+Monte-Carlo sample in the simulation is summarized
+as pseudo-code in Alg. 1.
+3 Note that such cases are likely to correspond configurations in
+which the initial state cannot be recovered.
+Algorithm 1 Numerical simulation
+initialize state
+t = 0
+while correctable with clustering decoder & no logical
+errors made do
+t ← t + 1
+apply pair-creation noise
+perform faulty measurements
+for k = 0 : kmax do
+if t mod U k = 0 then
+update level-k syndromes
+apply level-k transition rules
+end if
+end for
+end while
+memory lifetime = t
+VI.
+NUMERICAL RESULTS
+The Monte Carlo simulation described above were
+performed for various system sizes with p = q. The
+following parameters were used:
+Q = 3 ,
+b = 7 ,
+Fc = 0.7 ,
+Fn = 0.2 .
+The resulting average memory lifetimes for L = 3,
+L = 9 and L = 27 are shown below in Fig. 6(a).
+These results clearly indicate that the code pre-
+sented in this work is indeed fault-tolerant. Further-
+more, while the current data is not sufficient to demon-
+strate a clear-cut fault-tolerant threshold, it still ex-
+hibits threshold behavior and is remarkably similar to
+the results previously obtained for the toric code [2]
+and the Ising topological code [32]. We estimate that
+the fault-tolerant threshold for the Fibonacci topologi-
+cal with pair-creation noise and measurement noise lies
+between p = 10−4 and p = 5 · 10−4, which corre-
+sponds to an inverse temperature between β = 9.2/∆
+and β = 7.6/∆. This is comparable to the threshold
+found for the Ising topological code [32], and only one
+order of magnitude below that for the surface code un-
+der similar circumstances [2]. For physical error rates
+near p = q = 10−4, corresponding to a temperature
+1/β one order of magnitude below the spectral gap, a
+code of linear size L = 27 yields logical error rates of
+the order 10−8.
+
+8
+(a) Average memory lifetime in function of the error strength,
+with p = q, for various system sizes. Each data point
+represents the average over 1000 Monte Carlo samples. The
+blue line shows the coherence time of a single physical qubit.
+The average memory lifetimes for p ≤ 10−3 were fitted to a
+function of the form f(p) ∼ p−a. The results for L = 3,
+L = 9 and L = 27 are shown as the green, yellow and red
+lines respectively.
+(b) Average lifetime in function of the error strength with
+p = q for various system sizes and (unphysical)
+instantaneous corrections. Each data point represents the
+average over 1000 Monte Carlo samples. The blue line shows
+the coherence time of a single physical qubit.
+Figure 6
+A second round of simulations was performed to de-
+termine average memory lifetimes with the assumption
+that all corrections happen instantaneously. While this
+is akin to the Abelian topological codes, where dis-
+tant anyons can be fused using unitary string-operators,
+this scenario is unphysical for non-Abelian anyons as
+they do not admit unitary string-operators.
+Never-
+theless, it is worth studying to which extend the re-
+sults in Fig. 6(a) are influenced by the restriction to
+non-instantaneous recovery operations.
+In Ref. [32]
+it was conjectured that allowing instantaneous correc-
+tions does not significantly change the qualitative be-
+havior of the average memory lifetimes as a function
+of the error rate, but mostly just increases the memory
+lifetimes. Our results, shown in Fig. 6(b), confirm this
+hypothesis.
+VII.
+DISCUSSION AND OUTLOOK
+The results presented in this work demonstrate that
+fault-tolerant error correction is possible for non-
+Abelian topological quantum error correcting codes
+supporting a universal logical gate set within their code
+space.
+For a code consisting of Fibonacci anyons
+in hexagonal tiles on a two-dimensional torus, sub-
+jected to pair creation noise and measurement noise,
+we demonstrated that the cellular automaton decoder
+detailed in this work is fault-tolerant. In particular, for
+physical error rates p ≤ 10−3, it was found that the
+logical memory lifetime surpasses the physical coher-
+ence time for all system sizes. When interpreting the
+pair-creation noise as resulting from a non-zero tem-
+perature, this pseudo-threshold corresponds to an in-
+verse temperature β = 6.9/∆, where ∆ is the energy
+required to create a pair of Fibonacci anyons. Further-
+more, our results suggest that this code admits a fault-
+tolerant quantum error correction threshold around p =
+10−4, or β = 9.2/∆, which is similar to the fault-
+tolerant threshold found for the Ising topological code
+[32].
+Several future research directions present them-
+selves. First, more research on a possible fault-tolerant
+threshold is necessary. Wile the numeric results pre-
+sented in this work provide a strong indication that a
+fault-tolerant error correction threshold exists, they do
+not conclusively prove its existence, nor do they pro-
+vide a precise estimate of its value. Hence, an impor-
+tant open problem is the formulation of a mathematical
+proof of its existence. Such proofs were previously for-
+mulated for the toric code [2] and for non-cyclic non-
+Abelian anyon models such as in the Ising topological
+
+9
+code [32]. Due to the cyclic nature of Fibonacci anyons
+(or any universal anyon model), however, the existing
+proofs are not sufficient.
+Second, it would be interesting to study different
+decoders in an identical setting.
+This includes both
+different cellular-automaton decoders such as those in
+Refs. [31, 34], as well as new decoders tailored to the
+Fibonacci topological code.
+Third, while this work demonstrates that the Fi-
+bonacci topological code can be operated fault-
+tolerantly as a quantum memory, results regarding
+its use for fault-tolerant quantum computing are still
+lacking.
+We envisage that fault-tolerant topological
+quantum computing at non-zero temperatures could be
+achieved by combining the code and decoding proce-
+dure presented in this work with the scheme for per-
+forming Dehn twists presented in Refs. [35–37]. Alter-
+natively, one can also perform transversal logical gates
+in a folded Fibonacci code [38].
+Finally, it would be of great interest to expand the
+current results to microscopic models for non-Abelian
+topological quantum error correction, such as the Fi-
+bonacci Turaev-Viro code [24].
+ACKNOWLEDGMENTS
+The authors would like to thank Guillaume Dauphi-
+nais and Jim Harrington for enlightening discussions
+on the cellular automaton decoder. The computational
+resources (Stevin Supercomputer Infrastructure) and
+services used in this work were provided by the Flem-
+ish Supercomputer Center (VSC), funded by Ghent
+University, the Research Foundation Flanders (FWO),
+and the Flemish Government. AS was supported by a
+fellowship of the Belgian American Educational Foun-
+dation. LB was supported by a PhD fellowship from
+the FWO. GZ was supported by the U.S. Department
+of Energy, Office of Science, National Quantum In-
+formation Science Research Centers, Co-design Center
+for Quantum Advantage (C2QA) under contract num-
+ber DE-SC0012704.
+Appendix A: Fibonacci anyons
+Below, we give with a brief overview of the topolog-
+ical aspects of our model. A thorough exposition of the
+theory of anyon models [39–42] is beyond the scope
+of this work, and we restrict to a basic description of
+the aspects of the Fibonacci model that are required for
+the specific purpose of our simulations. We refer to
+Ref. [33] for an in-depth discussion of anyonic fusion
+states on the torus.
+The Fibonacci anyon model has two particle types,
+1 and τ, that satisfy the fusion rules
+1 × 1 = 1 ,
+1 × τ = τ × 1 = τ ,
+τ × τ = 1 + τ .
+(A1)
+It is a non-Abelian anyon model, as the fusion of two
+τ-anyons can yield two distinct outcomes. In this case,
+the fusion space V c
+ab associated to the fusion of anyons
+a and b to c is one-dimensional, and is spanned by the
+state vector |a, b; c⟩ which we will represent graphi-
+cally as
+|a, b; c⟩ →
+b
+a
+c
+.
+(A2)
+When fusing several anyons, their total charge is a col-
+lective property of the anyons that does not depend on
+the specific order in which they are fused. Mathemati-
+cally, this is expressed through the associativity of the
+fusion rules,
+(a × b) × c = a × (b × c) .
+(A3)
+If we consider the case of three anyons a, b and c that
+fuse to a total charge d then this fusion may be car-
+ried out in two distinct ways, implying the existence
+of two equivalent decompositions of the associated fu-
+sion space V d
+abc in terms of fusion states (A2). These
+equivalent decompositions are related through a uni-
+tary transformation called an F-move, which is repre-
+sented graphically as
+b
+c
+e
+d
+a
+=
+�
+f
+F abe
+cdf
+f
+d
+b
+c
+a
+.
+(A4)
+The coefficients F abe
+cdf in this expression are called F-
+symbols of the anyon model. For the Fibonacci model
+they are given by
+F ττ1
+ττ1 = 1
+φ ,
+F τττ
+ττ1 = F ττ1
+τττ =
+1
+√φ ,
+F τττ
+τττ = − 1
+φ ,
+(A5)
+where all other F-symbols consistent with the fusion
+rules (A1) are equal to 1, and 0 otherwise.
+In addition to this recoupling one can also consider
+the exchange or braiding of pairs of anyons, which pre-
+serves their total charge.
+At the level of the fusion
+
+10
+space, such an exchange corresponds to a basis trans-
+formation to a basis associated to a different linear or-
+dering of the anyons4. Within V c
+ab there are two such
+possible basis transformations,
+b
+a
+c
+= Rab
+c
+c
+a
+b
+=
+�
+Rba
+c
+�∗
+c
+b
+a
+,
+(A6)
+which we will refer to as a clockwise and a counter-
+clockwise swap respectively. For the Fibonacci model
+the R-symbols appearing in these expressions are given
+by
+Rττ
+1
+= e
+4πi
+5 ,
+Rττ
+τ
+= e− 3πi
+5 ,
+(A7)
+where all other Rab
+c
+allowed by the fusion rules are
+equal to 1, and 0 otherwise.
+As we are dealing with a system that allows for an
+extensive amount of anyonic excitations, we will be in-
+terested in fusion states of many anyons, a1, a2, ..., an,
+with some total charge c.
+This gives rise to an ex-
+ponentially large topological Hilbert space V c
+a1a2···an
+spanned basis states of the form
+a1
+a2
+c
+b1
+a3
+an−1
+an
+b2
+bn−3
+bn−2
+.
+(A8)
+When dealing with anyonic fusion states on surfaces of
+higher genus, one must take into account additional de-
+grees of freedom in these states that are related to the
+anyonic charge that runs along non-contractible cycles.
+On a torus, this results in two distinct descriptions of
+fusion states, which are related through a basis change.
+These are known as the inside and outside bases, and
+are depicted in Fig. 7. A detailed discussion can be
+found in Ref. [33]. We will simply refer to these ad-
+ditional degrees of freedom as the handle labels of the
+state.
+4 As opposed to the more conventional view of braiding as an active
+transformation that maps between different fusion spaces, we have
+opted for the equivalent framework of braiding as a passive basis
+transformation, as the latter is more appropriate in view of our
+specific model and numerical simulations.
+(a)
+(b)
+Figure 7: Two possible basis choices for anyons on a
+torus: a) the inside basis, b) the outside basis.
+For our current purpose, we won’t need the full de-
+scription of said handle labels. It is sufficient for us to
+pick one basis, and subsequently set the total charge of
+all anyons to the vacuum. We are left with only a single
+label (1 or τ) representing the anyonic charge flowing
+along the non-contractible cycle associated with our
+basis choice. This is precisely the origin of the twofold
+degeneracy of the anyonic vacuum on the torus, which
+we have taken to be our code space.
+In the this work we will always start from the code
+state corresponding to a trivial handle label.
+As a
+change in handle label at any point during the error cor-
+rection procedure must be the consequence of a topo-
+logically non-trivial process which constitutes a logi-
+cal error, any simulation is aborted at the occurrence
+of such an event, meaning that all handle labels can be
+safely ignored in the remainder of our discussion.
+A basic functionality required for the simulation of
+the error correction procedure is the ability to correctly
+sample a measurement of the total charge of a given set
+of anyons. Starting from a given fusion state, this can
+be achieved by first transforming to a basis in which the
+relevant anyons are fused sequentially. For a system
+of many anyons these reordering basis transformations
+are obtained by combining Eqs. (A4) and (A6), giving
+
+11
+rise to the clockwise swap
+aj
+b3
+b2
+b1
+aj+1
+=
+�
+b′
+2
+Bb1ajb2
+aj+1b3b′
+2
+aj
+aj+1
+b3
+b2
+b1
+,
+(A9)
+and the counterclockwise swap
+aj
+b3
+b2
+b1
+aj+1
+=
+�
+b′
+2
+�
+Bb1ajb2
+aj+1b3b′
+2
+�∗
+aj
+aj+1
+b3
+b2
+b1
+,
+(A10)
+where
+Bb1ajb2
+aj+1b3b′
+2 =
+�
+c
+F aj+1ajc
+b3b1b′
+2
+Rajaj+1
+c
+F b1ajb2
+aj+1b3c .
+(A11)
+By performing a certain set of these basis transforma-
+tions, the fusion order can always be made consistent
+with the group of anyons of which we want to mea-
+sure the total charge. Subsequently, the fusion state is
+recoupled such that the relevant group of anyons is con-
+nected to the rest of the state by a single edge c. For a
+charge measurement of a pair of anyons this recoupling
+takes the form
+aj
+b3
+b2
+b1
+aj+1
+=
+�
+c
+F b1ajb2
+aj+1b3c
+aj
+b3
+b1
+aj+1
+c
+,
+(A12)
+and charge measurements of larger groups of anyons
+simply require consecutive applications of the recou-
+pling identity (A4). Finally, the charge outcome is ob-
+tained sampling the total charge c from the probability
+distribution corresponding to the resulting superposi-
+tion of fusion states.
+Appendix B: Classical simulatbility
+It is well known that Fibonacci anyons are universal
+for quantum computation [3]. One might therefore be
+tempted to conclude that the classical simulation of the
+topological Fibonacci code described in Sec. II with
+pair-creation noise is unlikely to succeed. However,
+as was noted in Ref. [23], the simulation of noise and
+error correction processes does not require the simula-
+tion of general anyon dynamics. In particular, individ-
+ual noise processes create distinct connected groups of
+anyons with vacuum total charge (or extend such exist-
+ing groups). These groups correspond to anyons that
+have interacted at some point during their lifetime, and
+must thus only be merged whenever a noise or error
+correction process involves two members from discon-
+nected groups. Since each connected group has a triv-
+ial total charge, braiding between disconnected groups
+is trivial. Hence, the total fusion space factorizes into a
+tensor product of fusion spaces of individual connected
+groups, and we are only required to simulate anyon dy-
+namics within each of these groups separately. This
+factorization of the fusion space is illustrated in Fig. 8.
+The creation and subsequent merging of discon-
+nected groups of anyons by noise and recovery pro-
+cesses can be thought of as a kind of percolation pro-
+cess.
+Hence, below the percolation threshold, one
+expects that the size of the largest connected group
+scales as O(log(L)) (with variance O(1)), where L
+is the linear system size [43]. As this is a probabilis-
+tic statement, there will be instances where the largest
+connected group has a size larger than O(log(L), but
+the probability of such events is suppressed exponen-
+tially with the system size L. This logarithmic scal-
+ing of the largest cluster size s = O(log(L)) coun-
+ters the exponential scaling of the dimension of the fu-
+sion space d = O(exp(s)) for individual connected
+groups. Therefore, the fusion spaces of individual con-
+nect groups will have dimension dim = O(poly(L)),
+meaning that the dynamics within connected groups
+can be simulated efficiently.
+Exploiting the tensor product structure within the to-
+tal fusion space, requires the use of basis for the any-
+onic fusion space which reflects this structure and can
+be updated dynamically to keep track of noise and re-
+covery processes. This is best achieved by using the
+framework of curve diagrams, which were introduced
+in Ref. [23] (also see Ref. [44] for a more rigorous
+treatment using the language of modular functors), and
+also discussed extensively in Ref. [24]. Since these are
+merely a technical tool for keeping track of the most
+appropriate basis during the numerical simulations, we
+will refrain from discussing them here. Interested read-
+ers are referred to the aforementioned references for
+details.
+
+12
+V ab
+1 ⊗ V cde
+1
+⊗ V fg
+1
+V ab
+1 ⊗ V cde
+1
+⊗ V fg
+1
+(a)
+a
+b
+c
+d
+e
+f
+g
+x
+y
+V abcdefg
+1
+= ⊕x,y
+�
+V ab
+x ⊗ V x cde
+y
+⊗ V y fg
+1
+�
+(b)
+V ab
+1 ⊗ V cde
+1
+⊗ V fg
+1
+a
+b
+c
+d
+e
+f
+g
+(c)
+Figure 8: (a) A set of noise processes. (b) A generic
+state in the full fusion space of all anyons created by
+the noise processes involves superpositions in the
+labels x and y. (c) The factorized Hilbert space which
+is sufficient to represent the state.
+Appendix C: Transition rules for Q = 3
+Below, we give a full definition of the transition rules
+for Q = 3. It is possible to define more general tran-
+sition rules that apply for any colony size, examples of
+such rules can be found in Refs. [2] and [32]. However,
+since the numerical simulations performed in this work
+were performed to Q = 3 we have taken the freedom
+to tailor the transition rules to this case specifically.
+• North-West
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;
+else move south;
+• North
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else move south;
+• North-East
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
+else if sk,n(ρ + (1, 1)) ̸= 0, move south;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, −1)) ̸= 0, move west;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else move south-west;
+• East
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
+else if sk,n(ρ + (1, 1)) ̸= 0, move south;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;
+else move west;
+• South-East
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (−1, 1)) ̸= 0, move north-east;
+else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
+else if sk,n(ρ + (1, 1)) ̸= 0, move south;
+else if sk,n(ρ + (0, −1)) ̸= 0, move west;
+else move north;
+
+13
+• South
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
+else if sk,n(ρ + (1, 1)) ̸= 0, move east;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (0, 1)) ̸= 0, do nothing;
+else move north;
+• South-West
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (1, 0)) ̸= 0, move south;
+else if sk,n(ρ + (1, −1)) ̸= 0, move south-west;
+else if sk,n(ρ + (1, 1)) ̸= 0, move east;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 0)) ̸= 0, move north;
+else if sk,n(ρ + (0, 1)) ̸= 0, move east;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else move north-east;
+• West
+if sk,c(ρ) = 0, do nothing;
+else if sk,n(ρ + (0, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (1, 0)) ̸= 0, do nothing;
+else if sk,n(ρ + (1, −1)) ̸= 0, do nothing;
+else if sk,n(ρ + (−1, −1)) ̸= 0, do nothing;
+else move east;
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+arXiv:2301.03101v1  [cs.IT]  8 Jan 2023
+1
+Massive MIMO and NOMA
+Bits-per-Antenna Efficiency under Power
+Allocation Policies
+Thiago A. Bruza Alves, Taufik Abrão
+State University of Londrina (UEL), Department of Electrical Engineering, Londrina-PR, Brazil.
+Abstract—A comparative resource allocation analysis in terms
+of received bits-per-antenna spectral efficiency (SE) and energy
+efficiency (EE) in downlink (DL) single-cell massive multiple-input
+multiple-output (mMIMO) and non-orthogonal multiple access
+(NOMA) systems considering a BS equipped with many (M)
+antennas, while K devices operate with a single-antenna, and the
+loading of devices ρ = K
+M ranging in 0 < ρ ≤ 2 is carried out under
+three different Power Allocations (PA) strategies: the inverse
+of the channel power allocation (PICPA), a modified water-
+filling (∆-WF) allocation method, and the equal power allocation
+(EPA) reference method. Since the two devices per cluster are
+overlapped in the power domain in the NOMA system, the channel
+matrix requires transformation to perform the zero-forcing (ZF)
+precoding adopted in mMIMO. Hence, NOMA operating under
+many antennas can favor a group of devices with higher array gain,
+overcoming the mMIMO and operating conveniently in the higher
+loading range 0.6 < ρ < 2.0. In such a scenario, a more realistic
+and helpful metric consists in evaluating the area under SE and
+EE curves, by measuring the bit-per-antenna and bit-per-antenna-
+per-watt efficiency, respectively. Our numerical results confirm a
+superiority of NOMA w.r.t. mMIMO of an order of 3x for the
+SE-area and 2x for the EE-area metric.
+Index Terms—Non-Orthogonal Multiple Access (NOMA); mas-
+sive Multiple-Input Multiple-Output (mMIMO); Energy Efficiency
+(EE); Spectral Efficiency (SE).
+I. Introduction
+The beyond Fifth Generation (5G) of wireless communica-
+tion systems must allow ultra-dense connections with vastly
+heterogeneous requirements. The challenges in networks per-
+sist, including the Spectral Efficiency (SE) and the Energy
+Efficiency (EE) joint improvement, the increase in the SE-
+EE trade-off, and Quality of Service (QoS), always aiming
+to meet the growing number of devices connected to the
+network. Among the proposals to solve these challenges, the
+massive Multiple-Input Multiple-Output (mMIMO) system is
+the primary proposed system that allows the increase of the
+link capacity, exploring the propagation of multiple paths with
+the use of a large number of antennas at the Base Station
+(BS) [1], [2]. Another relevant enabling technology is the
+Non-Orthogonal Multiple Access (NOMA), which explores the
+power domain as an alternative way in terms of multiple ac-
+cess technology, helping to mitigate the spectrum exhaustion
+problem and serving more than one device per resource block
+[3].
+Although in many works mMIMO is classified as an or-
+thogonal technique, allocating the signal from devices in the
+same resource block, possible by spatial diversity, allows us to
+This work was partly supported by The National Council for Scien-
+tific and Technological Development (CNPq) of Brazil under Grants
+310681/2019-7, partly by the CAPES- Brazil - Finance Code 001, and
+the Londrina State University - Paraná State Government (UEL).
+T. A. Bruza Alves and Taufik Abrão are with the State University of
+Londrina (UEL), Department of Electrical Engineering, Londrina-PR, E-
+mails: [email protected], taufi[email protected]
+classify it as a non-orthogonal technique too [4]. There is a
+vast literature demonstrating the superior performance of the
+Spectral Efficiency of NOMA when compared to Orthogonal
+Multiple Access (OMA) techniques [5]. Previous aims to
+improve the communication system performance by combining
+MIMO (with a small number of antennas M) and NOMA have
+been discussed in [6]–[9].
+Studies comparing NOMA and mMIMO in a single cell are
+proposed in [4], [10], [11]. The acquisition of Channel State
+Information (CSI) through pilot acquisition to NOMA system
+is proposed in [10]. In [11], the application of NOMA in the
+mMIMO scheme is proposed, and better results are achieved
+in the proposed comparative. Moreover, in [4] is analyzed the
+performance of NOMA and mMIMO in line of sight and non-
+line of sight.
+The canonical mMIMO refers to the systems with BSs
+formed by a large number of antennas M when compared
+to the number of actives devices, K, succinctly M ≫ K is
+considered a mMIMO setup. The typical NOMA improves the
+SE by superposing the signals of the selected devices to form
+a cluster in the power domain, multiplexing it over the same
+signal and served by the same beamforming. Nonetheless,
+the success of NOMA depends on the Successive Interference
+Cancellation (SIC).
+Power-domain NOMA can be a candidate technology in
+dense networks [12]. To improve performance and minimize
+the impact to assume the perfect SIC [13], devices are divided
+into two groups. After grouping in pairs and forming a cluster,
+each pair forms a cluster with a high difference between
+channel conditions. The device with a higher channel condition
+can decode the signal sent to the device with the lower channel
+condition. The interference can thus be eliminated by SIC. The
+use of NOMA in BS equipped with a large number of antennas
+was investigated in terms of SE [4], [10] we propose in a similar
+configuration system increasing the loading up to 2 times the
+number of antennas in BS and analyze the SE, EE, and SE-EE
+trade-off.
+The EE metric is a popular figure of merit employed to
+analyze the balance between power consumption and data
+rate. The EE is the ratio between the effectively transmitted
+data rate and the total power expended during the transmis-
+sion process, including instantaneous and static components.
+With the EE metric, it is possible to evaluate the efficiency
+with which a system uses the limited energy resource to
+communicate data and optimize this ratio. Can show the
+tendency of energy consumption in the case of seeking justice
+among devices.
+The Zero Forcing (ZF) is simple and popular alternative
+interference suppression beamforming under perfect CSI con-
+dition and achieving a satisfactory condition in real situations
+when imperfect CSI, in this work, we adopt perfect CSI, for
+
+2
+that the pilots are needed. The adoption of NOMA system
+with a large number of antennas requires a defined equivalent
+channel to be deployed for interference mitigation; and ac-
+cording to the NOMA principle, makes the equivalent channel
+matrix smaller than the original one due to the exploration of
+power domain in NOMA.
+Various transmission topologies already deal with the EE
+problem in mMIMO, finding the optimal number of antennas,
+number of devices in a cell, and the maximal EE [2], [14]. The
+EE analysis in the NOMA system is carried out in [15], and its
+superiority is demonstrated when compared with conventional
+orthogonal multiple access (OMA) systems. Recent researches
+seek to improve the NOMA performance, e.g. in [16], the
+minimum pairing distance is defined and compared to the
+OMA, while in [17] it’s presented a comparison between OMA
+and cell-free system equipped with mMIMO-NOMA. An EE
+analysis in Terahertz (THz)-NOMA-Multiple-Input Multiple-
+Output (MIMO) was proposed in [18]. Still, the number of
+active devices is much smaller than the number of antennas
+in the BS, and [19] is a survey about Power Domain NOMA
+and makes clear the vacuum of EE analysis and comparison
+between NOMA with many antennas and mMIMO.
+Recent works propose the deployment of NOMA combined
+with other techniques a more effective transmission scheme;
+e.g., in [20] NOMA and mMIMO are jointly considered in a
+two-tier network for accommodating colossal traffic. Further-
+more, in [21], authors apply NOMA in Distributed Antenna
+Systems (DAS), aiming to achieve better performance when
+compared to the conventional NOMA or DAS technique alone.
+While [22] shows an in-depth survey of the state-of-the-art
+of power-domain NOMA variants; moreover, several open
+issues and research challenges of NOMA-based applications
+are systematized. The NOMA system presents drawbacks,
+such as hardware (including SIC) complexity, channel feed-
+back, receiver design, and careful power and pilot allocation
+strategies [12], [19], [23].
+This work focus on revealing the advantages of applying
+the mMIMO scheme versus NOMA scheme with a massive
+number of BS antennas, and varying the loading of devices,
+i.e., the ratio of the number of mobile devices to the number
+of BS antennas, ρ =
+K
+M , while we change the PA strategy.
+Besides, we adopt a realistic model for the system’s power
+consumption as in [2] but adapted to our needs, aiming at
+providing a suitable analysis of the system resource allocation.
+Contributions: the contributions of this work are fourfold.
+a) an extensive and comparative analysis on the spectral
+efficiency (SE) performance of mMIMO system against NOMA
+system, varying the system loading under specific (three dif-
+ferent) power allocation methods and making use of the area
+under the SE (Ssyst) curve of the system as an effective, useful
+and fair metric of performance and efficiency; b) we develop
+an energy efficiency (EE) analysis using a detailed model of
+energy consumption, with fixed and variable terms related to
+circuitry power consumption with number of antennas and
+devices, respectively, providing an extensive and comparative
+analysis on both the NOMA and mMIMO systems under
+realistic operation scenarios and making use of the area under
+the EE (Esyst) curve of system; c) an analysis on the SE-EE
+trade-off is developed considering a wide range of loading of
+devices, verifying the fairness between devices; d) finally, under
+mild conditions, we provide evidences for the NOMA’s ability
+to serve a greater number of devices than mMIMO system.
+The remainder of the paper is organized as follows. Section
+II describes the system models for NOMA and mMIMO
+adopted in this work. In Section III we present the proposed
+EE-SE formulation for NOMA and massive MIMO systems.
+Numerical results are analyzed in IV. Section V concludes the
+paper.
+Notation. In this work, boldface lower case and upper case
+characters denote vectors and matrices, respectively. The
+operator (x)+ = max(0, x). The operators [·]T, E[·] and
+|·| denote transpose, expectation and cardinality, respectively.
+A random vector x ∼ CN {0, Im} is circularly symmetric
+Gaussian distributed with mean 0 and covariance matrix Im.
+Im is m × m identity matrix.
+II. System Models
+Let us consider a multi-user single-cell Downlink (DL)
+transmission operating in a Time Division Duplex (TDD) with
+K single-antenna actives devices, communicating with one BS,
+which is equipped with M transmit antennas in Non-Line-Of-
+Sight (NLOS). The set K is formed by K devices, these devices
+are randomly distributed in a radius disk dmax, the disk area
+is formed by two sub-disk with the same number of devices
+in each sub-area; both subsets are identified as KH and KL.
+In the first subset, KH represents devices’ indexes having the
+higher channel coefficient and sort in descending order, while
+the other subset KL are formed by the devices with lower
+channel coefficient and sort in ascending order; the indexes
+k ∈ KH and k ∈ KL such that:
+K = KH ∪ KL,
+where
+KH = {1, ..., K/2} and KL = {K/2 + 1, ..., K}.
+(1)
+The channel vector modeling of device k can be described liked
+as:
+hk =
+�
+βkh′
+k,
+k = 1, ..., K,
+(2)
+where βk is the large-scale fading coefficient and satisfy
+βj > βi,
+∀j ∈ KH, ∀i ∈ KL.
+(3)
+Herein, the pathloss model in [dB] is defined as:
+βk = β0 + 10 · ξ · log10(dk),
+(4)
+where dk is the distance of user k to BS, ξ is the pathloss
+coefficient, and β0 is the attenuation at the distance of
+reference.
+In each coherence interval, h′
+k in (2) for device k is an
+independent random small-scale fading realization from an in-
+dependent Rayleigh fading distribution, h′
+k ∼ CN(0, IM), k =
+1, ..., K . The transmitted signal xk ∈ CM is the beamformed
+data symbol of device k:
+xk = gk
+√pksk,
+(5)
+where gk is a normalized beamforming vector, pk normalized
+transmission power and sk ∼ CN(0, 1) is the data symbol of
+device k, and period Ts. The signal received at the k-th device:
+yk =
+K
+�
+k′=1
+hT
+k xk′ + nk,
+=
+�
+βkh′T
+k
+K
+�
+k′=1
+gk′√pk′sk′ + nk,
+(6)
+=
+�
+βkpkh′T
+k gksk +
+�
+βkh′T
+k
+K
+�
+k′̸=k
+k′=1
+gk′√pk′sk′ + nk,
+where nk ∼ CN(0, 1) is the additive noise. Notice that this
+modeling applies to both NOMA and mMIMO systems, but
+beamforming is selected differently, and this topic will be
+addressed in the next sections.
+
+3
+A. Prior Actions
+Because the BS needs to know a priori crucial information
+related to the channel and devices distributed in the cell,
+including device location, rate demanded, and channel coeffi-
+cient, such required a priori information may differ depending
+on the multiple access scheme considered [12].
+The option for the mMIMO and NOMA systems was carried
+out with the guarantee that the needs of the devices would
+be met, and this initial step was carried out successfully.
+Subsection III-E briefly discusses the preliminary information
+required to proceed with different Power Allocation (PA)
+procedures in both mMIMO and NOMA systems.
+B. Pilot Overhead for Channel State Information
+Fig. 1 compares the pilot-data transmission structure along
+the one-channel coherence interval for both mMIMO and
+NOMA systems considered. Notice that Ts is the time required
+to transmit a data symbol (Data). Moreover, the channel
+coherence time interval T is assumed to be a multiple of the
+Data symbol period, T = ι · Ts. The power allocated to each
+pilot in the training step is enough. In contrast, the number of
+pilots, and the dedicated portion of coherence interval for data
+transmission are assumed to be the same in both systems.
+!"
+!"
+#"
+$%&'(
+#)()
+#"
+#"
+*+,+-
+.-+/
+-0123'41510612,0(157)&
+(%*1
+Figure 1: Coherence time interval structure: the training and
+data transmission structure for mMIMO and NOMA schemes
+under TDD NLOS setup.
+Notice that in NOMA transmission, the pilot transmission
+step is split into two portions, half for the UL transmission
+pilots and receive the DL pilot confirmation; this happens
+because to perform SIC, the cell-center devices need to learn
+the effective channels that are established by the beamforming.
+Additionally, the beamforming vectors are based on cell-center
+devices, producing limited rates achieved in cell-edge devices.
+On the other hand, in the mMIMO scheme, a significant
+advantage is that there is no need for DL pilots since the
+effective channels created by the beamforming are highly
+predictable, i.e. nearly deterministic gain and phase due to
+channel hardening effect [4].
+Assumption 1: In the NOMA system, the power allocated to
+each downlink pilot is sufficient to reach the destination device.
+C. Beamforming for NOMA and mMIMO systems
+At the mMIMO system, each device is served by a sin-
+gle beamforming vector. The ZF technique is a popular
+interference-suppressing beamforming scheme in the mMIMO
+system since it eliminates all inter-user interference using
+individual beamforming for each device, while the favorable
+propagation facilitates such interference suppressing in massive
+MIMO configurations. Besides, to perform ZF precoding in
+NOMA system, it is essential to understand the NOMA user-
+pairing concept.
+User-pairing: Inherent to the NOMA system, user clustering
+can be performed in several ways after the user-sorting and
+the user classification in center-users and edge-users subsets.
+Because we know that the SE of NOMA is directly proportional
+to the difference between the pathloss of the users, a natural
+choice consists in pairing users with as higher as possible
+pathloss differences [6]:
+∆βk = βk − βK+1−k,
+(7)
+forming the cluster k for k = 1, ..., K/2. With the pair formed,
+carefully beamforming vectors selection is required. Hence, in
+NOMA we assume that the beamforming vector for paired
+users is the same, i.e., gk = gK+1−k for all k = 1, ..., K/2.
+Assumption 2: In user-pairing procedure, we assume that
+the paired users are aligned with the BS so that the same
+beamforming can serve all paired users simultaneously. Hence,
+by admitting that each pair of devices is spatially aligned with
+the BS, and using localizing tools described, for instance, in
+[23], [24], one should assume further a priori user-pairing step
+in NOMA systems.
+Assumption 3: In NOMA system, beamforming serves more
+than one aligned device simultaneously; specifically, in this
+paper, two aligned devices per cluster are admitted according
+to the user-pairing step, while eliminating the inter-cluster
+interference (favorable propagation) under adopted perfect
+CSI conditions.
+In this work we adopt the linear ZF precoding as defined by
+the vector:
+gk = h′k(h′T
+k h′k)−1,
+(8)
+and satisfying h′T
+i gk
+=
+0, ∀ i
+̸=
+k, i.e., the favorable
+propagation effect between users belonging to distinct clusters.
+III. SE-EE in NOMA and mMIMO systems
+We discuss the SE and EE configurations in the NOMA and
+mMIMO systems. The operation of the NOMA system requires
+pairing devices so that the channel coefficients of the devices
+in the same cluster must be appropriately different, enabling
+power domain usage. As already mentioned, the interference
+cancellation process via beamforming presents problems that
+we will demonstrate below.
+A. Data Rates in NOMA with ZF
+Devices are divided into two sets like described in Eq. (1),
+and these groups are represented by Eq. (9) by their large-
+scale fading coefficient and are grouped into pairs forming a
+cluster as Fig. 2, the cluster k is formed by one device in cell
+center set KH and one device in cell edge set KL. Hence, the
+devices are grouped into two subsets:
+KH ={β1 > β2 > ... > βk/2},
+(center devices set)
+(9)
+KL ={βK < βk−1 < ... < βk/2+1}. (edge devices set)
+The user-pairing adopted in Eq. (9) is the same as proposed
+in [6], creating the largest possible difference in channel
+coefficients for devices not yet paired.
+Assumption 4: In this paper, we assume perfect SIC, and only
+one perfect SIC stage per cluster is performed, since just 2
+devices per cluster are admitted.
+
+4
+!"
+Figure 2: System Model indicating the Paring formation in
+NOMA system. Both mMIMO and NOMA systems deploy the
+same massive number of antennas at base-station, M.
+The instantaneous Signal to Interference plus Noise Ratio
+(SINR) of devices in cluster k is defined as:
+SINRk =
+βkpk|h′T
+k gk|2
+βk
+�K
+k′̸=k pk′|h′T
+k gk′|2 + 1
+.
+(10)
+In each cluster, the cell-edge devices treat the interference as
+noise and decode their data symbols, whereas the cell-center
+device can decode the data symbols of the cell-edge device
+and perform SIC, hence effectively removing the interference
+due to the cell-edge device under Assumption 3.
+To perform SIC, the cell-center device needs to be able to
+decode data signal intended for the cell-edge device, i.e., the
+ergodic SINR of the cell-edge device, sinrK+1−k, at device
+k, defined as sinrk,K+1−k, must be greater than or equal
+to the ergodic SINR of the k-th cell-center device. Hence,
+given the uplink (mMIMO and NOMA) and downlink (NOMA)
+pilot overhead and assuming perfect CSI in all receivers, and
+admitting Assumption 4, the following condition must be
+satisfied [4], [10]:
+E[SINRk,K+1−k] ≥ E[SINRk],
+(11)
+where
+SINRk,K+1−k =
+βkpK+1−k|h′T
+k gk|2
+βk
+�K
+k′̸=K+1−k pk′|h′T
+k gk′|2 + 1
+.
+(12)
+Herein, the condition in (11) must be satisfied by selecting the
+transmit powers appropriately.
+The achievable ergodic rate of devices in cluster k, i.e.
+device k in KH subset and device K+1−k in KL subset, under
+Assumptions 1–4, is given by the ergodic rate contribution of
+user-center device:
+Rnoma
+k
+= τE [log2 (1 + SINRk)] ,
+∀k ∈ KH
+(13)
+in [bits/s/Hz], and for the user-edge device:
+Rnoma
+K+1−k = τE [log2 (1 + SINRK+1−k)] ,
+∀k ∈ KL (14)
+where τ = (1− K·Ts
+T ), is the portion of each channel coherence
+interval (T) that is used for data transmission.
+Assuming perfect channel state information, ZF precoding
+for inter-clusters interference elimination, and using random
+matrix theory results [25], the k-th cluster NOMA achievable
+rate is obtained plugging eq. (10), (13) and (14):
+Rnoma
+cl-k
+= τE
+�
+log2
+�
+1 + ¯
+Mβkpk
+��
++
+(15)
+τE
+�
+log2
+�
+1 + βK+1−kpK+1−k
+βK+1−kpk + 1
+�
+,
+�
+∀k ∈ K and ¯
+M = M + 1 − K/2. Hence, the NOMA system
+can operate until K < 2M − 1. A detailed derivation of the
+expressions on this section can be found in [4] and [10].
+B. Data Rates in mMIMO with ZF
+In the mMIMO system with ZF precoding the ergodic
+achievable rate for device k is given by:
+Rm-mimo
+k
+= τE [log2 (1 + SINRk)] ,
+[bits/s/Hz]
+(16)
+where SINRk is defined in (10). Hence, the above mMIMO
+achievable rate equation becomes:
+Rm-mimo
+k
+= τE [log2 (1 + (M − K)pkβk)] ,
+[bits/s/Hz],
+(17)
+where (M − K) is obtained using random matrix theory,
+representing the coherent array gain of the received signal [25].
+Under linear precoding and combiners, the mMIMO system
+operates consistently when K < M. Finally, the average
+system sum-rate (avg-sum-rate) is defined simply by:
+Rm-mimo =
+K
+�
+k=1
+Rm-mimo
+k
+and
+Rnoma =
+KH
+�
+k=1
+Rnoma
+cl-k .
+The mMIMO system equations have been thoroughly investi-
+gated in literature and can be found in [25] and [26].
+C. Energy Efficiency
+EE metric is the ratio of the number of effective bits of
+information received over the total energy consumed by the
+overall system to transmit and receive/decode such informa-
+tion. The system data rate can determine the number of
+effective information bits received at the destination. Power
+consumption required for processing the signal at the trans-
+mitter and receiver side is often neglected; in this sense, it
+is calculated just as proportional to the radiated transmitted
+power. The growth in the number of antennas in the BS and
+the increased number of devices in 5G systems can lead to
+unattainable EE goals. In general, the average EE can be
+expressed as:
+EE =
+�K
+k=1 Rk
+Ptot
+,
+[bits/Joule/Hz],
+(18)
+where Ptot is the total power consumption across the com-
+munication system. It should consider transmission power
+consumption, such as RF power amplifier inefficiency, base-
+band signal processing, and cooling, among others. Therefore,
+a more realistic and detailed energy consumption model is
+required.
+Based on [2], the adopted power consumption model in our
+work considers two power terms: a) fixed-term; b) terms scaled
+with the number of antennas M and the number of devices
+K. The scaled terms occur because of the transceiver chains,
+coding/decoding, channel estimation, and precoding. Let the
+computational efficiency be L operations per joule in BS. We
+describe it as follows:
+RF Power: Prf is the power consumed to transmit the signal
+to active devices achieved the SINR target and 0 < ̟ ≤ 1 is
+the efficiency of the power amplifier.
+
+β1dcenl
+50mβ2K/2
+β
+K
++1
+2KOK-1
+umac5
+Fixed consumption: Pfixed is the power consumed at the
+BS which is independent of the number of transmit antennas
+and devices in the cell, is formed of term P0 including the
+power consumption of backhaul infrastructure, control signal-
+ing, baseband processor, and term Psyn a single oscillator used
+in all BS.
+Pfixed = P0 + Psyn
+Dependence only on K: PK is formed by the consumption
+to coding and modulation of information symbols to devices,
+represented by Pcod, the consumption to BS decoded the K
+sequences of information symbols, defined by Pdec, and the
+received power, represented by PRX, still composes this term,
+multiplied by K as well. In addition, a portion of the ZF
+precoding cost [27] depends only on K3.
+PK = K(Pcod + Pdec + PRX) + K3
+2
+3LT
+Dependence only on M. The term PM represents the
+transmitted power (PTX), hence
+PM = MPTX
+Dependence on K and M: the term PKM is the cost of the
+ZF precoding (due to LU-based matrix inversion) [27], which
+depends on the number of devices, the number of antennas,
+and the vector information symbol.
+PKM = MK 3 + T
+T L
++ MK2 2
+T L
+Adding the portions, we obtain the overall power consump-
+tion of the system:
+Ptot = Prf
+̟ + Pfixed + PK + PM + PKM
+[W].
+(19)
+D. Power Allocation Strategies
+In the sequel, we present three well-known and frequently
+applied strategies for power allocation. Still, due to the in-
+herent characteristics of NOMA, we propose modifications on
+the classical water-filling (WF) algorithm to enable application
+in the NOMA system. Such modifications, namely ∆-WF,
+ensure that none of the paired devices are dropped-out without
+undoing the pairing of devices. To guarantee a certain level of
+power disparities in each paired device, the power allocation ∆-
+WF procedure in the NOMA system has two steps: a) first, we
+allocated power for the clusters; b) we allocate power between
+paired devices. Thereby, we could analyze the behavior of the
+systems and compare their results.
+Notice that both mMIMO and NOMA systems deploy the
+same massive number of antennas at base-station, M. Hence,
+due to the channel hardening effect [1], [25] inherent to
+massive MIMO configuration, the small-scale fading vanishes
+across the M antennas equipped with a linear ZF precoding
+with vector as eq. (8). Hence, one can consider just the
+pathloss coefficients βk as the main parameter in the power
+allocation policies of systems based on a massive number of
+antennas.
+1) Equal Power Allocation (EPA): Equal Power Allocation
+(EPA) power allocation is deployed as a simple, naive strategy,
+where all devices are served with the same power. In mMIMO,
+all devices are served with the same transmission power
+regardless of their distance from the BS. In NOMA, power
+allocation has two steps. In the first step, the power is allocated
+equally between the pairs, and then we allocate each device’s
+power equally. The EPA strategy applied to mMIMO can be
+defined by:
+pk = Prf
+K
+[W],
+∀ k ∈ K.
+(20)
+In the case of EPA procedure applied to NOMA, it is composed
+of two steps: in the first step, power reference to each cluster
+can be defined simply as:
+pcl
+ref = 2 · Prf
+K
+[W],
+∀ k ∈ KH.
+(21)
+In the second step, the power allocation among the devices in
+the same cluster is defined as:
+pcl-k
+k
+= pcl-k
+K+1−k = pcl
+ref
+2 .
+(22)
+2) Proportional
+Channel
+Inversion
+Power
+Allocation
+(PICPA): is another power allocation technique adopted in
+this study. Unlike the EPA technique, which applies the same
+power to all devices, PICPA applies more power to devices
+with the worst channel conditions, favoring fairness across
+the devices. Such power allocation penalizes the average sum
+rate in favor of fairness among all users.
+The Proportional to the Inverse of the Channel Power Al-
+location (PICPA) strategy applied to mMIMO can be defined
+as:
+pk = Prf
+β−1
+k
+�K
+k=1 β−1
+k
+[W],
+∀k ∈ K,
+(23)
+while the PICPA procedure applied to NOMA follows two
+steps; in the first step, the power is allocated equally among
+the K/2 clusters:
+pcl
+ref = 2 · Prf
+K
+[W],
+∀k ∈ KH,
+(24)
+after that, the power of each device within the k-th cluster is
+defined by allocating more power to the device with smaller
+large-scale fading βk:
+pcl-k
+k
+= pcl
+ref
+βK+1−k
+βk − βK+1−k
+,
+and
+pcl-k
+K+1−k = pcl
+ref − pcl-k
+k
+,
+(25)
+where pcl-k
+k
+is the power allocated to the device k in the cl-k
+cluster, and pcl-k
+K+1−k is the power allocated to the device (K +
+1 − k), also belonging to the k-th cluster.
+3) Classical Water-Filling (WF) Algorithm: The application
+of the Water-Filling (WF) algorithm in mMIMO system results
+in an optimal (maximum) system sum-rate solution. However,
+some devices are dropped out of the service due to the dete-
+riorated channel condition. The WF power allocation strategy
+for mMIMO is described as:
+µ = 1
+|K|
+
+Prf +
+|K|
+�
+k=1,k∈K
+1
+βk
+
+ ,
+(26)
+Prf =
+|K|
+�
+k=1,k∈K
+pk ,
+where
+pk =
+�
+µ − 1
+βk
+�+
+, ∀k ∈ K
+and
+p = [p1, p2, ..., p|K|],
+with the operator (z)+ = max(0, z). Notice that the con-
+strained value for the total power available is set to Prf [W].
+The Algorithm 1 describes the classical WF procedure.
+On the other hand, the direct application of WF algorithm
+in the NOMA system implies harming the pair formation, i.e.,
+devices present in the KL set are effectively dropped-out of
+the service, undoing the pair. We propose a modification in
+classical WF like the following to allow some comparison.
+
+6
+Algorithm 1: Classical Water Filling (WF) for mMIMO
+Input: K,Prf, K = |K|
+1 NP ̸= ⊘;
+2 while (NP ̸= ⊘) do
+3
+solve Eq. (26) → p;
+4
+NP ← identify null positions in p;
+5
+K/{k}NP ← exclude from p devices labeled as NP
+6 end
+Output: p = [p1, p2, . . . , p|K|]
+4) ∆-WF for NOMA: The application of classical WF in
+NOMA in the same way as it is applied to mMIMO causes
+some formed pairings to be broken, since after WF algorithm
+application, some devices are dropped-out from the system,
+making the NOMA power difference (∆) in the devices of the
+same cluster vanish. Hence, we suggest modifying the classical
+WF procedure to be applied to NOMA accordingly. The steps
+of the ∆-WF algorithm are described as follows.
+In NOMA, the power allocation has two steps, in the first
+step, the allocation is between clusters. Hence, to prevent the
+formed pairs from being broken, we propose the application of
+WF based on the large-scale fading differences of the paired
+devices, as defined in eq. (7): ∆βk = (βk − βK+1−k) inside
+each KL and KH subsets, eq. (9). In the second step of
+the procedure, the power is allocated between the devices
+inside the group, assuming perfect successive interference
+cancellation (SIC); for this to be possible, the condition in Eq.
+(11) must be satisfied. The new water-level in the modified
+���-WF power allocation strategy for NOMA is defined by
+µ = 2
+KH
+
+Prf +
+|KH|
+�
+k=1,k∈KH
+1
+∆βk
+
+ ,
+(27)
+Prf =
+|KH|
+�
+k=1,k∈KH
+pcl-k,
+∀k ∈ KH
+where
+pcl-k =
+�
+µ −
+1
+∆βk
+�+
+,
+and
+pcl-k = [p1, p2, ..., p|KH|],
+In the second step, the power allocation to both devices in the
+k-th cluster is defined as:
+pcl-k
+K+1−k = pcl-k
+k
+= pcl-k
+2
+(28)
+Algorithm 2 summarize the proposed ∆-WF power allocation
+procedure, aiming to improve the SE of NOMA systems.
+Algorithm 2: ∆-WF (modified) for NOMA systems
+Input: KH, KL, Prf
+1 NP ̸= ⊘;
+2 while (NP ̸= ⊘) do
+3
+solve Eq. (27) → pcl-k;
+4
+NP ← identify null position in pcl-k;
+5
+KH/{k}NP ← exclude from pcl-k devices labeled as
+NP
+6 end
+Output: pcl-k = [p1, p2, . . . , p|KH|]
+Complexity analysis: In a comparative analysis of complexity,
+the ∆-WF algorithm for power allocation in NOMA system
+(Algorithm 2) performs two simple additional operations com-
+pared to the classical WF procedure (Algorithm 1): a) in eq.
+(27) the subtraction in (βk − βK+1−k); and b) the division
+by two in (28). Besides, NOMA vs. mMIMO systems require
+different a priori information to proceed accordingly with the
+PA procedure.
+E. Prior Information for Power Allocation Step
+For implementing the PA policies, prior information is re-
+quired at the BS, as defined in Table I. Some of this necessary
+information can be obtained through the dedicated pilot trans-
+mission step, at the cost of some overhead, as described in
+Section II-B. Moreover, a preliminary step is known, in which
+the spatial localization and path loss estimation of all devices
+must be realized. With such a priori information availability,
+the user-sorting and user-pairing steps can be performed.
+Table I: Prior information required to PA procedure
+PA
+βk
+h′
+k
+K
+KH
+KL
+pk, eq.
+EPA
+mMIMO
+−
+−
+▲∗
+−
+−
+(20)
+NOMA
+−
+−
+▲
+−
+−
+(21), (22)
+PICPA
+mMIMO
+▲∗
+−
+−
+▲∗
+▲∗
+(23)
+NOMA
+▲
+−
+−
+▲∗
+▲
+(24), (25)
+WF
+mMIMO
+▲∗
+−
+−
+▲∗
+▲∗
+(26), Alg. 1
+∆-WF
+NOMA
+▲
+−
+−
+▲∗
+▲
+(27, 28), Alg. 2
+▲ information needed a prior
+∗ obtained via Pilot Overhead
+− Information not needed
+IV. Numerical Results
+The numerical evaluations for the proposed analyses of
+NOMA and mMIMO systems are presented in this section.
+The simulation system and channel parameter values de-
+ployed along this section are depicted in Table II. The BS
+is located at the center of cell and equipped with massive
+M BS transmit antennas in typical non-line-of-sight (NLOS)
+channel propagation scenario. At the same time, the devices
+are randomly distributed in the cell area and split into two
+subsets, KL and KH, as illustrated in Fig. 2. In all simulations,
+we consider a block fading model where the time-frequency
+resources are divided into coherence time intervals (T), in
+which the channels remain constant and frequency flat, and
+it is measured in multiple of symbol transmit period (Ts).
+The system and channel scenarios have been simulated using
+Matlab 2019 software running under one Intel HD Graphics
+6000 GPU, Intel(R) Dual-Core(TM) I5 CPU @ 1.6 GHz, and
+8 GB RAM.
+Table II:
+Simulation Parameters
+Parameter
+Value
+BS antennas
+M = 64, 128 and 256
+Max. # Devices in the cell
+K = ζ · M (NOMA)
+K = M (mMIMO)
+Cell loading
+ρ = K/M
+Total RF power available
+Prf = 1W
+Pairs NOMA / Clusters
+N = K/ζ = K/2
+NOMA devices per cluster
+ζ = 2
+# antennas per device
+1
+Cell edge length
+dmax = 350 m
+Strong device position
+[dmin; d1] ∈ [50; 100] m
+Weak device position
+[d2; dmax] ∈ [150; 350] m
+Array gain MIMO device
+M − K
+Array gain NOMA kH
+M + 1 − K/2
+Data symbol period
+Ts
+Coherence time interval
+T = 512 · Ts,
+ι = 512
+Channel
+Pathloss exponent
+ξ = 3.78
+Attenuation at a d0 reference
+β0 = 130 [dB]
+# Monte-Carlo realizations
+1000
+
+7
+A. Spectral Efficiency Comparison
+The mMIMO and NOMA SE performance analysis is carried
+out in this subsection, by increasing the number of devices
+two by two until the loading limit ρ = 2. The results consider
+M = 64, 128 and 256 BS antennas. In Fig. 3. (a) the results
+of SE are achieved when the RF power available is allocated
+following the EPA strategy, where each device receives the
+same PA values. The mMIMO system overcame NOMA in all
+situations when the loading ρ < 0.6. However, the NOMA
+system achieves a higher SE than the mMIMO for each M
+scenario when the loading of devices increases, ρ > 0.6. The
+maximum avg-SE is 373 [bits/s/Hz], being attained with ZF-
+NOMA M = 256 antennas and ρ ≈ 0.76. Besides, one can
+infer that the mMIMO does not work with a loading higher
+than 1, due to the array gain reaching 0 at full loading of
+devices, while NOMA works suitably until the loading attains
+M · ζ, where ζ is number of devices per cluster.
+Fig. 3.(b) depicts the results of SE achieved in the mMIMO
+and NOMA systems when the PICPA method is applied to
+allocate the available RF power per device along the BS an-
+tennas. The maximum avg-SE in mMIMO system overcomes
+the NOMA counterpart until the loading ρ exceeds ≈ 0.62 for
+the three BS antenna configurations, M = 64, 128, and 256.
+This PA technique provides more power to devices with the
+worst channel condition, making the SE result reach maximum
+values below the EPA.
+Fig. 3.(c) depicts the conventional WF algorithm applied
+to mMIMO. Under such a power allocation approach, we
+highlight that forming pairs is unfeasible in the NOMA sys-
+tem. Indeed, the WF algorithm can maximize the system SE
+since it allocates more power to devices with better channel
+conditions. In contrast, devices under bad channel conditions
+(below the water level) are dropped out of the service.
+The classical WF algorithm has been adapted to the NOMA
+system dropped-out always a pair of devices. Such adaptation
+reveals substantial improvements of avg-sum-rate when M
+is low compared to classical WF PA in mMIMO. The ∆-
+WF power allocation procedure preserves the pairs clustering
+formation in the NOMA system, allocating more power to
+the cluster with a higher difference between coefficients of
+large-scale fading. Fig. 3.(c) shows that the maximum avg-SE
+≈ 361 [bits/s/Hz], which is achieved under ρ = 1 (K ≈ 256
+devices) when the modified WF is deployed in NOMA system.
+Moreover, when the number of BS antennas is lower (M = 64
+or 128), the NOMA achieved a peak higher than mMIMO,
+e.g., for M = 64 antennas, the peak of SE mMIMO occurs
+at loading ρ ≈ 0.7, while the NOMA SE peaks at ρ ≈ 1.2.
+However, as the number of BS antennas grows, the NOMA
+SE advantage decreases.
+Number of active devices after PA procedure. Fig. 4 shows
+the number of actives device after applying PA methods: in
+the EPA and PICPA algorithms, all devices remain activated.
+However, in classical WF mMIMO system when the number
+of device increases beyond ρ ≈ 0.25, half of the devices are
+dropped-out; while in ∆-WF NOMA PA, the percentage of
+active devices is always higher, e.g. higher than 70% for ρ ≈
+1.1 and M = 256 antennas, the worst case.
+Fig. 5 summarizes avg-sum-rate surfaces in terms of SE
+×ρ × M results achieved by NOMA with EPA, mMIMO
+with WF, and NOMA with modified ∆-WF. In the initial
+loading part, ρ < 0.65, the classical WF PA in ZF-mMIMO
+achieves better results until the loading (pink surface). When
+the number of antennas is low as M = 64 and ρ is in
+between 0.7 and 1.8, the EPA PA applied to NOMA (green
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+50
+100
+150
+200
+250
+300
+350
+400
+Average Sum Rate (bits/s/Hz)
+ZF-mMIMO
+ZF-NOMA
+M = 256
+M = 128
+M = 64
+(a) EPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+30
+60
+90
+120
+150
+180
+Average Sum Rate (bits/s/Hz)
+ZF-mMIMO
+ZF-NOMA
+M = 256
+M = 128
+M = 64
+(b) PICPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+50
+100
+150
+200
+250
+300
+350
+400
+450
+Average Sum Rate (bits/s/Hz)
+ZF-mMIMO-WF
+ZF-NOMA-
+-WF
+M = 256
+M = 128
+M = 64
+(c) Classical WF in ZF-mMIMO and ∆-WF in ZF-NOMA
+Figure 3: The average sum-rate with the loading of devices 0 <
+ρ ≤ 2, considering four power allocation methods: EPA, PICPA,
+WF, and ∆-WF The average is obtained over 1000 random devices
+locations.
+
+8
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+1.9
+2
+Loading  = K/M
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Actives Users
+ZF-mMIMO-WF - M = 64
+ZF-NOMA-
+-WF - M = 64
+ZF-mMIMO-WF - M = 128
+ZF-NOMA-
+-WF - M = 128
+ZF-mMIMO-WF - M = 256
+ZF-NOMA-
+-WF - M = 256
+ALL-EPA
+ALL-PICPA
+Figure 4: The average of active devices after PA procedure
+versus loading of devices in the range 0 < ρ ≤ 2. The average
+is obtained over 1000 random devices locations.
+Figure 5: Average sum rate with loading ρ and M.
+surface) achieved superior results. Moreover, when M = 128
+and 0.8 < ρ < 1.6, the ZF-NOMA-EPA achieve superior
+results (green surface). For a higher number of antennas in
+BS, i.e., M = 256 only in a short loading of devices range,
+0.86 < ρ < 0.97, the ZF-NOMA-EPA achieves superior SE
+results. Finally, when ρ > 0.97, the modified ∆-WF achieves
+competitive results (blue surface).
+B. Jain’s Fairness Index
+Another critical analysis developed was to analyze the
+fairness between the devices, i.e., to know the difference in
+the transmission rate achieved by active devices in the cell. For
+this measure, we use the Jain’s Fairness index like described
+in [28] and can be defined as:
+Fsyst
+m
+=
+��M
+k=1 Rk
+�2
+M �M
+k1 R2
+k
+.
+(29)
+The Fig. 6. depicts the fairness curves attainable by NOMA
+and mMIMO systems with EPA, PICPA, WF, and ∆-WF PA
+procedures when the loading of devices grows until ρ = 2. Fig.
+6.(a) shows the Jain’s Fairness Index when EPA policy is used,
+the mMIMO system performs better F results than NOMA for
+ρ < 1, on the other hand, NOMA can attain Fnoma
+m
+≈ 0.5 in
+almost every loading of devices, independent of M.
+Fig. 6.(b) reveals the Jain’s Fairness Index when the PICPA
+method is applied, despite the SE result being lower in this PA
+method, the mMIMO obtains the best fairness result, keeping
+the Fmmimo
+m
+consistently above 0.85, still the NOMA under
+0.5.
+The Jain’s Fairness Index when WF and δ-WF are depicted
+in Fig.6(c), It is intrinsic to these algorithms to allocate more
+power to devices with better channel conditions, which causes
+fairness between devices to be impaired. In this method, it is
+possible to observe a significant influence of the number of
+antennas M in the BS and the fairness result.
+C. Energy Efficiency Comparison
+Energy efficiency (EE) is another important figure of merit
+used to compare systems’ performance. In this section, a power
+consumption model based on fixed circuitry power part and
+that one varying according to the number of antennas M and
+the number of devices K has been adopted, following eq. (19).
+Table III [2] presents the adopted parameter values for the
+EE analysis and comparison discussed in this subsection. Fig.
+7 depicts the performance of EE with EPA, PICPA, WF, and
+∆-WF PA procedures, considering the exact three quantities
+of antennas. Fig. 7.(a) shows the EE performance with EPA. In
+this method, all devices receive the same power. The avg-EE
+mMIMO overcomes the NOMA around 13% to 20%. It was
+possible to observe that adding antennas in the BS increases
+the power consumption, harming the EE result. Again, under
+loading of devices 0.6 < ρ ≤ 2.0, the NOMA overcomes the
+mMIMO system.
+Table III: Adopted Parameters values for EE analysis [2]
+Parameter
+Value
+Backhaul Infrastruture
+P0 = 2 W
+Single oscillator
+Psyn = 2 W
+Coding and modulation
+PCOD = 4 W per device
+Decoding and demodulation
+PDEC = 0.5 W per device
+Receive power
+PRX = 0.3 W per device
+Transmitted power
+PT X = 1 W per antenna
+Efficiency of Power Amplifier
+̟ = 0.3
+Operations/Joule
+L = 109 oper. per joule
+Fig. 7.(b) reveals the EE performance with PICPA. In this
+method, more power is allocated to devices with poor channel
+coefficients, resulting in reduced poor EE performance for both
+NOMA and M-MIMO systems, attaining a maximum of 0.25
+and 0.16 [bits/W] for mMIMO and NOMA, respectively. The
+maximum EE attained by mMIMO is generally around 50%
+higher than NOMA. However, for loading of devices ρ ≥ 0.65,
+NOMA overcomes mMIMO EE performance.
+Fig. 7.(c) depicts the EE performance in the mMIMO with
+classical WF and in the NOMA with ∆-WF algorithm. It
+is possible to confirm the superiority of energy efficiency of
+mMIMO within the range where it operates consistently, i.e.,
+0 < ρ < 1. Notice that the maximum EE achieved by mMIMO
+is about 70 % higher than NOMA for different BS antennas.
+Finally, NOMA becomes more energy efficient than mMIMO
+only when the loading of devices is high, ρ > 0.95.
+In all analyzed system scenarios, the mMIMO equipped with
+classical WF PA procedure achieves higher maximum EE. The
+mMIMO attains better EE results than NOMA for ρ < 1. On
+the other hand, NOMA can serve a more significant number
+of devices (twice) than mMIMO.
+Fig. 8 summarizes the best EE results in a surface plotting
+for the mMIMO with WF overcoming NOMA across the entire
+
+IOMA-Z-VWF
+ZF-NOMA-EPA450
+n Rate (bits/s/Hz)
+400
+350
+300
+200256128
+M
+64Average Sur
+150
+100
+50
+0
+2
+1.8
+1.6
+1.4
+1.2
+p
+0.8
+0.6
+0.4
+0.29
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Jain Fairness Index
+ZF-mMIMO - M = 64
+ZF-NOMA - M = 64
+ZF-mMIMO - M = 128
+ZF-NOMA - M = 128
+ZF-mMIMO - M = 256
+ZF-NOMA - M = 256
+(a) EPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Jain Fairness Index
+ZF-mMIMO - M = 64
+ZF-NOMA - M = 64
+ZF-mMIMO - M = 128
+ZF-NOMA - M = 128
+ZF-mMIMO - M = 256
+ZF-NOMA - M = 256
+(b) PICPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Jain Fairness Index
+ZF-mMIMO - M = 64
+ZF-NOMA - M = 64
+ZF-mMIMO - M = 128
+ZF-NOMA - M = 128
+ZF-mMIMO - M = 256
+ZF-NOMA - M = 256
+(c) WF and ∆-WF
+Figure 6: The fairness of NOMA and mMIMO system under three
+power allocation procedures: (a) EPA; b) PICPA; (c) WF and ∆-
+WF. The average is obtained over 1000 random device locations.
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+EE (bits/Joule/Hz)
+ZF-mMIMO - M=64
+ZF-NOMA - M=64
+ZF-mMIMO - M=128
+ZF-NOMA - M=128
+ZF-mMIMO - M=256
+ZF-NOMA - M=256
+(a) EPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0.03
+0.06
+0.09
+0.12
+0.15
+0.18
+0.21
+0.24
+0.27
+EE (bits/Joule/Hz)
+ZF-mMIMO - M=64
+ZF-NOMA - M=64
+ZF-mMIMO - M=128
+ZF-NOMA - M=128
+ZF-mMIMO - M=256
+ZF-NOMA - M=256
+(b) PICPA
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+EE (bits/Joule/Hz)
+ZF-mMIMO-WF - M=64
+ZF-NOMA-
+-WF - M=64
+ZF-mMIMO-WF - M=128
+ZF-NOMA-
+-WF - M=128
+ZF-mMIMO-WF - M=256
+ZF-NOMA-
+-WF - M=256
+(c) WF and ∆-WF
+Figure 7: EE for NOMA vs. mMIMO under three power allocation
+procedures: (a) EPA; b) PICPA; (c) WF and ∆-WF. Average EE
+obtained over 1000 random devices locations.
+
+10
+loading range where it operates consistently. For device loading
+ρ > 1, the NOMA operates under lower EE until the loading
+ρ = 2. Moreover, considering the smallest number of antennas
+in the BS, the NOMA with EPA overcame the NOMA with
+modified ∆-WF; despite that, as the number of antennas in
+the BS increases, the NOMA with ∆-WF achieves marginal
+superior EE results.
+Figure 8: EE with loading ρ and M.
+D. Area Under Curves SE and EE
+For a fair comparison, one can consider a wide range
+of average SE and EE along the loading of devices, and
+normalized per antenna, which can be attainable by NOMA
+and mMIMO systems. Hence, let us consider the corresponding
+areas under the SE and EE curves in Fig. 3 and Fig. 7, such
+that:
+Ssyst
+M
+= 1
+M ·
+� ρ=2
+0
+SE(ρ) dρ
+�bits/antenna
+s · Hz
+�
+and
+Esyst
+M
+= 1
+M ·
+� ρ=2
+0
+EE(ρ) dρ
+�bits/antenna
+Joule · Hz
+�
+,
+respectively, where SE(ρ) is the average overall system sum-
+rate, and EE(ρ) is the average overall system energy efficiency
+achieved under specific loading of devices ρ. Hence, comparing
+the values of corresponding areas under the SE and EE curves
+of Fig. 3 and Fig. 7, we obtained Fig. 9.
+From the SE perspective, and considering EPA policy, the
+higher area-under-SE-curve ratio gain is achieved when the
+number of BS antennas is M = 64:
+Snoma
+M=64 ≈ 2.7 · SmMIMO
+M=64 .
+Notice that when the number of antennas M grows, the ratio
+above decreases. In the same way, considering WF policy, the
+gain trend remains. In contrast, considering the PICPA policy,
+the ratio practically remains the same.
+Furthermore, considering now the EE perspective, under
+EPA policy, the higher ratio is achieved when the BS is
+equipped with M = 256 antennas:
+Enoma
+M=256 ≈ 1.8 · EmMIMO
+M=256 .
+As one can conclude, in almost all scenarios, NOMA is more
+spectrally and energetically efficient than mMIMO over an
+extensive range of loading of devices 0 < ρ ≤ 2, roughly,
+in average, 80% in terms of energy efficiency, and 170% more
+efficient in terms of spectral efficiency.
+EPA-64
+EPA-128
+EPA-256
+WF-64
+WF-128
+WF-256
+PICPA-64
+PICPA-128
+PICPA-256
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+ZF-NOMA
+mMIMO
+EPA
+WF
+PICPA
+(a) S - Bar chart of Area Under SE curves.
+EPA-64
+EPA-128
+EPA-256
+WF-64
+WF-128
+WF-256
+PICPA-64
+PICPA-128
+PICPA-256
+0
+0.002
+0.004
+0.006
+0.008
+0.01
+0.012
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+ZF-NOMA
+mMIMO
+EPA
+WF
+PICPA
+(b) E - Bar chart of Area Under EE curves.
+Figure 9: The Area Under curve of NOMA and mMIMO
+system under three power allocation procedures: (a) SE curves;
+b) EE curves. The average is obtained over 1000 random
+devices locations.
+E. Resource Efficiency (SE-EE Trade-off)
+The NOMA and mMIMO are analyzed in terms of SE
+and EE trade-off, namely resource efficiency (RE), considering
+loading of devices increasing up to 2. From Fig. 10, one
+can find graphically the best loading of devices range that
+maximizes the SE-EE trade-off for each BS antenna configu-
+ration M. The left y-axis depicts avg-SE, and the right y-axis
+shows the avg-EE. Table IV summarizes the optimal loading
+of devices that maximizes the SE-EE trade-off and shows the
+percentage of active users after Power Allocations and Jain’s
+Fairness Index. Fig. 10.(a) reveals the results when M = 64,
+NOMA with EPA achieved SE-EE trade-off with the highest
+loading of devices and the highest SE in trade-off; on the
+other hand, mMIMO with classical WF achieved the highest
+SE-EE trade-off; however, the percentage of actives devices is
+around 0.5. Fig. 10.(b) depicts results when M = 128, NOMA
+with EPA achieved SE-EE trade-off with the highest loading
+of devices, in contrast, mMIMO with classical WF with lower
+loading of devices achieved higher values of SE and EE in
+trade-off with half of the active devices. Fig 10.(c) showed
+the results when M=256, NOMA with ∆-WF achieved SE-EE
+trade-off with the highest loading of devices, and one more
+time mMIMO with WF achieved higher SE-EE trade-off with
+47% of active devices. It is possible to demonstrate that the
+
+ZE.I0.8
+0.7256128
+M
+64
+0.6
+0.4
+0.2
+0EE(bits/Joule
+0.4
+0.3
+0.2
+0.1 ~
+0
+2
+1.8
+1.6
+1.4
+1.2
+1
+0.8
+p11
+increase of antennas in the BS improves the SE result, on the
+other hand, it worsens the EE result.
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+134
+0
+0.781
+ZF-mMIMO-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+158
+0
+0.475
+ZF-NOMA-
+-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+163
+Average Sum rate (bits/s/Hz)
+0
+0.476
+EE (bits/Joule/Hz)
+ZF-NOMA-EPA
+(a) M = 64 BS antennas
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+249
+0
+0.767
+ZF-mMIMO-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+251
+0
+0.449
+ZF-NOMA-
+-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+267
+Average Sum rate (bits/s/Hz)
+0
+0.453
+EE (bits/Joule/Hz)
+ZF-NOMA-EPA
+(b) M = 128 BS antennas
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+412
+0
+0.706
+ZF-mMIMO-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Loading  = K/M
+0
+361
+0
+0.405
+ZF-NOMA-
+-WF
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+0
+373
+Average Sum rate (bits/s/Hz)
+0
+0.412
+EE (bits/Joule/Hz)
+ZF-NOMA-EPA
+(c) M = 256 BS antennas
+Figure 10: SE-EE trade-off points when M = 64, 128 and 256.
+V. Conclusion and Future Works
+This work proposes a comparative SE and EE analysis in
+DL single-cell between mMIMO and NOMA with BS equipped
+with three antenna configurations. Under the SE perspective,
+mMIMO with the classical WF algorithm achieved better low-
+and medium-loading results. On the other hand, when the
+Table IV: SE-EE Trade-off
+M = 64
+ρ
+SE
+EE
+Actives Users
+F
+mMIMO-WF
+0.652
+131.15
+0.762
+.50
+.485
+NOMA-EPA
+0.875
+147.45
+0.428
+1.0
+.485
+NOMA-∆-WF
+0.844
+143.48
+0.441
+1.0
+.475
+M = 128
+ρ
+SE
+EE
+Actives Users
+F
+mMIMO-WF
+0.625
+243.21
+0.745
+.50
+.475
+NOMA-EPA
+0.734
+241.54
+0.411
+1.0
+.49
+NOMA-∆-WF
+0.703
+226.32
+0.405
+.98
+.45
+M =256
+ρ
+SE
+EE
+Actives Users
+F
+mMIMO-WF
+0.578
+404.84
+0.691
+.47
+.43
+NOMA-EPA
+0.523
+344.70
+0.380
+1.0
+.495
+NOMA-∆-WF
+0.594
+333.31
+0.375
+.85
+.398
+system loading is higher as ρ > 0.6 the NOMA achieves better
+results in the range 0.6 < ρ ≤ 2.
+The analyzed PA methods applied to the NOMA system,
+(EPA, PICPA, and ∆-WF) result in different SE performance.
+Indeed, when the channel hardening condition is fully attained,
+and the amount of BS antennas increases(M = 128 and 256),
+the best SE results are attained with the proposed ∆-WF
+algorithm, but, as expected, the fairness index is harmed.
+Under the EE perspective, the mMIMO achieved better
+results when employing the three EPA, PICPA, and WF PA
+methods under K < M. However, the NOMA can operate
+under higher system loading, i.e., K < 2M − 1.
+In terms of area-under-SE-curve and EE-curve metrics, S
+and E, respectively, the NOMA system attained better results,
+due to its ability to serve a larger number of users than
+mMIMO. Such numerical results confirm NOMA’s ability to
+operate with high loading of devices. On the other hand,
+achieving high fairness with NOMA is impossible.
+From the perspective of SE-EE trade-off, mMIMO achieved
+the best results, because of the superiority in EE; always
+achieved in loading of devices ρ = 0.6 in all M setups.
+NOMA systems present exciting features and have been
+intensively investigated as a promising technique in devising
+future wireless generations. As future works, hybrid NOMA
+systems and alternative techniques such as rate-splitting mul-
+tiple access (RSMA) can improve the overall EE of massive
+MIMO systems.
+Acknowledgement
+This work was partly supported by The National Council for
+Scientific and Technological Development (CNPq) of Brazil
+under Grants 310681/2019-7, partly by the CAPES- Brazil -
+Finance Code 001, and the Londrina State University - Paraná
+State Government (UEL).
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+

AtFQT4oBgHgl3EQf9DeI/content/tmp_files/2301.13449v1.pdf.txt

@@ -0,0 +1,1205 @@
+Certification Design for a Competitive Market
+Andreas Haupt
+Nicole Immorlica
+Brendan Lucier∗
+February 1, 2023
+Abstract
+Motivated by applications such as voluntary carbon markets and edu-
+cational testing, we consider a market for goods with varying but hidden
+levels of quality in the presence of a third-party certifier. The certifier
+can provide informative signals about the quality of products, and can
+charge for this service.
+Sellers choose both the quality of the product
+they produce and a certification. Prices are then determined in a compet-
+itive market. Under a single-crossing condition, we show that the levels of
+certification chosen by producers are uniquely determined at equilibrium.
+We then show how to reduce a revenue-maximizing certifier’s problem to
+a monopolistic pricing problem with non-linear valuations, and design an
+FPTAS for computing the optimal slate of certificates and their prices. In
+general, both the welfare-optimal and revenue-optimal slate of certificates
+can be arbitrarily large.
+1
+Introduction
+Many markets sell goods that have different features but appear identical to
+consumers.
+Examples include the market for carbon credits, the market for
+contract workers such as electricians or plumbers, or even some markets for
+physical goods like milk and eggs. In each of these markets, the goods have
+hidden, hard-to-verify properties that distinguish them. Nature-based carbon
+credits, for example, are sourced from a variety of forests with different pro-
+tection and longevity properties.
+Contract workers have differing skill levels
+and amount of knowledge. And physical goods are produced in a variety of
+circumstances, some more ethical and sustainable than others.
+In such settings, sellers seek to distinguish their products through costly
+certification. In the example of carbon credits, large certifiers issue certification
+standards for carbon emissions.
+Contract workers can enroll in certification
+programs or take exams to document their skill level. And farms can reach out
+to third-party organizations that certify the conditions, such as free-range or
+vegetarian-fed, under which their food is produced.
+∗We thank Robert Maddox, Yanika Meyer-Oldenburg and a seminar audience at Harvard.
+1
+arXiv:2301.13449v1  [cs.GT]  31 Jan 2023
+
+In each of these cases, a downstream market allocates goods based on cer-
+tification.
+Carbon exchanges allow for the trade of carbon certificates, and
+platforms for local services allow the trade of electrical or plumbing services.
+These markets have price formation that is independent of, and not controllable
+by, the certifier. Therefore, both welfare- and revenue-maximizing certifiers will
+need to reason about how the certification options they provide affect trade,
+and how this in turn influences the demand for certification.
+What impact does the presence of certification have on the downstream
+market? We answer this question in the context of a competitive market with
+production. In our model there is a mass of heterogeneous producers each creat-
+ing a single unit of a vertically-differentiated product. Producers can select the
+quality level of their products. The corresponding production cost is determined
+by the producer’s type and is increasing in quality. A mass of unit-demand con-
+sumers purchases these goods. Consumers also have varying types which deter-
+mine the strength of their preference for quality. Types are totally ordered and
+satisfy a single-crossing condition: higher-type producers are relatively more
+efficient at producing higher-quality goods, which higher-type consumers have
+relatively stronger preference for.
+Quality cannot be verified directly; instead, a third-party certifier offers a
+menu of certifications, each with its own requirements and certification price.
+Producers can purchase certifications of their products’ qualities from this menu.
+If their product surpasses the quality level of the certificate, it will be marketed
+as such. The certified goods, together with indistinguishable uncertified ones,
+are then sold in a competitive market.
+As in the Economics literature on certification, we assume that market par-
+ticipants are able form correct beliefs about the quality implications of a certifi-
+cate Milgrom (1981); Grossman (1981); Lizzeri (1999); DeMarzo et al. (2019). A
+big strand of empirical work considers the question of whether certifications are
+correctly interpreted in various markets (Wimmer and Chezum (2003) for race
+horses, Tadelis and Zettelmeyer (2015) on used-car auctions, Ramanarayanan
+and Snyder (2012) for dialysis screening centers, Luca (2016) for restaurant
+reviews, Elfenbein et al. (2015) for seller ratings on Ebay, Conte and Kotchen
+(2010) for voluntary carbon credits). In our model, certificates are based on ver-
+ifiable certification requirements, and at perfect Bayesian equilibrium all market
+participants hold rational beliefs about the distribution of quality implied by
+any given certificate. We note that such rational beliefs need not be aligned
+with official certificate descriptions. For example, in voluntary carbon markets,
+several empirical contributions point out that carbon certificate descriptions
+might not correspond to counterfactual mitigation outcomes, compare West
+et al. (2020, 2023); Guizar-Couti˜no et al. (2022); Clarke and Barratt ([n. d.]);
+Greenfield ([n. d.]b,n).
+The presence of certificates clearly impacts the market equilibrium because
+it influences consumers’ beliefs and hence willingness-to-pay. Our first result
+shows that, for any menu of certificates offered by the certifier, the equilibrium
+choice of certificates and the allocation outcome of the downstream market
+equilibrium is unique. Furthermore better producers (that is, those who can
+2
+
+produce higher quality at lower cost) always purchase higher (more restrictive)
+levels of certification.
+This implies that the producer-consumer matching in
+equilibrium is assortative. Better producers sell to consumers with higher value
+for quality, and at higher quality levels.
+This analysis immediately implies that the welfare-optimal menu offers every
+certificate at cost, and suggests a dynamic program for finding the welfare-
+optimal menu subject to a cardinality restriction on the number of certification
+levels that can be offered. But in many markets, certification is provided by
+third-party firms with profit-maximizing incentives.
+Our main result shows
+how to approximately construct the revenue-optimal menu in polynomial time.
+Specifically, we provide an FPTAS: we show how to compute a menu whose
+revenue for the certifier is an additive ϵ less than the optimal revenue in time
+polynomial in 1/ϵ. Our algorithm can optionally take as input a cardinality
+constraint k on the menu size (i.e., a maximum number of certification levels),
+in which case the revenue benchmark is the optimal slate of at most k certificates
+and their prices.
+Our proof contains technical insights that may be of independent interest.
+Namely, we reduce the certifier’s problem to one of a seller who wishes to sell
+a divisible good, facing buyers whose valuations are non-linear and potentially
+non-monotone in quantity. The seller corresponds to the certifier, and the buy-
+ers correspond to producer-consumer pairs. This projection of producers and
+consumers into a single economic agent is enabled by the fact that the matching
+in equilibrium is unique and assortative. The buyer valuations in this reduced
+problem are concave but not necessarily monotone as a function of quantity,
+meaning that for different buyer types the valuation may reach its maximum at
+different quantities. The valuations do satisfy a single-crossing property, which
+implies that types are totally ordered and higher-type buyers have pointwise
+higher valuations and have weakly higher preferred quantities.
+Under these
+conditions, we show that the revenue-optimal menu may be non-linear but will
+exhibit prices that are monotone in quantity. Monotonicity of prices enables
+the use of dynamic programming to construct an approximately optimal menu,
+though some care must be taken when discretizing the space of quantities and
+prices to ensure that buyer preferences do not change substantially. This non-
+linear pricing problem generalizes prior work that specifically studied the case
+of quadratic demand Bergemann et al. (2012a,b), and may be of independent
+interest.
+The menu of certifications offered by a revenue-maximizing third-party cer-
+tifier can distort welfare and result in inefficient levels of trade.
+That said,
+we show that a certifying agent entering into the market can never lead to a
+loss of welfare, no matter what menu of certification options they make avail-
+able. More specifically, we can imagine that a collection of (perhaps costly)
+certification options are available to the market as a baseline, perhaps provided
+by a public or tightly regulated agency. If a third-party certifier then enters
+the market and makes available any additional certification options into the
+market, at any price, we show that the total utility of all market participants
+(even excluding the new entrant) can only increase. A policy implication is that
+3
+
+a regulatory body concerned about inefficient certification arising from profit-
+seeking motives need not prevent certifiers from entering the market. Rather, it
+suffices to make sure that there exists some set of certifications available, either
+through third-party providers or public options, that enable an acceptable level
+of welfare from trade.
+1.1
+Related Literature
+This work contributes to the literature on certification. The early results Mil-
+grom (1981); Grossman (1981) produce unraveling type results: In equilibrium,
+the quality of a good is fully revealed. The main intuition for these results in
+models of certification is that certifying non-informatively will be interpreted
+by the market as a sign of bad quality—adverse selection is extreme. In our
+model of a revenue-maximizing certifier, there might be other reasons for certi-
+fication less informatively, as the price in the competitive market may depend
+on not only an individual seller’s quality, but all the sellers in the market. Later
+contributions Lizzeri (1999); DeMarzo et al. (2019); Acharya et al. (2011) allow
+for unsuccessful certifications, restricting the stark result obtained in Milgrom
+(1981); Grossman (1981).
+More broadly, our work is related to mechanism and information design in
+the presence of an exogenously given game played after the design. We rec-
+ommend Bergemann and Morris (2019) for a general overview of information
+design. Our work is especially related to the design of information provided to
+buyers and/or sellers of a good. Bergemann et al. (2017) considers the design of
+information for a first-price auction, where a third party can reveal a signal cor-
+related with a buyer’s valuations, and fully characterizes the achievable revenue
+and payoffs. Alijani et al. (2022) extends the analysis to a scenario with mul-
+tiple buyers. In a general mechanism design framework, Candogan and Strack
+(2021) develop an optimal disclosure policy for action recommendations in a
+game with private types and a hidden state. Dworczak (2020) designs mech-
+anisms in a setting where players participate in a finite Bayesian game after
+participating in the mechanism, so that game outcomes are impacted by infor-
+mation revealed over the course of the mechanism. He finds a cutoff structure
+of optimal mechanisms in the first stage.
+For-profit certification relates to the sale of hard information, which has
+been studied in the context of competitive markets. Ali et al. (2022) consider
+a seller holding a good of uncertain quality. The seller can purchase a quality-
+correlated signal from a revenue-maximizing intermediary before bringing the
+good to market. In general the resulting equilibria are not unique and can vary
+substantially, but by employing noisy signals the intermediary can robustly
+guarantee high revenue. Our model differs in that any certification options are
+made available to the entire market and product quality is endogenous, so the
+certifying agent can impact welfare. More generally, our work relates to the
+problem of how to sell payoff-relevant hard information to a potential buyer.
+Bergemann et al. (2018) solve for the revenue-maximizing mechanism in binary
+environments, and Bergemann et al. (2022) establish when full disclosure is
+4
+
+approximately optimal under more general spaces of actions. In contrast, our
+certifying agent is selling a signal that is valuable in that it conveys information
+to other participants in a subsequent game.
+In our mathematical analysis, we reduce the certifier’s problem to a pricing
+problem with a non-linear valuation. The non-linear pricing literature following
+Mussa and Rosen (1978) (see also treatment in Dewatripont et al. (2005) and
+(B¨orgers and Krahmer, 2015, Chapter 2.3)) studies a non-linear concave valu-
+ation with a quadratic cost. The functional form assumptions in these papers
+allow to characterize the optimal mechanism in closed form. Typically, the op-
+timal menu of offered goods is a continuum, in contrast to the linear screening
+problem first studied in the influential Myerson (1981). Our analysis will show
+that also the class of models we consider may feature infinite menus.
+In our polynomial-time approximation scheme, we use an approximation of
+a non-linear pricing model. The papers Bergemann et al. (2012a,b) consider
+approximation of non-linear single- and multi-dimensional pricing environments
+with finite menus (in the papers called “finite information”). The papers make
+functional form assumptions similar to the ones in Mussa and Rosen (1978), and
+derive rates of approximation by finite menus in these finite menus. The present
+paper allows for a general class of utility functions that satisfy a single-crossing
+condition, and, in addition to showing approximation by finite menus, shows
+that the finite-sized menu can be computed efficiently.
+Finally, our main assumption guaranteeing uniqueness of our equilibrium is
+a single-crossing condition. Single-crossing conditions are important in several
+domains, among them the interdependent private values auctions (Milgrom and
+Weber (1982)) and social choice and voting (Saporiti and Tohm´e (2006)). A re-
+cent line of work in algorithmic mechanism design has employed single-crossing
+conditions to enable approximately optimal designs in interdependent value set-
+tings Roughgarden and Talgam-Cohen (2013); Chawla et al. (2014). Closest to
+the present paper, another implication of single-crossing is adverse selection in
+markets Mirrlees (1971); Spence (1974).
+1.2
+Outline
+The rest of this article is structured as follows.
+We formalize our model in
+section 2. In section 3 we analyze the structure of equilibria given the certifier’s
+offerings. Section 4 contains our main results, a reduction of revenue-maximizing
+certification to a non-linear pricing problem, the FPTAS for its computation.
+Section 5 contains our results on welfare maximization and explores the welfare
+implications of third-party certification.
+2
+Model
+Market
+We consider a continuum market between producers and consumers.
+Producers are unit-supply and parameterized by types ψ ∈ R+ with measure G.
+Consumers are unit-demand and parameterized by types φ ∈ R+ with measure
+5
+
+F. The type measures F and G are atomless and continuous with compact
+support.
+Goods can be produced at different levels of quality, denoted q ∈ [0, 1].
+Goods of higher quality are more valuable to consumers but more costly to
+produce. We write g(q; ψ) for the cost incurred by a producer of type ψ when
+producing a good of quality q. We assume g is weakly convex and non-decreasing
+in q for every ψ and normalized so that g(0; ψ) = 0. We also write f(q; φ) for
+the value enjoyed by a consumer of type φ for a good of quality q, where f
+is weakly concave and non-decreasing in q for every φ and normalized so that
+f(0; φ) = 0. We will scale valuations so that f(q; φ) ≤ 1 for all q and φ, which
+is without loss for bounded values.
+We will assume that costs and valuations satisfy single-crossing with respect
+to the producer and consumer types, respectively. Roughly speaking, this means
+that producers (consumers) of higher types have lower marginal cost (higher
+marginal value) for producing higher-quality goods. More formally, for all φ1 <
+φ2 and q1 < q2, we have
+f(q2; φ2) − f(q1; φ2) > f(q2; φ1) − f(q1; φ1).
+Likewise, for all ψ1 < ψ2 and q1 < q2, we have
+g(q2; ψ2) − g(q1; ψ2) < g(q2; ψ1) − g(q1; ψ1).
+Transfers between producers and consumers are permitted. We assume that
+producers and consumers are risk-neutral and have quasi-linear preferences with
+respect to money. That is, if consumer φ purchases a product of quality q from
+producer ψ at a price of p, then the consumer enjoys utility
+uC((q, p); φ) = f(q; φ) − p
+and the producer’s utility is
+uP ((q, p); ψ) = p − g(q; φ).
+Certification
+Crucially, producers and consumers cannot contract on quality.
+This means that a producer cannot credibly commit to the quality of the good
+they produce, and a consumer cannot verify quality at the point of trade. But
+there is a third-party certifier who is able to determine the quality of a producer’s
+good. This verification is costly to the certifier, with cost c ≥ 0.
+After verifying the quality of a good, the certifier is able to assign to that
+good a signal (or certificate) σ ∈ Σ, where Σ is an arbitrary space of potential
+certificates.
+This certificate is visible to all producers and consumers.
+The
+certifier is permitted to collect payments from producers for this service, and
+these transfers can depend arbitrarily on the certificate produced.
+The certifier can commit to a certification menu M, which is a collection
+of certificate / transfer pairs (σ, t(σ)) along with quality requirements for each
+certificate σ. Following the literature on information design and persuasion,
+6
+
+we note that it is without loss of generality to associate each certificate signal
+σ with the set of quality levels that are eligible for that certificate. We will
+therefore assume without loss of generality that Σ = 2[0,1], the collection of all
+subsets of [0, 1], and each σ is a subset of [0, 1]. The interpretation is that a
+producer can select an item from this menu, in which case she pays the certifier
+price (or transfer) t(σ), the certifier verifies the good’s quality q, and as long
+as q ∈ σ the producer will receive certification σ. We will assume for technical
+convenience that each σ in the certifier’s menu contains a minimum element,
+meaning that inf σ ∈ σ.1
+We will write σ0 = [0, 1] for the trivial signal that conveys no additional
+information about quality. A good with certificate σ0 is functionally equivalent
+to a good that has not been certified.
+It will be notationally convenient to
+assume that the certifier always offers signal σ0 at cost 0. This allows us to
+think of every good as being certified, albeit possibly at the trivial level. If a
+producer attempts to purchase a certificate but does not satisfy that certificate’s
+requirements, they will instead be assigned certificate σ0; i.e., the certifier will
+not certify the good.
+The Competitive Market
+All certification is assumed to occur simultane-
+ously, and in advance of any trading between producers and consumers. Given
+the menu M of certificates and prices offered by the certifier, the producers’
+(production and certification) strategy is a mapping from producer type ψ to
+a choice of quality level q and certification σ. We will denote such a strategy
+Γ : ψ �→ (q, σ), and restrict attention to measurable functions Γ. In a slight
+abuse of notation, we will also use Γ to denote the measure over pairs (q, σ) of
+products with corresponding quality and certification that result when produc-
+ers apply strategy Γ.
+Goods that are assigned the same certificate are indistinguishable by the
+consumers. After all certification is complete, each producer has a single unit
+of a good marked with a certification σ. For any given certification σ, write Γσ
+for the marginal distribution over quality q of Γ restricted to certificate σ. That
+is, fixing the choices of the producers, Γσ is the distribution of levels of quality
+for a product with certification σ. Then the value of a consumer of type φ for
+a good with certificate σ can be evaluated as
+f(σ; φ) = Eq∼Γσ[f(q; φ)].
+That is, each consumer rationally evaluates the expected quality of each product
+given its certification level and the choices of the producers.
+1This excludes certificates of the form “the quality q is strictly greater than 1/2,” as
+opposed to “...at least 1/2.” Certificates of the former type are inconvenient because there is
+no single least-cost choice of quality that satisfies the certification requirement, and hence no
+utility-maximizing choice of quality for the producer. We could handle such certificates in a
+straightforward but tedious way by relaxing our equilibrium notion and assuming that each
+producer selects an ϵ-approximately utility maximizing choice of quality for some arbitrarily
+small ϵ. For the remainder of the paper we will ignore such technical issues and simply assume
+that each σ includes a minimum element.
+7
+
+Since goods with the same certification are indistinguishable to consumers,
+we can view the competitive market between producers and consumers as a
+market for certificates σ.
+A Walrasian (or Competitive) equilibrium of the
+resulting market is an allocation x(φ) of a certificates to each consumer φ, along
+with a price pσ for each certificate, such that:
+• Demand satisfaction: every consumer purchases her most-preferred good.
+That is, for every consumer type φ, f(x(φ); φ) − px(φ) ≥ f(σ; φ) − pσ for
+every σ ∈ Σ.
+• Market Clearing: every good with a positive price is sold. That is, for all
+σ, the measure of consumers φ such that x(φ) = σ is at most the measure
+of producers ψ who select level of certification σ. If pσ > 0 then these
+measures are equal.
+Since buyers (consumers) are unit-demand and hence their preferences satisfy
+the gross substitutes condition, a Walrasian equilibrium is guaranteed to ex-
+ist Gul and Stacchetti (2000).
+We will therefore assume that trade occurs
+between consumers and producers at competitive equilibrium prices given the
+choices made by the producers.
+Timeline
+To summarize, the timing of the market with certification is as
+follows:
+1. The certifier commits to a menu of certificates with corresponding prices.
+2. Each producer ψ simultaneously and privately chooses whether to produce
+a good, and if so at what level of quality.
+3. Each producer that chose to produce decides whether to certify their prod-
+uct, and at which certificate. These decisions are made simultaneously for
+all producers.
+4. The certifier verifies the products of producers who choose to certify and
+assigns certificates. Any producer who does not successfully certify re-
+ceives certificate σ0.
+5. Producers and consumers trade goods in a competitive market. I.e., trade
+occurs at market-clearing prices for the chosen levels of certification.
+Since Walrasian equilibria are not unique in general, one might wonder if
+the outcome described in the final step is well-defined. We will show in the
+next section that the competitive market equilibrium described the final step
+exists and its resulting allocation is unique for any certificate menu chosen by
+the certifier and any choice of certification levels chosen by the producers.
+8
+
+3
+Equilibrium Characterization
+In this section we describe the market outcome that will occur for any given
+menu of certificates offered by the certifier. We show that it is without loss
+of generality for the certifier to restrict to offering threshold certificates that
+guarantee that a product is at least a certain level of quality. We characterize
+the unique Walrasian market equilibrium allocation that results from any as-
+signment of such certificates to producers. We then use that characterization
+to solve for each producer’s utility-maximizing choice of certificate from the
+certifier’s menu, which will also be unique.
+3.1
+Certifications as Minimum Quality Thresholds
+A first simple observation is that since producer costs are increasing in quality
+level, and since goods at different quality levels but with the same certification
+are indistinguishable to consumers (and hence must sell at the same price),
+a producer who is assigned certificate σ will always choose to produce at the
+minimum quality level eligible for that certificate.
+Observation 3.1. Fix any certifier menu M and any production and certifica-
+tion strategy of the producers. Then for any producer ψ, selecting certificate σ
+and producing at quality q > min σ is dominated by selecting certificate σ and
+producting at quality min σ.
+Given this observation, we know that for any menu M, any equilbrium
+strategy Γ for the producers, and any certificate σ, the marginal distribution
+over quality Γσ will be a point mass at min σ. In particular, any two certificates
+with the same minimum will induce the same equilibrium beliefs over quality and
+hence have indistinguishable value to all consumers. It is therefore without loss
+of generality for the certifier to only offer certificates of the form σq = [q, 1]; i.e.,
+certificates that are differentiated only with respect to their minimum values.
+If a producer selects certificate σq, then that producer’s chosen quality level
+at equilibrium will necessarily be q. Any given certification menu M therefore
+reduces to a (possibly infinite) collection of quality levels in [0, 1] to certify.
+Motivated by this observation, we will assume for the remainder of the paper
+that all certificates are of the form [q, 1], and associate each σ = [q, 1] with its
+quality threshold q. We can then think of a certification menu M as a collection
+of pairs {(qi, ti)}, where qi is a quality threshold and ti is a corresponding price
+for certifying that quality is at least qi.
+3.2
+Uniqueness and Assortativeness of Competitive Mar-
+ket Allocations
+We now turn to an analysis of the competitive market outcome that will result
+given the strategies of the certifier and producers. Recall that we can restrict
+attention to certificates of the form σq = [q, 1] and that any good with certificate
+σq will have quality q with probability 1, so for the remainder of the section we
+9
+
+will think of a market outcome as an allocation x and prices p of quality levels.
+That is, x(φ) ∈ [0, 1] for all consumers φ, and for each q ∈ [0, 1] in menu M
+there is an associated market price pq. We emphasize that x is a mapping from
+consumers to the certified goods they buy at market, whereas Γ is a mapping
+from producers to the certificates that they choose from the certifier.
+The following lemma shows that for any choice of certification menu M and
+production and certification strategy Γ of the producers, all competitive market
+equilibria in the resulting market have the same uniquely-determined allocation.
+This allocation will be assortative, with higher-type consumers purchasing the
+higher-quality certificates.
+Lemma 3.2. Fix any certifier menu M and any strategy Γ of the produc-
+ers. Then in every competitive market outcome (x, p), the allocation x satisfies
+x(φ1) ≤ x(φ2) and px(φ1) ≤ px(φ2) for all φ1 ≤ φ2.
+Proof. Since consumers are unit-demand and each producer has a single unit of
+good, a Walrasian equilibrium (x, p) of the market is guaranteed to exist. By
+the first welfare theorem, any such equilibrium must maximize the total welfare,
+�
+φ
+f(x(φ); φ)dF(φ).
+Suppose there exist types φ1 < φ2 with x(φ1) > x(φ2). By the single-crossing
+condition, we have that
+f(x(φ1); φ2) − f(x(φ2); φ2) > f(x(φ1); φ1) − f(x(φ2); φ1)
+and hence
+f(x(φ1); φ2) + f(x(φ2); φ1) > f(x(φ1); φ1) + f(x(φ2); φ2)
+which contradicts the supposed welfare optimality of allocation x.
+We now turn to prices. Fix any consumer types φ1 < φ2, so in particular
+we know x(φ1) ≤ x(φ2), and suppose for contradiction that px(φ1) > px(φ2). By
+monotonicity of the value function f, we must have f(x(φ1); φ1) ≤ f(x(φ2); φ1).
+But this then means f(x(φ2); φ1)−px(φ2) > f(x(φ1); φ1)−px(φ1), which violates
+the competitive market condition that consumer type φ1 is choosing her most-
+preferred good. We therefore conclude that px(φ1) ≤ px(φ2), as claimed.
+3.3
+Uniqueness and Assortativeness of Certificate Selec-
+tion
+Given that market outcomes are well-defined, we next turn to the equilibrium
+choices of the producers when selecting quality levels and their corresponding
+certifications. We again show that for any menu M of certificates offered, the
+quality choices of producers are unique at equilibrium. Moreover, higher-type
+producers will always select (weakly) higher certificates. In the Lemma 3.2, we
+saw that higher-type consumers also purchase (weakly) higher certificates. As
+10
+
+we discuss later, this means the matching in any equilibrium will be assortative
+and hence constrained-efficient given the available certificates.
+Recall that a producer strategy Γ is a mapping from producer type ψ to a
+choice of certification and quality, which we know from will always coincide. We
+will therefore write Γ(ψ) = q to mean that producer ψ produces at quality level
+q and purchases certificate σq. In particular, we must have Γ(ψ) ∈ M for all ψ.
+Lemma 3.3. Fix any certification menu M offered by the certifier. Then there
+is a unique equilibrium strategy Γ for the producers, and Γ(ψ) is weakly increas-
+ing in ψ.
+Proof. Fix strategy Γ, which implies the measure of certificates chosen by the
+collection of producers. Let (x, p) denote a Walrasian equilibrium in the result-
+ing competitive market, and recall that x is uniquely determined.
+We first show that Γ is weakly increasing is ψ. Assume for contradiction
+that there exist ψ1 < ψ2 with q1 = Γ(ψ1) and q2 = Γ(ψ2) with q2 < q1. Then
+by the single-crossing condition for producers, we have g(q1; ψ1) − g(q2; ψ1) >
+g(q1; ψ2) − g(q2; ψ2).
+But then, if we let pq1 and pq2 denote the Walrasian
+equilibrium prices of q1 and q2 given Γ, we have
+(pq1 − g(q1; ψ1)) + (pq2 − g(q2; ψ2)) < (pq1 − g(q1; ψ2)) + (pq2 − g(q2; ψ1))
+which means that either
+pq1 − g(q1; ψ1) < pq2 − g(q2; ψ1)
+or
+pq2 − g(q2; ψ2) < pq1 − g(q1; ψ2).
+In other words, either producer ψ1 or ψ2 (or both) would strictly improve their
+utility by switching their choice of quality and certification. As such a swap has
+measure zero and does not influence the competitive equilibrium, this would be
+an improving deviation for the producer(s), violating the assumption that Γ is
+an equilibrium strategy for the producers.
+We have shown that Γ is weakly increasing in ψ. On the other hand, we
+know from Lemma 3.2 that the market allocation of quality levels to consumers is
+weakly increasing in φ. This means that any equilibrium outcome of production
+and trade is equivalent to one in which consumers and producers are matched
+assortatively, with higher-type consumers trading with higher-type producers.
+In other words, for any producer type ψ, there is a consumer type φ = φ(ψ) such
+that ψ always trades with φ(ψ). Specifically, φ(ψ) is such that F(φ(ψ)) = G(ψ)
+(treating F and G as cumulative distribution functions).
+Given this, we claim that Γ(ψ), the certification selected by producer ψ
+at equilibrium, will always be a certificate qi from the certifier’s menu M =
+{(qi, ti)} that maximizes f(qi; φ(ψ)) − g(qi; ψ) − ti. To see why, suppose for
+contradiction that the producer instead chooses some other certificate q′ at price
+t′ such that f(q′; φ(ψ))−g(q′; ψ)−t′ < f(qi; φ(ψ))−g(qi; ψ)−ti−ϵ for some ϵ > 0,
+11
+
+and sells to consumer φ(ψ) at an assumed market-clearing price pq′.2 Then, the
+producer ψ could instead deviate to purchasing qi at a price of ti, and offering
+it on the competitive market at a price of pq′ +g(qi; ψ)−g(q′; ψ)+(ti −t′)+ϵ/2.
+Note that if consumer φ(ψ) were to purchase from producer ψ at this price,
+then her utility would be
+f(qi; φ(ψ)) − [pq′ + g(qi; φ(ψ)) − g(q′; φ(ψ)) + (ti − t′) + ϵ/2]
+= (f(qi; φ(ψ)) − g(qi; ψ) − ti − ϵ) + (g(q′; ψ) + t′) + ϵ/2 − pq′
+> f(q′; φ(ψ)) − (g(q′; ψ) + t′) + (g(q′; ψ) + t′) − pq′ + ϵ/2
+> f(q′; φ(ψ)) − pq′.
+But since (q′, pq′) is the most-demanded offering to consumer φ(ψ) in the market
+equilibrium, this means that the offering of qi at the proposed price would be the
+most-demanded offering to consumer φ(ψ) under this deviation. In particular
+this means that some consumer would want to purchase qi at the suggested price,
+and therefore in the adjusted market equilibrium after this proposed deviation
+the price of qi must be at least this high.
+We conclude that the utility of producer ψ under this deviation is at least
+[pq′ + (g(qi; ψ) − g(q′; ψ) + (ti − t′) + ϵ/2] − g(qi; ψ) − ti = pq′ − g(q′; ψ) − t′ + ϵ/2
+> pq′ − g(q′; ψ) − t′
+and hence this deviation is strictly utility-improving for the producer, contra-
+dicting the assumption that Γ is an equilibrium.
+We therefore conclude that at equilibrium, each producer ψ chooses whichever
+certificate qi from the menu maximizes f(qi; φ(ψ)) − g(qi; ψ) − ti. The choice of
+each producer is therefore unique, up to tie-breaking on sets of measure zero.
+An immediate corollary of Lemma 3.2 and Lemma 3.3 is that the equilibrium
+outcome for a given menu M is not only essentially unique (up to the choice of
+market-clearing prices), but also has a natural assortative interpretation. Each
+producer in the market has a corresponding consumer with whom they will
+always trade. The producer will select whichever certification level maximizes
+the gains from trade between themselves and their partner consumer, less the
+price of the certification.
+Corollary 3.4. For any menu M = {(qi, ti)} of the certifier, the resulting
+equilibrium market outcome has producer ψ trade with consumer φ = φ(ψ) where
+G(ψ) = F(φ). The level of quality at which φ and ψ trade maximizes f(qi; φ) −
+g(qi; ψ) − ti, and is weakly increasing in ψ.
+2Note that as we showed Γ is weakly increasing in ψ, the deviation can not change the
+ordering of firms in terms of the certificate they purchase and hence does not change the
+consumer to which they sell.
+12
+
+4
+Revenue-Optimal Certification
+In the previous section we solved for the equilibrium outcome of the game played
+between producers and consumers given a certification menu M chosen by the
+certifier. With that characterization in hand, we now turn to the problem faced
+by a certifier who wishes to construct a revenue-maximizing slate of certificates.
+A priori, it would appear that the choice of which certificates to offer in-
+fluences the downstream competitive market allocation and prices. After all,
+producers compete for consumers and the certificates they purchase determine
+how they fare in this competition. Hence one might expect the menu of avail-
+able certificates to influence how much each producer would be willing to pay
+for any given certificate. However, as we will show, the assortative characteriza-
+tion of market outcomes means that the certifier can reason about the behavior
+of each producer type separately, without worrying about how they will jointly
+interact in the competitive market. Thus, the certifier is essentially facing a
+single buyer with an unknown type, and must simply maximize revenue subject
+to this buyer. This leads us to define a reduction from the certifier’s problem
+to that of a seller facing a (non-linear) buyer.
+4.1
+Reduction to a Non-linear Pricing Problem
+We will relate the certifier’s revenue-maximization problem to a non-linear pric-
+ing problem between a single seller and a single buyer. The buyer seeks to buy
+a perfectly divisible good. The seller may commit to a menu of quantities and
+prices. If a buyer of type θ purchases quantity q ∈ [0, 1] of the good at a total
+price of t, then the buyer utility is
+u((q, t); θ) = v(q; θ) − t,
+where v is concave in quantity q but not necessarily non-decreasing, and v(0; θ) =
+0 for all θ. The valuations v satisfy a single-crossing condition, which is that
+for θ1 < θ2 and q1 < q2, we have
+v(q2; θ2) − v(q1; θ2) > v(q2; θ1) − v(q1; θ1).
+The principal seeks to maximize revenue subject to a constant cost of production
+c > 0 for any non-zero quantity given a prior over buyer types. Given a menu
+M, we will write Rev(M) to denote the expected revenue obtained from M. We
+will also write OPT for the revenue of the revenue-maximizing choice of menu
+M.
+Proposition 4.1. The certifier’s revenue-maximization problem is equivalent
+to an instance of the non-linear pricing problem described above.
+Proof. The certifier’s problem is to design a certificate menu M = {(qi, ti)}
+that maximizes the total revenue collected, less the certification cost c paid
+to verify any non-trivial certificate qi > 0. By Corollary 3.4, given menu M,
+13
+
+each producer ψ will purchase whichever certificate qi maximizes f(qi; φ(ψ)) −
+g(qi; ψ) − ti.
+For q ∈ [0, 1], we can interpret ψ as a buyer type and define valuation
+function v(q; ψ) = f(qi; φ(ψ))−g(qi; ψ). Then since f and g both satisfy single-
+crossing with respect to their corresponding types, valuation function v does
+as well. Moreover, v is concave and v(0; ψ) = 0. By definition, the producers’
+choices of certificates from menu M corresponds precisely to the buyer’s choice
+of quantity when facing the same menu, interpreting each certificate quality
+threshold as a quantity.
+Thus the outcomes, and hence revenue, in the two
+settings are equivalent.
+Note that an immediate implication of Proposition 4.1, given Corollary 3.4,
+is that for any menu M chosen by the seller, the choice of quantity purchased
+by the buyer is weakly increasing in buyer type.3
+4.2
+Optimizing Revenue in the Non-linear Pricing Prob-
+lem
+By Proposition 4.1, to solve the certifier’s revenue maximization problem it
+suffices to optimize revenue in the non-linear pricing problem. As a first step, we
+note that in the special case where the valuations v are linear in quantity (which
+happens, for example, if the cost g and value f functions in our certification
+problem are both linear in q), this problem reduces to a standard pricing problem
+in mechanism design. A characterization due to Myerson Myerson (1981) then
+immediately establishes that it is revenue-optimal to choose a menu with only
+a single non-trivial item (q, p) with q = 1.
+Observation 4.2. If v(q; θ) is linear in q for all θ, then there is a revenue-
+optimal menu that offers only quantity q = 1 at some price p. This price p will
+be chosen to maximize p × Prθ[v(1; θ) > p].
+However, in general, non-linearity substantially changes the problem relative
+to the linear case. In particular, it is not necessarily optimal to offer a single
+menu item.
+Proposition 4.3. There are problem instances in which posting any single
+menu item is an arbitrarily poor approximation to the optimal revenue. The ap-
+proximation factor can be as large as Ω(log(H)), where H = maxθ1,θ2
+maxq v(q;θ1)
+maxq v(q;θ2)
+is the ratio between the highest and lowest maximum values across buyer types.
+Proof. Choose c = 0 and consider the valuation function v(q; θ) = q if q ≤ θ,
+and v(q; θ) = 2θ − q if q ≥ θ. That is, v is piecewise linear for each θ, with
+maximum value θ occurring at q = θ. Fix some H ≥ 1 and suppose the type
+distribution is such that Pr[θ > h] = 1/h for all h ∈ [1, H]. That is, the type
+distribution is equal-revenue on range [1, H].
+3Alternatively, this is a direct consequence of the single-crossing condition on valuation
+functions v.
+14
+
+This valuation function is concave and satisfies v(0; θ) = 0 for all θ. More-
+over, it satisfies the single-crossing condition. Indeed, for any q and any θ1 < θ2,
+we note that
+d
+dqv(q; θ1) ≤
+d
+dqv(q; θ2), since for any θ this derivative is 1 for q < θ
+and −1 for q > θ. Since v is also continuous in both q and θ, the single-crossing
+condition is implied.4 Finally, we note that this valuation function v can in-
+deed arise in our reduction from the certification problem with producers and
+consumers.5
+We now show the desired gap in approximation. Consider any menu M with
+a single non-trivial menu item (q, p). Then the revenue achieved by the seller
+is at most the welfare generated by the optimal allocation of quality level q.
+Since v(q; θ) ≤ q for all θ, this is certainly at most q times the probability that
+v(q; θ) > 0, which is q Pr[θ > q/2] ≤ q(2/q) = 2.
+On the other hand, the seller could offer a menu that includes every quality
+level q ∈ [1, H] at a price of q/2.
+A buyer of type θ would then choose to
+purchase quality level q = θ for a utility of θ/2, generating revenue θ/2.6 The
+total revenue is then E[θ/2] = O(log H).
+Posting a single menu item is therefore at best an O(log H) approximation
+to the optimal revenue, as claimed.
+We therefore know that any approximately revenue-optimal mechanism must
+sometimes have multiple non-trivial offerings on its menu. What should such a
+menu look like? Note that since buyers do not have free disposal in our setting
+(i.e., valuations are non-monotone), it is not even immediately obvious that
+higher quantities should be sold for higher prices in a revenue-optimal menu.
+We next establish that, in fact, there is always a revenue-optimal choice of menu
+for which higher quantities are sold at higher prices.
+Definition 4.4. We say a menu M = {(qi, pi)} is monotone if for all (qi, pi),
+(qj, pj) ∈ M such that qi < qj, we have pi ≤ pj.
+Lemma 4.5. For any instance of the non-linear pricing problem there is a
+revenue-optimal menu that is monotone.
+Proof. Let M be a revenue-optimal menu, and write M = {(qi, pi)}i∈Λ where
+Λ is some (possibly uncountable) index set. It is without loss to assume Λ is a
+subset of [0, 1] such that qi < qj for all i < j (e.g., by reindexing so that the
+index of qi is equal to qi). We can further assume without loss of generality that
+every item in M is purchased by some buyer type, as any element that is never
+purchased could be removed from M without impact. By Corollary 3.4, quality
+4Technically our construction only satisfies weak single-crossing since the inequality in
+derivatives is not strict. We can make the example strict by perturbing the slope of the initial
+line segment by an infinitesimal amount so that it depends on the type θ. We omit these
+details for expositional clarity.
+5In particular, take φ and ψ to be supported on [1, H], define f(q; φ) = q for all φ and
+g(q; ψ) = max{0, 2(q − ψ)} for all ψ. Then f is concave (in fact, linear), g is convex, the
+single-crossing conditions are satisfied, and v(q; θ) = f(q; φ(θ)) − g(q; θ) as required.
+6Buying a higher quality level q′ > θ is worse for the buyer because it generates less value at
+a higher price, whereas buying any quality level q′ < θ generates utility q′−q′/2 = q′/2 < θ/2.
+15
+
+levels purchased will be monotone non-decreasing in buyer type. This means
+that every item (qi, pi) is purchased by some contiguous interval of buyer types
+Ii (which may have measure zero).
+Suppose that menu M is not monotone. This means that either there is an
+element i < sup Λ such that pi > pj for all j > i, or else there exists a pair of
+menu items (qi, pi) and (qj, pj) with j > i such that (a) there exists some ℓ ∈ Λ
+with i < ℓ < j, and (b) pℓ < min{pi, pj} for all i < ℓ < j.
+Consider the former case, where there is an element i < sup Λ such that
+pi > pj for all j > i. In particular there must exist some j ∈ Λ with j > i. Let
+M ′ be the menu {(qℓ, pℓ)}ℓ∈Λ,ℓ≤i. I.e., M ′ is M with all elements with quantities
+greater than qi removed. Note that for all j ≤ i, since the types Ij preferred
+element (qj, pj) to any other element in M, they prefer element (qj, pj) to any
+other element in M ′ as well. Moreover, since purchase decisions are monotone in
+buyer type, we conclude that all types θ ∈ Iℓ with ℓ > i will purchase element
+(qi, pi) from menu M ′. But since pi > pℓ for all ℓ > i, this means that the
+revenue generated by menu M ′ is strictly greater than the revenue generated
+by menu M, contradicting the supposed optimality of menu M.
+Next consider the other case, there exists a pair of menu items (qi, pi) and
+(qj, pj) such that pℓ < min{pi, pj} for all i < ℓ < j.
+Let M ′ be the menu
+{(qℓ, pℓ)}ℓ∈Λ,ℓ≤i ∪ {(qℓ, p��)}ℓ∈Λ,ℓ≥j. That is, M ′ is menu M with all elements
+strictly between (qi, pi) and (qj, pj) removed. Then as in the previous case, for
+all ℓ ≤ i and ℓ ≥ j, types Iℓ all still prefer to purchase (qℓ, pℓ). In particular,
+types Ii purchase (qi, pi) and types Ij purchase (qj, pj). By monotonicity of
+purchasing decisions due to Corollary 3.4, all intermediate types θ ∈ Iℓ for
+i < ℓ < j must purchase either (qi, pi) or (qj, pj). As pi and pj are both larger
+than the prices of the elements those types were purchasing under menu M, the
+revenue of menu M ′ must be strictly larger, which is again a contradiction.
+We conclude that the prices in menu M must be monotone non-decreasing
+in quality levels, as claimed.
+4.3
+An FPTAS for Revenue
+We are now ready to consider the problem of constructing an approximately
+revenue-optimal menu for the non-linear pricing problem. For this we will make
+one further technical assumption, which is that the derivative of the valuations
+v(q; θ) with respect to q is bounded at 0. I.e., there exists some λ > 0 such
+that
+d
+dqv(0; θ) < λ for all θ. In other words, buyers are not infinitely sensitive
+to product quantity. In the context of certification, this is implied by having
+d
+dqf(0; φ) < λ for all φ, meaning that consumers are not infinitely sensitive to
+quality at 0.
+Assumption 4.6. There exists some λ > 0 such that
+d
+dqv(0; θ) < λ for all θ.
+Given this assumption, the following result provides an FPTAS for the op-
+timal menu with k items. We also show that by taking k = 1/ϵ one can obtain
+an FPTAS for the optimal (unrestricted) menu.
+16
+
+Theorem 4.7. A menu with revenue at least OPT −λϵ can be found in time
+polynomial in λ and 1/ϵ. The menu can optionally be constrained to contain at
+most k quality levels, in which case OPT is the optimal revenue achievable with
+at most k quality levels.
+Our first step to proving Theorem 4.7 is to show that we can restrict attention
+to menus with at most 1/ϵ entries at only a small loss of revenue.
+Lemma 4.8. For any monotone menu M, there exists a menu M ′ of size at
+most O(1/ϵ) such that Rev(M ′) ≥ Rev(M) − O(ϵ).
+Proof. Fix the revenue-optimal menu M = {(qi, pi)}i∈Λ. By Lemma 4.5 we can
+assume M is monotone.
+We define M ′ to be the following subset of M. Let A = {ℓϵ : 0 ≤ ℓ ≤ ⌊1/ϵ⌋}.
+Then for each a ∈ A, we add to M ′ the element (q, p) from M with smallest p
+such that p ≥ a. Then we note that M ′ contains at most ⌈1/ϵ⌉ items. Moreover,
+for each element (q, p) ∈ M there is some (q′, p′) ∈ M ′ such that p′ ≥ p − ϵ.
+Any element in M ′ will still be purchased by the types that purchased that
+element in M, as the menu of alternative options has only gotten smaller. Since
+prices are monotone in quantity, and quantities purchased are monotone in buyer
+type, we can conclude that any type that was purchasing (q, p) in the original
+menu M will then purchase an element with price at least p − ϵ in menu M ′.
+The total loss in revenue conditional on any given buyer type is therefore at
+most ϵ, and hence the total revenue loss is at most ϵ as well.
+Next, we show that we can discretize the possible (quality, price) pairs that
+appear in our menu without losing too much revenue.
+Lemma 4.9. For any monotone menu M of size k, there exists a menu M ′
+such that
+• M ′ has at most k elements,
+• for each (q, p) ∈ M ′, p is a multiple of ϵ and q = ϵ(1+ϵ)ℓ for some integer
+ℓ ≥ 0, and
+• Rev(M ′) ≥ Rev(M) − O((k + λ)ϵ).
+Proof. Let us first give the high-level idea for the construction, which is illus-
+trated in Figure 1. We will round each (quantity, price) pair on menu M in
+three steps. First, we will round each quantity down to an appropriate dis-
+cretized grid, and then also lower the corresponding price to keep constant the
+ratio of quantity to price. The concavity of the value functions will then imply
+that buyer utilities cannot be reduced by too much, multiplicatively, as a result
+of this change. This first step discretized the quantities; in the second step we
+discretize prices by rounding each price down to an appropriate grid, which can
+only increase utilities. At this point we would be almost done, except for one
+complication: we must make sure that the (small) changes in buyer utility we in-
+duce do not result in buyers switching from more expensive items to significantly
+17
+
+Figure 1: Illustration of the discretization procedure from Lemma 4.9.
+Red
+curves denote buyer valuation functions.
+Menu item (p, q) is discretized in
+quantities to (˜p, q′) by shifting along a line to the origin, then discretized in
+price to (ˆp, q′). A final discount is applied to obtain the adjusted menu item
+(p′, q′).
+cheaper items from the menu. It is here where we use the price-monotonicity of
+the menu. Since the higher-quanitity items are the more expensive ones, in our
+third step we will provide discounts for the higher-quantity menu items, which
+will offset any utility perturbations due to discretization. These discounts are
+what lead to the loss term being proportional to kϵ, rather than ϵ, in our error
+bound.
+We now move on to the formal construction. Fix menu M of size k, so that
+M = {(q1, p1), . . . , (qk, pk)} where q1 < q2 < . . . < qk. We can assume without
+loss of generality that p1 ≤ p2 ≤ . . . ≤ pk, and that each element of M is
+purchased with positive probability.
+We construct a new menu M ′ in the following sequence of steps. First, for
+each (qi, pi) with qi ≥ ϵ, let q′
+i be qi rounded down to the nearest value of ϵ(1+ϵ)ℓ
+where ℓ ≥ 0 is an integer. Then define ˜pi = pi × (q′
+i/qi), noting that we chose ˜pi
+so that ˜pi/q′
+i = pi/qi. This element (q′
+i, ˜pi) corresponds to rounding quantity but
+keeping the price to quantity ratio constant in our intuitive description above.
+For our second step, we take ˆpi to be ˜pi rounded down to the nearest multiple
+of ϵ. Finally, in our third step, we introduce our discounts. For this we define
+p′
+i = ˆpi − 3iϵ, noting that the difference between p′
+i and ˆpi is increasing in i.
+This completes our discretization procedure, so we add (q′
+i, p′
+i) to menu M ′. We
+note that menu M ′ is not necessarily monotone, may contain elements that are
+preferred by no types, and may contain elements with negative prices.
+We claim that for every type θ, if θ purchased item (qi, pi) in menu M with
+qi ≥ ϵ, then θ will purchase some (q′
+j, p′
+j) in menu M ′ such that j ≥ i. To see
+why, first consider what would happen if (q′
+i, ˜pi) were on the menu. What is
+u((q′
+i, ˜pi); θ)? Since valuation function v is concave and v(0; θ) = 0, we must
+18
+
+price
+3E
+2e:
+ (p,q)
+(p,q').
+(p,q').
+E
+quantity
+E E(1 +e) e(1+)2
+(3 + L)3
+-2E-
+(p',q').have v(q′
+i; θ) ≥ q′
+i
+qi v(qi; θ). But since q′
+i ≥
+1
+1+ϵqi, this implies
+u((q′
+i, ˜pi); θ) = v(q′
+i; θ) − ˜pi
+≥ q′
+i
+qi
+v(qi; θ) − ˜pi
+= q′
+i
+qi
+(v(qi; θ) − pi)
+≥
+1
+1 + ϵu((qi, pi); θ)
+≥ u((qi, pi); θ) − ϵ.
+Since we also know that ˜pi − ϵ ≤ ˆpi ≤ ˜pi and p′
+i = ˆpi − 3iϵ, we have
+u((q′
+i, p′
+i); θ) = u((q′
+i, ˆpi); θ) + 3iϵ
+≥ u((q′
+i, ˜pi); θ) + 3iϵ
+≥ u((qi, pi); θ) + (3i − 1)ϵ.
+On the other hand, for any j < i, the utility of purchasing (q′
+j, ˜pj) is at most
+the utility of purchasing (qj, ˜pj), which is at most ϵ more than the utility of
+purchasing (qj, pj) (since the ratio between pj and ˜pj is no greater than (1+ϵ)).
+We therefore have
+u((q′
+j, p′
+j); θ) = u((q′
+j, ˆpj); θ) + 3jϵ
+≤ u((q′
+j, ˜pj); θ) + (3j + 1)ϵ
+≤ u((qi, pi); θ) + (3j + 2)ϵ.
+Since we know that u((qi, pi); θ) ≥ u((qj, pj); θ) by assumption that θ purchases
+item (qi, pi) from menu M, we conclude that u((q′
+i, p′
+i); θ) ≥ u((q′
+j, p′
+j); θ) as well,
+since (3j + 2) ≤ (3i − 1) for i > j.
+We conclude that each type θ that purchases (qi, pi) from M with qi ≥ ϵ
+will purchase a menu item (q′
+j, p′
+j) from M ′ such that j ≥ i. Since prices are
+monotone in menu M, we conclude that the total loss in revenue can be at most
+the difference in price between pi and p′
+i for any i. This is at most O(kϵ).
+Finally, consider a type θ that purchases (qi, pi) from M with qi < ϵ. By
+Assumption 4.6, the maximum willingness to pay for any agent for quality level
+ϵ is λϵ.
+These types therefore generate revenue at most λϵ, thus regardless
+of their purchase behavior they account for a total loss in revenue of at most
+O(λϵ).
+With Lemma 4.9 in hand, we can complete the proof of Theorem 4.7 by
+employing dynamic programming to determine the revenue-optimal mechanism
+with a given maximum-quality entry. One subtlety in the construction is that
+we must be careful to account for potential cannibalization by higher-quality
+elements in the menu. We handle this by insisting that the menu we construct
+contains only elements that are selected by a non-zero measure of buyer types,
+and we check this condition when recursively applying the dynamic program.
+19
+
+Proof of Theorem 4.7. We show how to compute the optimal menu with qual-
+ities and prices chosen from a discrete indexed set of possible options, using
+dynamic programming. In general, given a quantity q and price p that lie in
+our discrete set of options, we will use Q and P to denote the integer indexing
+of q and p, respectively.
+Given any choice of Q and P and some k ≥ 1, write M[Q, P, k] for the
+optimal revenue that can be obtained using a menu with at most k elements,
+of which the one with highest quality is the one indexed by Q and P. We will
+also write L[Q, P, k] for the lowest type θ that purchases quality level Q in this
+optimal menu. We can compute M[Q, P, k] and L[Q, P, k] recursively as follows.
+If k = 1 then M[Q, P, k] is precisely p times the probability that v(q; θ) ≥ p,
+and L[Q, P, k] is precisely the infimum of types θ for which v(q; θ) ≥ p.
+For k > 1, we will consider all possible options for the next-highest quality
+level on the menu given our discretization, say (q′, p′) with Q′ < Q. For each
+choice of (Q′, P ′), we let θ(Q′, P ′) denote the type that is indifferent between
+menu items (Q′, P ′) and (Q, P), if any. Recall from the single-crossing condition
+that this choice of θ(Q′, P ′) is unique if it exists. If there is no such θ(Q′, P ′),
+then we disqualify menu item (Q′, P ′) from consideration. Otherwise, we con-
+sider L[Q′, P ′, k−1], the lowest type that purchases element (Q′, P ′) in the opti-
+mal menu with highest quality level Q′ at price P ′. If L[Q′, P ′, k−1] ≥ θ(Q′, P ′),
+then again we disqualify menu item (Q′, P ′) from consideration, as this means
+that the optimal menu containing menu item (Q, P) does not include any types
+that would purchase menu item (Q′, P ′).
+Otherwise, we have that L[Q′, P ′, k−1] < θ(Q′, P ′). We can therefore calcu-
+late the revenue from the optimal menu with highest and second-highest quality
+levels (Q, P) and (Q′, P ′) as R(Q′, P ′) = M[Q′, P ′, k − 1] + (P − P ′)Pr[θ >
+θ(Q′, P ′)]. That is, the additional revenue gain or loss due to including menu
+item (Q, P) is (P − P ′)Pr[θ > θ(Q′, P ′)], the difference due to agents with type
+greater than θ(Q′, P ′) switching from menu item (Q′, P ′) to menu item (Q, P).
+Finally, consider also the revenue that would be obtained by using only
+menu item (Q, P); call this R. If all potential choices of (Q′, P ′) were elim-
+inated or if R > R(Q′, P ′) for all potential choices of (Q′, P ′), then we set
+M[Q, P, k] = R and set L[Q, P, k] to be the infimum type θ such that v(Q; θ) ≥
+P. This corresponds to the case that the optimal menu contains only the el-
+ement (P, Q). Otherwise, let (Q′, P ′) be the choice that maximizes R(Q′, P ′),
+which by assumption is larger than R. Then we take L[Q, P, k] = θ(Q′, P ′) and
+M[Q, P, k] = R(Q′, P ′).
+We conclude that we can fill tables M and L, with each entry taking time
+proportional to ϵ−2 (the time needed to consider every possible choice (Q′, P ′)).
+As there are kϵ−2 entries in total, the total time to fill the tables is at most
+kϵ−4. The revenue-optimal mechanism with at most k menu items can then be
+obtained by taking the maximum of M[Q, P, n] over all choices of Q and P.
+Finally, by Lemma 4.8, we can take k = 1/√ϵ and our dynamic program will
+obtain a menu M such that Rev(M) is at most O(λ/√ϵ) less than the optimal
+revenue. An appropriate change of variables, taking k = 1/ϵ and discretizing to
+multiples of ϵ2, then implies that our resulting menu is at most O(λϵ) less than
+20
+
+that of the optimal menu.
+5
+Welfare Maximization
+We have studied the problem faced by a revenue-maximizing certifier. But what
+about a certifier that wishes to maximize the welfare enjoyed by the consumers
+and producers? We could think of such a certifier as a government agency who
+is offering certification services not to generate profit, but rather to maximize
+the efficiency of production and trade.
+We define the welfare of a market outcome as the sum of utilities of the
+consumers, the producers, and the certifier, taking into account all transfers
+between parties. Given a menu M of options provided by the certifier, we will
+write Wel(M) for the welfare that results in the unique market outcome resulting
+from menu M.
+An immediate consequence of Corollary 3.4 is that the welfare-optimal choice
+of menu is to offer all possible certification levels, at the cost of verification c.
+Theorem 5.1. The welfare-optimal menu of certificates offers every possible
+certification level q > 0 at a cost of c, and level 0 at a cost of 0.
+Proof. If all quality levels were visible, the welfare-maximizing outcome is would
+be for each producer ψ to trade with consumer φ(ψ) at whichever quality level
+q maximizes their gains from trade f(q; φ(ψ)) − g(q; ψ). However, since quality
+levels are hidden, producers and consumers can trade at a positive level of
+quality only if the cost of verification is paid. So if the maximum gains from
+trade is less than c, then it is preferable to trade at level 0. However, we observe
+that this is precisely the outcome implemented at equilibrium from the proposed
+certification menu, so it must be welfare-optimal over all possible menus.
+The welfare-optimal menu described above includes an arbitrarily large num-
+ber of certificates. In practice it may be helpful to find a welfare-optimal slate of
+at most k certificates. It turns out that a minor variation on the dynamic pro-
+gram described in the previous section can be used to compute an approximately
+welfare-optimizing slate, under Assumption 4.6.
+Theorem 5.2. Let M be the welfare-maximizing menu with at most k elements.
+A menu with M ′ with at most k elements and such that Wel(M ′) ≥ Wel(M) −
+O(λϵ) can be found in time polynomial in 1/ϵ.
+Proof. The proof is very similar to the one for Theorem 4.7, and strictly simpler,
+so we only briefly describe the differences here. First, it is without loss to restrict
+attention to menus that post price c for every non-trivial level of quality.
+Given any such menu M, we can discretize potential levels of quality by
+rounding down to the nearest multiple of ϵ. By Assumption 4.6 this reduces
+welfare by at most λϵ, as the welfare from each menu item is reduced by at most
+this much and each producer selects her gains-from-trade-maximizing element
+from the menu.
+21
+
+Given such a discretization, one can express the welfare-optimal menu re-
+cursively via dynamic programming as in Theorem 4.7, with the simplification
+that we need only index on quality rather than (quality, price) pairs (since all
+prices will be set to c). Rather than defining M[Q, k] (and respectively L[Q, k])
+to be the maximum revenue of a menu with k elements and maximum quality
+indexed by Q, it will be the maximum welfare of a menu with k elements and
+maximum quality indexed by Q. Our method of recursively computing M[Q, k]
+(and L[Q, k]) then remains nearly unchanged relative to Theorem 4.7.
+The
+only change to note is the actual welfare calculation, relative to the revenue
+calculation. In Theorem 4.7 we used that the revenue obtained when agents of
+type θ > θ′ purchase certificate q at price p is p times Pr[θ > θ′]. In contrast,
+the welfare obtained when producers of type ψ > ψ′ all purchase certificate q
+at price c can be calculated in closed form as
+� ψ′
+ψ (f(q; φ(η)) − g(q; η) − c)dη.
+Substituting this welfare calculation for the revenue calculation completes the
+necessary changes.
+Is the revenue-maximizing choice of menu also approximately welfare-maximizing?
+As it turns out, the welfare that results from the certifier’s reveneu-optimal menu
+can be an arbitrarily small fraction of optimal welfare. This is inherited from
+standard examples of monopolistic distortion in the linear settings, where a mo-
+nopolist might be incentivized to sell much less of their good (i.e., certification)
+than what would be efficient in order to inflate prices.
+Proposition 5.3. There is a sequence of instances for the problem where welfare
+in the revenue-optimal solution is an arbitrarily small fraction of first-best wel-
+fare. The approximation can be as bad as Ω(log H), where H = maxθ1,θ2
+maxq v(q;θ1)
+maxq v(q;θ2)
+is the ratio between the highest and lowest maximum values across buyer types.
+Proof. Take c = 0 and suppose f(q; φ) = φq and g(q; ψ) = 0 for all q and ψ. Our
+distribution over consumer types φ is an equal-revenue distribution supported
+on [1, H]; that is, F(φ) = 1 − 1
+φ for all φ ∈ [1, H]. Recalling Observation 4.2, it
+is revenue-optimal for the certifier to offer contract q = 1 at a price of H, for
+an expected welfare (and revenue) of 1. However, the optimal welfare, log(H),
+can be achieved by offering contract q = 1 at price 0.
+However, one implication of our equilibrium analysis is that adding addi-
+tional certification options to a menu of certificates cannot reduce the sum of
+utilities of the consumers and producers, regardless of the prices selected. One
+interpretation of this is that welfare cannot be harmed by a revenue-maximizing
+certifier entering a market for certification in which some certification options
+are already available.
+Proposition 5.4. Consider two certification menus M and M ′ with M ⊆ M ′.
+Then Wel(M ′) − Rev(M ′) ≥ Wel(M) − Rev(M).
+Proof. If producer ψ selects option (q, p) from certification menu M, then the
+welfare generated for the producer ψ and corresponding consumer φ, less the
+revenue raised by the certifier, is f(q; φ(ψ)) − g(q; ψ) − p. By Corollary 3.4,
+22
+
+producer ψ purchases precisely whichever menu item from M maximizes this
+quantity.
+Providing additional items can therefore only increase the welfare
+jointly enjoyed by producer type ψ and corresponding consumer φ(ψ). As this
+holds pointwise for every ψ, it holds in aggregate over all types as well.
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+25
+

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+Classical Algorithm for the Mean Value problem over
+Short-Time Hamiltonian Evolutions
+Reyhaneh Aghaei Saem1 and Ali Hamed Moosavian1
+1Phanous
+Abstract
+Simulating physical systems has been an important application of classical and quantum computers. In
+this article we present an efficient classical algorithm for simulating time-dependent quantum mechanical
+Hamiltonians over constant periods of time. The algorithm presented here computes the mean value of an
+observable over the output state of such short-time Hamiltonian evolutions. In proving the performance of
+this algorithm we use Lieb-Robinson type bounds to limit the evolution of local operators within a lightcone.
+This allows us to divide the task of simulating a large quantum system into smaller systems that can be
+handled on normal classical computers.
+1
+Introduction
+Understanding physical systems with quantum mechanical interactions have been an increasingly challenging
+and important task in the past century. Because nature is inherently quantum, many naturally occurring or
+artificially implemented phenomena cannot be explained without quantum mechanics. Some early examples
+include understanding atoms [1], diatomic molecules [2], the Meissner effect in superconductors [3] and
+black-body radiation [4]. With the advent of better theoretical and computational tools, the list of phenomena
+that have been shown to require quantum mechanics has grown enormously in the past few decades. To name
+a few, some of the more notable examples include photosynthesis [5], topological ordered phases such as
+fractional quantum Hall effect [6], isomerization of diazene [7] and non-Abelian lattice gauge theories [8].
+Indeed, because of the challenging nature of simulating quantum systems on classical computers, one of the
+earliest motivations for quantum computers has been to use them to study other quantum systems [9].
+Besides the obvious practical applications, the problem of simulating quantum systems has some critical
+complexity theory value too. On the one hand, it is known that the problem of simulating several quantum
+systems belongs to the Bounded-error Quantum Polynomial-time complete (BQP-complete) class [10, 11,
+12]. Also, there are specific problems where the complexity class of simulating them varies as the runtime
+increases, and exhibit a dynamical phase transition [13, 14].
+In this paper, we present a classical algorithm for computing the expectation value of an observable that
+can be written as a tensor product of local operators acting on each qubit. In the literature this problem is
+sometimes called the quantum mean value problem [15] (not to be confused with a similarly named open
+problem in mathematics [16]). Our algorithm evaluates the mean value of an operator where the state is
+generated by evolving a product state under a geometrically local time-dependent Hamiltonian for a short
+period of time. The setup of our problem is motivated by the state of current quantum technologies. On one
+1
+arXiv:2301.11420v1  [quant-ph]  26 Jan 2023
+
+hand, analog simulators lack fault-tolerance, which means they have a limited decoherence time, and on the
+other hand, we still do not have access to fault-tolerant universal quantum computers either.
+Suppose that we have a bounded-norm time-dependent Hamiltonian H(t) that acts on n-qubits. The
+initial state of the system is assumed to be a product state, typically |ψ(0)⟩ = |0⟩⊗n. The quantum mean value
+problem wants to compute the expectation value of an operator O with respect to |ψ(T)⟩. The expectation
+value is represented with µ:
+µ ≡ ⟨ψ(T)|O|ψ(T)⟩ .
+(1)
+We consider geometrically local time-dependent Hamiltonians that are defined on 2D or 3D lattices. 1D
+systems with gapped Hamiltonians have been thoroughly studied before [17, 18].
+The unitary evolution operator corresponding to this Hamiltonian can be written as:
+U(t) = T :
+�
+exp
+�
+− i
+¯h
+� t
+0
+H(t′)dt′
+��
+,
+(2)
+where T : [.] means the operators respect time ordering. Assuming the observable can be written as the tensor
+product of single-qubit Hermitian operators Oj acting on individual sites, O = O1 ⊗ O2 ⊗ · · · ⊗ On, we
+can write the mean value of the observable O as:
+µ = ⟨0n| U†(T)O1 ⊗ O2 ⊗ · · · ⊗ OnU(T) |0n⟩ .
+(3)
+In this paper, we introduce a classical algorithm that can approximate the mean value problem for time-
+dependent Hamiltonians with an additive error. The structure of the rest of this paper is as follows. In Sec. 2
+we give an overview of the algorithm. Section 3 explains how the dynamics of each operator is approximately
+restricted to within a lightcone. In Sec. 4 we analyze the problem of classically calculating the local unitary
+operators and compare different numerical algorithms for doing the task and in Sec. 5 we conclude.
+2
+Algorithm
+We provide a classical algorithm for approximating µ within an additive error, δ, for the special case of the
+mean value problem where the time-dependent Hamiltonian is geometrically local in 2D or 3D.
+Conceptually, the algorithm can be broken into two parts. First we use a Lieb-Robinson bound [19, 20,
+21] to limit the unitary evolution corresponding to each qubit within a lightcone. This allows us to classically
+compute the unitary operator. Then we follow the steps in [15] to divide the lattice into pseudo 1D slices, that
+can be efficiently simulated either using Matrix Product State algorithms [22, 23, 24] (for a nice review of
+these algorithms see [18]) or the algorithm ascribed to [15].
+Theorem 1 Let H(t) be a bounded-norm time-dependent lattice Hamiltonian that acts on n-qubits. Suppose
+the observable O is a tensor product of operators , O = O1 ⊗ O2 ⊗ · · · ⊗ On. For constant evolution times,
+there exists a classical algorithm that estimates µ within an additive error δ,
+|˜µ − µ| ≤ δ .
+(4)
+The error δ includes three different parts. The first contribution comes from the Lieb-Robinson bound where
+we used it to approximately limit the evolution within the lightcone of each qubit. Second, numerical methods
+such as trotterization are used to calculate the local unitary evolutions; these methods are not exact and incur
+some errors. The third part comes from the additive error in simulating a constant depth quantum circuit [15].
+In [15], they provide a classical algorithm for constant depth circuits in 2D or 3D which can approximate the
+2
+
+mean value to an additive error. In our work, the short-time evolution of a geometrically local Hamiltonian
+which is limited by the Lieb Robinson bound is comparable with the shallow quantum circuit with a constant
+depth.
+Suppose that the error of localizing the time evolution of the Hamiltonian, classical simulation of time-
+dependent unitaries and simulating shallow circuits are given by εLR, εCS and εSSC respectively. Then the
+total error is as follows:
+|˜µ − µ| ≤ εLR + εCS + εSSC = δ .
+(5)
+In Eq. (10), we will see how the first error is dictated by the simulation time, the lightcone radius and the
+Hamiltonian. One should note that εLR is bounded by εLR ≤ nε(LR)(L, T), where ε(LR)(L, T) is defined
+later in Eq. (10) as the Lieb Robinson error for a single site. Section 4 derives the dependency of the second
+error term on the simulation parameters.
+Algorithm 1: High level overview of the algorithm
+input :a √n × √n lattice of qubits, a geometrically local time-dependent Hamiltonian H(t), the
+operator O = O1 ⊗ O2 ⊗ · · · ⊗ On where each ∥Oj∥ ≤ 1, an upper bound for error δ, a
+simulation time T.
+output :an approximation for µ = ⟨ψ(T)|O|ψ(T)⟩, where |ψ(0)⟩ = |0⊗n⟩ and
+i¯h d
+dt |ψ(t)⟩ = H(t) |ψ(t)⟩
+1 Initialization:
+2 Use T and δ to calculate the lightcone radius, L, from the Lieb-Robinson bound.
+3 As in Fig. 1 partition the lattice into 4L × √n strips twice, let us call each set {Ai}i and {Bi}i. Also,
+define the sets
+�
+A0
+i
+�
+i and
+�
+B0
+i
+�
+i as the central part of the strips.
+4 Also, group sites into 2L × 2L super-sites to form a coarse-grained lattice.
+5 Calculations:
+6 for Ai ∈ {Ai}i do
+7
+initialize an MPS.
+8
+for Oj in A0
+i do
+9
+Use a classical ODE solver to calculate the local unitary operator Uj corresponding to Oj
+which includes the terms that are in the lightcone of the jth qubit.
+10
+Transform OjUj into a Matrix Product Operator.
+11
+Add the necessary sites from the current super-site to the active memory and apply the MPO
+on it.
+12
+Measure any sites that will no longer be needed.
+13
+Let us call the (not normalized) MPS outcome |ΨAi(T)⟩ .
+14 Repeat the Line 6 loop for Bjs.
+15 return ˜µ =
+��
+j ⟨ΨBj(T)|
+���
+i |ΨAi(T)⟩
+�
+The runtime of classical simulation of the time-dependent unitaries is related to the complexity of matrix
+multiplication. We can find the lightcone radius L from Section 3 and then find the number of qubits m in
+each lightcone which is O
+�
+L2�
+. For most practical cases, the fastest matrix multiplication algorithm is the
+famous Strasson algorithm with asymptotic complexity of O
+�
+(2m)log2 7�
+[25], however, the best known
+asymptotic complexity for matrix multiplication is O
+�
+(2m)2.373�
+[26]. For most physical Hamiltonians
+3
+
+Figure 1: A: It shows the relationship between simulation time and the lightcone radius. B:shows the sites
+inside the lightcone for various lightcone radii. C and D:show the set of strips {Ai}i and {Bi}i as well as
+the
+�
+A0
+i
+�
+i and
+�
+B0
+i
+�
+i sets on a 2D grid. The yellow squares represent the Oi operators that are to be applied
+on either set of strips.
+where we have translational symmetry in the bulk, we only need to calculate the unitary evolution matrix for
+each geometrical configuration of the sites once. This means that the complexity of this part of the algorithm
+would be O
+�
+22.373m�
+. But for generic Hamiltonians the number of configurations could grow linearly with
+the system size and the complexity would be O
+�
+n22.373m�
+.
+For 2D and 3D systems with n qubits, the runtime of simulating a shallow circuit is related to the last error
+term with O
+�
+nL226L2/(εSSC)2�
+and O
+�
+nL326L2n1/3/(εSSC)2�
+respectively [15]. We have provided a
+high-level overview of the 2D algorithm in Algorithm 1. Consequently the general total complexity of the algo-
+rithm for 2D and 3D systems is O
+�
+n22.373m + nL226L2/(εSSC)2�
+and O
+�
+n22.373m + nL326L2n1/3/(εSSC)2�
+respectively.
+4
+
+B.
+1.00000
+0.10000
+lerror
+0.01000
+0.00100
+Simulation time
++0.01
++ 0.02
+0.00010
++0.03
++0.04
+0.00001
+0
+2
+Lightcone radius3
+Local Unitary Operators
+According to [20], we know that a local operator OA which is defined inside a region A, remains local after a
+short-time evolution under a local Hamiltonian. Suppose that the Hamiltonian has the form
+H(t) =
+�
+e
+ue(t)he ,
+(6)
+where he acts non-trivially only on the two vertices of edge e of the graph G. Suppose that ∥ue(t)he∥ ≤ g
+for 0 ≤ t ≤ T and the maximum degree of the graph to be ∆. The Hamiltonian for terms in region A and the
+set of vertices in the L-boundary of it has the form
+HA(t) =
+�
+e⊂A∪∂L(A)
+ue(t)he .
+(7)
+The time evolution operator of this local Hamiltonian acts non-trivially only on the region A and its L-
+boundary. The time evolution operator is given by
+V(t) = T :
+�
+exp
+�
+− i
+¯h
+� t
+0
+HA(t′)dt′
+��
+.
+(8)
+This is known as Dyson series and T : [.] represents time-ordering [27, p. 551]. According to [20] the
+following Lieb-Robinson bound holds:
+||U†(T)OAU(T) − V†(T)OAV(T)|| ≤ ε(LR)(L, T) ,
+(9)
+where ε(LR)(L, T) is defined as:
+ε(LR)(L, T) =
+�
+2
+π |A| ||OA|| exp
+�
+−L(log L − log T − log(4g(δ − 1))) − 1
+2 log L
+�
+.
+(10)
+For each local operator in O = O1 ⊗ O2 ⊗ · · · ⊗ On, we only consider the Hamiltonian terms inside the
+lightcone of it. We replace the global unitary that acts on the entire system with these local unitary operators
+and apply the algorithm in Sec. 2 to approximate the mean value.
+The only remaining problem would be to find a classical algorithm for classically calculating the local
+unitary operators V(t). We analyze different approaches for doing so in Sec. 4.
+4
+Classical Simulation of the time-dependent unitaries
+4.1
+Trotterization
+Theorem 2 Let HA(t) be a bounded-norm time-dependent Hamiltonian and OA an observable inside region
+A. Let V(t) be the unitary evoluion of HA(t). We can approximate the operator V†(T)OAV(T) with
+W†(T, N)OAW(T, N) where W(t, N) is defined as
+W(t, N) =
+N= t
+δt
+�
+j=1
+exp
+�
+− i
+¯hδt HA(jδt)
+�
+,
+(11)
+5
+
+such that
+��W†(T, N)OAW(T, N) − V†(T)OAV(T)
+�� ≤ 6T 2
+N¯h ∥OA∥ ∥H′
+A(t∗)∥ = εCS,
+(12)
+where
+∥H′
+A(t∗)∥ = max
+0≤t≤T ∥H′
+A(t)∥.
+(13)
+This is a well-known textbook result that can be found in the literature. For instance see chapter IX of [28].
+4.2
+Numerical Differential Equation Solvers
+Another approach for approximating the unitary time evolution would be to derive a differential equation
+from Schrodinger’s equation and solve that numerically by using a suitable classical algorithm.
+i¯h d
+dt |ψ(t)⟩ = H(t) |ψ(t)⟩ ,
+i¯h d
+dtU(t) |ψ(0)⟩ = H(t)U(t) |ψ(0)⟩ ,
+i¯h d
+dtU(t) = H(t)U(t) .
+(14)
+If there are m qubits inside the lightcone, the matrix representation of U and H will be 2m ×2m operators
+and Eq. (14) will constitute a set of Ordinary Differential Equations (ODEs). The upside of using a classical
+ODE solver is that they can consistently attain much lower errors than what is possible from Eq. (11) or a
+higher order Suzuki-Trotter solution [30, 31] that is fine-tuned for a time-dependent problem [32, 33]. The
+downside is that they are typically orders of magnitude slower and require more memory than the straight
+forward trotterization as in Eq. (11). Assuming our lightcone is small enough, many numerical ODE solvers
+will be able to handle Eq. (14). See Fig. 2 for a comparison between multiple different ODE solvers.
+5
+Conclusions
+To conclude we have provided a classical algorithm for the mean value problem on outcomes of short time
+dependent Hamiltonian evolutions. These mean values are typically used in other algorithms such as the
+Variational Quantum Algorithm [34].
+The quantum mean value problem, almost by definition, belongs to the BQP-complete class for polynomial
+times. So, naturally we do not expect to be able to solve the polynomial time problem on a classical computer
+efficiently. Nonetheless, an important open question would be to find the minimum simulation time for getting
+a quantum speedup. The current work shows us that in order to benefit from the quantum speedup we need
+simulation times that are at least greater than constant [35]. This is in accordance with a wide variety of other
+results that target specific problems too [15, 36, 13, 37, 38, 20]. In general the problem of mapping out the
+entire dynamical complexity phase diagram is theoretically interesting.
+Another direction for future research would be to improve or generalize the current algorithm. If the
+algorithm cannot be further improved or generalized, then proving these limitations is another open question.
+6
+
+Figure 2: A comparison of different ODE solvers. The top figure shows average minimum time used by each
+ODE solver to solve Eq. (14) for different number of qubits. The bottom plot shows the average memory
+used by each solver to solve the same differential equation. The benchmark was done on a personal PC and
+each data point was repeated at least 100 times over 20 randomly generated time-dependent Hamiltonians.
+All of the methods except Trotter were picked from Julia’s DifferentialEquations.jl roster of state of the art
+ODE solvers [29], and compared to the Trotter method, they consistently had at least 10 orders of magnitude
+less error of the form Eq. (12). The Trotter solution was also implemented in Julia, and used N = 30.
+7
+
+1.0×109
+Solver
+(su)
+1.0×108
+Trotter
+runtime (
+1.0×107
+MMidpoint
+1.0×106
+MLeapfrog
+1.0×105
+1
+2
+3
+4
+5
+6
+7
+MGL4
+1.0×1010
+MNC6
+1.0×109
+(bytes)
+MGL6
+1.0×108
+memory
+1.0×107
+MNC8
+1.0×106
+MGL8
+1.0×105
+1
+2
+3
+4
+5
+6
+7
+Number of Sites6
+Acknowledgements
+We would like to thank Salman Beigi for overseeing this project and commenting on the draft. We also thank
+Leila Taghavi and Erfan Abedi for helpful discussions.
+References
+[1]
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+arXiv:2301.01513v1  [hep-th]  4 Jan 2023
+HU-EP-23/01
+Remarks on conformal invariants for
+piecewise smooth curves and Wilson loops
+Harald Dorn 1
+Institut f¨ur Physik und IRIS Adlershof, Humboldt-Universit¨at zu Berlin,
+Zum Großen Windkanal 6, D-12489 Berlin, Germany
+Abstract
+This short note is some obvious mathematical addendum to our papers on Wilson
+loops on polygon-like contours with circular edges [1, 2].
+Using the technique of
+osculating spheres and circles we identify the conformal invariants characterising the
+kinks (cusps) of generic piecewise smooth curves in 3-dimensional space.
+[email protected]
+
+1
+Introduction
+The metrical invariants of smooth curves in Euclidean D-dimensional space are length,
+curvature and (D −2) torsion parameters as functions along the curve. For piecewise
+smooth curves with cusps 2, in addition each cusp is characterised by the Euler angles
+specifying the orthogonal transformation needed to rotate the Frenet-Serret frames
+on both sides of a cusp to one another.
+What concerns the issue of conformal invariants, there is for smooth curves a
+considerable amount of mathematical papers. In analogy to the metrical invariants
+length s, curvature κ and torsion τ one finds for 3-dimensional curves, see e.g. [3],
+conformal length ω,3
+dω = √νds,
+ν =
+�
+(κ′)2 + κ2τ 2 ,
+(1)
+conformal curvature Q
+Q = 4(ν′′ − κ2ν)ν − 5(ν′)2
+8ν3
+,
+(2)
+and conformal torsion T
+T = 2(κ′)2τ + κ2τ 3 + κκ′τ ′ − κκ′′τ
+ν
+5
+2
+.
+(3)
+Use of these invariants has been made in the physical literature to characterise the
+boundary conditions for the treatment of minimal surfaces in AdS via Pohlmeyer
+reduction [4].
+But what concerns the extension to piecewise smooth curves, we did not find any
+paper in the mathematical literature. Therefore, we decided to write up what one gets
+by straightforward application of one of the various techniques used for the smooth
+case: the kinematics of osculating spheres, see e.g. [5–8] and refs. therein.
+This note is organised as follows. In the next section we find formulas expressing
+the conformal invariants for cusps, with legs made of generic smooth pieces of curves,
+in terms of their metrical invariants. Then in section 3 we consider the very special
+curves studied in our papers [1,2], i.e. polygon-like curves with circular edges. These
+curves need a separate discussion, since along the circular edges all the conformal
+invariants (1),(2),(3) are zero or ill-defined.
+2
+Conformal invariants of generic cusps
+In 3-dimensional space the osculating circle (sphere ) at a generic point x on a smooth
+curve is a circle (sphere) having contact of second (third) order with the curve at x.
+Since the order of contact is preserved under conformal transformations, osculating
+circles (spheres) at a given point of a curve are mapped to those for the image under
+2We follow the language in the Wilson loop literature and use the word cusp for a kink with
+nonzero opening angle.
+3The prime denotes derivative with respect to s.
+1
+
+a conformal map. Let us denote by ⃗t,⃗n,⃗b the unit tangent, normal and binormal
+vectors at x. Then the center a of the osculating circle S1 is given by
+a = x + 1
+κ ⃗n
+(4)
+and the center c of the osculating sphere S2 by
+c = x + 1
+κ ⃗n −
+κ′
+τκ2 ⃗b .
+(5)
+We now turn to the case where x is a point of discontinuity, i.e. the tip of a cusp.
+The cusp is then characterised by two osculating circles S1
+−, S1
++ and two osculating
+spheres S2
+−, S2
++. The index ” ± ” indicates the limits one gets by approaching x along
+the both respective legs of the cusp.
+Let us first count the number of conformal invariants we can expect for the cusp.
+There are 9 metrical invariants at hand: κ±, κ′
+±, τ± and the 3 Euler angles for the
+rotations from the Frenet frame {⃗t−,⃗n−,⃗b−} to {⃗t+,⃗n+,⃗b+}. The difference of the
+numbers of parameters of the 3-dimensional conformal and isometry group is equal
+to 4. This should result in 9 − 4 = 5 conformal parameters. Now of course two
+of them are the corresponding limits of the differential of the conformal length (1).
+Hence there remain 3 conformal parameters to be attributed genuinely to the cusp.
+To find explicit formulas for them, we start with the conformal invariants of pairs
+of spheres (Sm, Sn) of the same or of different dimension. There is a rigorous math-
+ematical treatment for arbitrary dimensions in ref. [9]. Applied to our case it means
+that to each of the pair (S1
+−, S1
++), (S1
+−, S2
++), (S2
+−, S1
++), (S2
+−, S2
++) belongs just one con-
+formal parameter. To proceed, we first study these four invariants and will show
+afterwards, that only three of them are independently.
+(S1
+−, S1
++)
+The tangents of the osculating circles agree with those of the curve.
+Therefore the related conformal invariant is
+A11 = ⃗t−⃗t+ =
+− cos α .
+(6)
+Here α is the cusp angle (understood as the opening angle, i.e. α = π in the smooth
+case).
+(S2
+−, S2
++)
+Now the conformal invariant is given by the so-called inversive product,
+see e.g. [5,9]
+A22 = R2
+− + R2
++ − (c− − c+)2
+2R−R+
+= (c− − x)(c+ − x)
+R−R+
+.
+(7)
+R− and R+ are the radii of S2
+− and S2
++. Obviously A22 is the cosine of the angle
+between the vectors pointing from x to the centers of the two osculating spheres.
+Strictly speaking, only its absolute value is invariant, since A22 changes sign under
+those special conformal transformations for which the preimage of infinity is situated
+inside just one of the spheres. The same comment applies to A12 and A21 below.
+(S2
+−, S1
++) and (S1
+−, S2
++) For a sphere and an intersecting circle the conformal invariant
+2
+
+is the scalar product of the unit vector pointing from x to the center of the sphere
+with the unit tangent of the circle
+A12 =
+⃗t−(c+ − x)
+R+
+,
+A21 = (c− − x) ⃗t+
+R−
+.
+(8)
+Using (4),(5) for both legs of the cusp we get from (7) and (8)
+A22 = κ−τ−κ+τ+ ⃗n−⃗n+ + κ′
+−κ′
++ ⃗b−⃗b+ − κ−τ−κ′
++ ⃗n−⃗b+ − κ+τ+κ′
+− ⃗b−⃗n+
+�
+κ2
+−τ 2
+− + κ′2
+−
+�
+κ2
++τ 2
++ + κ′2
++
+,
+(9)
+A12
+=
+κ+τ+ ⃗t−⃗n+ − κ′
++ ⃗t−⃗b+
+�
+κ2
++τ 2
++ + κ′2
++
+,
+(10)
+A21
+=
+κ−τ− ⃗t+⃗n− − κ′
+− ⃗t+⃗b−
+�
+κ2
+−τ 2
+− + κ′2
+−
+.
+(11)
+All the scalar products in the above formulas can be expressed in terms of three Euler
+angles ϕ, ϑ, ψ needed to rotate the Frenet frame {n−, b−, t−} to {n+, b+, t+}. Then
+we get
+A11 = ⃗t−⃗t+ = cosϑ ,
+i.e. ϑ = π − α ,
+(12)
+A22
+=
+1
+�
+κ2
+−τ 2
+− + κ′2
+−
+�
+κ2
++τ 2
++ + κ′2
++
+�
+κ−τ−κ+τ+ (cosϕcosψ − sinϕsinψcosϑ)
++κ′
+−κ′
++(cosϕcosψcosϑ − sinϕsinψ) + κ−τ−κ′
++ (sinϕcosψ + cosϕsinψcosϑ)
+−κ+τ+κ′
+− (cosϕsinψ + sinϕcosψcosϑ)
+�
+,
+(13)
+A12
+=
+sinϑ
+�
+κ2
++τ 2
++ + κ′2
++
+�
+κ+τ+ sinϕ − κ′
++ cosϕ
+�
+,
+(14)
+A21
+=
+sinϑ
+�
+κ2
+−τ 2
+− + κ′2
+−
+�
+κ−τ− sinψ + κ′
+− cosψ
+�
+.
+(15)
+The part of A22 containing the factor cosϑ is related to the product of A12 and A21
+in an obvious manner. With a bit more careful inspection one gets for the whole A22
+A22(ϕ, ϑ, ψ) =
+1
+sin2ϑ
+�
+A12(ϕ + π
+2 ) A21(ψ + π
+2 ) − cosϑ A12(ϕ)A21(ψ)
+�
+.
+(16)
+Now A12 and A21 for arguments shifted by π
+2 are not independent, but related by
+�
+A12(ϕ)
+�2 +
+�
+A12(ϕ + π
+2)
+�2 = (A21(ψ)
+�2 +
+�
+A21(ψ + π
+2 )
+�2 = sin2ϑ .
+(17)
+This means that a complete set of independent conformal invariants attributed to the
+cusp is given by the three parameters 4
+α ,
+B12 = κ+τ+ sinϕ − κ′
++ cosϕ
+�
+κ2
++τ 2
++ + κ′2
++
+,
+B21 = κ−τ− sinψ + κ′
+− cosψ
+�
+κ2
+−τ 2
+− + κ′2
+−
+.
+(18)
+4Remember ϕ, ϑ, ψ Euler angles, α = π − ϑ opening angle of the cusp.
+3
+
+Let us add a warning. Inserting by brute force ϕ = ψ = 0 into B12 and B21 one
+could reach the wrong conclusion that
+κ′
+√
+κ2τ 2+κ′2 could be an invariant for smooth
+curves. But since putting ϕ or ψ to zero is not a conformal invariant statement, this
+conclusion is not allowed and also straightforwardly proven to be wrong.
+We continue with some casual comment on the conformal invariants in 4-dimensio-
+nal space. To fix all the osculating spheres from S1 to S3 at a smooth point one needs
+κ, κ′, κ′′, τ1, τ ′
+1, τ2. This for the limits from both sides of the cusp, together with 6 Euler
+angles, needed in 4D for the rotation of the Frenet frame, gives 18 metrical parameters.
+The difference of the number of parameters oft the conformal and isometry group is
+now 5, hence we tentatively reach 13 conformal parameters. Now on both sides the
+limits of the differential of the conformal length and the first conformal torsion5 are
+not related to the cusp. Hence we can expect 13 − 4 = 9 conformal parameters to be
+attributed genuinely to the cusp.
+On the other side, among the nine pairs of osculating spheres
+(Si
+−, Sj
++), i, j = 1, 2, 3, the pairs (S2
+−, S2
++), (S2
+−, S1
++), (S1
+−, S2
++) have two invariants and
+all other only one [9]. This yields 12 conformal parameters. We keep it as an open
+question, whether there are indeed three relations of the 3D type (16),(17) to reach
+the minimal number of 9 independent parameters seen in the previous paragraph.
+We close this section with a comment on the cusp anomalous dimension for Wilson
+loops. It is generally believed to depend on the cusp angle α only, and therefore
+calculations have been done using straight edges. While in field theoretic perturbation
+theory this can be justified by power counting in the corresponding Feynman integrals,
+a rigorous proof for the generic situation at strong coupling is still lacking. We have
+presented a proof for the planar case with generically curved edges in [10]. For full
+generality in 3D, a proof of its independence from B12 and B21 is still lacking.
+3
+Comments on Wilson loops on polygon-like
+curves with circular edges
+The polygon-like curves with circular edges, whose related Wilson loops have been
+studied in our papers [1, 2], are not covered by the setting for generic curves as
+presented in the previous section. Along their edges one has constant curvature κ
+and zero torsion τ, resulting in zero conformal length (1) and undefined conformal
+curvature and torsion. In mathematical language these edges are conformal vertices.6
+Although such single edges carry no conformal data, their combination to a poly-
+gon does7 It has still no extension in the sense of conformal length, but there are of
+course the cusp angles at each cusp of the polygon and for more than 3 cusp points
+the corresponding cross ratios. In addition, like generic curves, these polygons can
+wind themselves out of a plane and exhibit torsion. In our paper [1] we have this
+issue parameterised by the introduction of torsion angles βj, defined at a given cusp
+5In 3D the conformal torsion (3) cannot be build out of the metrical parameters needed to fix all
+the osculating spheres at the cusp. But in 4D the osculating S3 inherits all the information.
+6See e.g. [3,5,6].
+7In a sense one could call it a conformal vertex with internal conformal substructure.
+4
+
+point xj by
+βj = ∡({xj, xj+1}, ccj) ,
+(19)
+where {xj, xj+1} denotes the circular edge between xj and xj+1 and ccj the circle
+fixed by the three cusp points xj−1, xj, xj+1.
+However, in contrast to B12 and B21 for cusps of generic curves, these torsion
+angles are not attributed to the local properties at the corresponding cusp. This is
+simply seen by changing in (19) the neighbouring cusp point xj+1 along the circle
+of which the edge {xj, xj+1} is a part . Then the tangent of the edge at xj remains
+the same, but the circle ccj and its tangent at xj changes. By this manipulation βj
+changes, although the local situation at the cusp at xj remains the same as before.
+We end with a remark on some setting for Wilson loops on piecewise smooth
+curves intermediate between those of full generality considered in section 2 and the
+polygons with circular edges in [1]. In conformal invariant gauge field theories as in
+e.g. N = 4 SYM there will be an anomalous conformal Ward identity of the type
+derived in [1], imposing for the Wilson loop the structure of a conformally covariant
+factor depending on the distances of the tips of the cusps times a conformally invariant
+remainder factor. In the generic case this remainder factor is a function of the cross
+ratios formed out of cusp points and the conformal cusp parameters identified in
+section 2, but in addition also a functional of the conformal invariants as functions
+along the edges.
+For the case of polygons with circular edges the remainder is a
+function of only a finite number of conformal parameters.
+To have a curve characterised by a finite number of conformal parameters but
+nevertheless having nonzero conformal length, one could consider polygons with edges
+which are pieces of curves with constant conformal curvature and torsion. Such curves
+have been classified in [11]. Among them are loxodromes on rotational surfaces.
+Acknowledgement
+I thank the Quantum Field and String Theory Group at Humboldt University for
+kind hospitality.
+5
+
+References
+[1] H.
+Dorn,
+“On
+anomalous
+conformal
+Ward
+identities
+for
+Wilson
+loops
+on
+polygon-like
+contours
+with
+circular
+edges,”
+JHEP
+03
+(2020),
+166
+doi:10.1007/JHEP03(2020)166 [arXiv:2001.03391 [hep-th]].
+[2] H. Dorn, “Wilson loops for triangular contours with circular edges,” J. Phys. A
+54 (2021) no.22, 225402 doi:10.1088/1751-8121/abe311 [arXiv:2010.14822 [hep-
+th]].
+[3] G. Cairns, R. Sharpe, and L. Webb, “Conformal Invariants for Curves and Sur-
+faces in Three Dimensional Space Forms”, Rocky Mountain J. Math. Volume 24,
+Number 3 (1994), 933-959.
+[4] Y. He, C. Huang and M. Kruczenski, “Minimal area surfaces in AdSn+1
+and Wilson loops,”
+JHEP 02 (2018),
+027 doi:10.1007/JHEP02(2018)027
+[arXiv:1712.06269 [hep-th]].
+[5] M.C. Romero-Fuster and E. Sanabria-Codesal,“Generalized evolutes, vertices
+and conformal invariants of curves in Rn+1”, Indag. Mathem., N.S., 10 (2), 297-
+305
+[6] A. Montesinos Amilibia’, M.C. Romero Fuster and E. Sanabria -Codesal, “Con-
+formal curvatures of curves in Rn+1”, lndag. Mathem., N.S., 12 (3) 369-382
+[7] R. Langevin, J. O’Hara and S. Sakata, “Space of subspheres and conformal
+invariants of curves” arXiv:1102.0344 [math.DG]
+[8] R. Langevin, J O’Hara, S Sakata, “Application of spaces of subspheres to con-
+formal invariants of curves and canal surfaces”, Annales Polonici Mathematici,
+January 2013 , doi: 10.4064/ap108-2-1
+[9] R. Sulanke, “M¨obius Invarints for Pairs of Spheres (Sm
+1 , Sl
+2) in the M¨obius Space
+Sn”, Contributions to Algebra and Geometry, Vol. 41 (2000), No.1, 233-246
+[10] H. Dorn, “Wilson loops at strong coupling for curved contours with cusps,”
+J. Phys. A 49 (2016) no.14, 145402 doi:10.1088/1751-8113/49/14/145402
+[arXiv:1509.00222 [hep-th]].
+[11] R. Sulanke, “ Submanifolds of the M¨obius space II, Frenet formulas and curves
+of constant curvatures.” Mathematische Nachrichten 100.1 (1981): 235-247.
+6
+

G9AzT4oBgHgl3EQfjP2K/content/tmp_files/load_file.txt

@@ -0,0 +1,171 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf,len=170
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='01513v1  [hep-th]  4 Jan 2023  HU-EP-23/01  Remarks on conformal invariants for  piecewise smooth curves and Wilson loops  Harald Dorn 1  Institut f¨ur Physik und IRIS Adlershof, Humboldt-Universit¨at zu Berlin,  Zum Großen Windkanal 6, D-12489 Berlin, Germany  Abstract  This short note is some obvious mathematical addendum to our papers on Wilson  loops on polygon-like contours with circular edges [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Using the technique of  osculating spheres and circles we identify the conformal invariants characterising the  kinks (cusps) of generic piecewise smooth curves in 3-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  1dorn@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='de  1  Introduction  The metrical invariants of smooth curves in Euclidean D-dimensional space are length,  curvature and (D −2) torsion parameters as functions along the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' For piecewise  smooth curves with cusps 2, in addition each cusp is characterised by the Euler angles  specifying the orthogonal transformation needed to rotate the Frenet-Serret frames  on both sides of a cusp to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  What concerns the issue of conformal invariants, there is for smooth curves a  considerable amount of mathematical papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In analogy to the metrical invariants  length s, curvature κ and torsion τ one finds for 3-dimensional curves, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [3],  conformal length ω,3  dω = √νds,  ν =  �  (κ′)2 + κ2τ 2 ,  (1)  conformal curvature Q  Q = 4(ν′′ − κ2ν)ν − 5(ν′)2  8ν3  ,  (2)  and conformal torsion T  T = 2(κ′)2τ + κ2τ 3 + κκ′τ ′ − κκ′′τ  ν  5  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (3)  Use of these invariants has been made in the physical literature to characterise the  boundary conditions for the treatment of minimal surfaces in AdS via Pohlmeyer  reduction [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  But what concerns the extension to piecewise smooth curves, we did not find any  paper in the mathematical literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Therefore, we decided to write up what one gets  by straightforward application of one of the various techniques used for the smooth  case: the kinematics of osculating spheres, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [5–8] and refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  This note is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In the next section we find formulas expressing  the conformal invariants for cusps, with legs made of generic smooth pieces of curves,  in terms of their metrical invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Then in section 3 we consider the very special  curves studied in our papers [1,2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' polygon-like curves with circular edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' These  curves need a separate discussion, since along the circular edges all the conformal  invariants (1),(2),(3) are zero or ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  2  Conformal invariants of generic cusps  In 3-dimensional space the osculating circle (sphere ) at a generic point x on a smooth  curve is a circle (sphere) having contact of second (third) order with the curve at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Since the order of contact is preserved under conformal transformations, osculating  circles (spheres) at a given point of a curve are mapped to those for the image under  2We follow the language in the Wilson loop literature and use the word cusp for a kink with  nonzero opening angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  3The prime denotes derivative with respect to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  1  a conformal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Let us denote by ⃗t,⃗n,⃗b the unit tangent, normal and binormal  vectors at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Then the center a of the osculating circle S1 is given by  a = x + 1  κ ⃗n  (4)  and the center c of the osculating sphere S2 by  c = x + 1  κ ⃗n −  κ′  τκ2 ⃗b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (5)  We now turn to the case where x is a point of discontinuity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' the tip of a cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  The cusp is then characterised by two osculating circles S1  −, S1  + and two osculating  spheres S2  −, S2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' The index ” ± ” indicates the limits one gets by approaching x along  the both respective legs of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Let us first count the number of conformal invariants we can expect for the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  There are 9 metrical invariants at hand: κ±, κ′  ±, τ± and the 3 Euler angles for the  rotations from the Frenet frame {⃗t−,⃗n−,⃗b−} to {⃗t+,⃗n+,⃗b+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' The difference of the  numbers of parameters of the 3-dimensional conformal and isometry group is equal  to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' This should result in 9 − 4 = 5 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Now of course two  of them are the corresponding limits of the differential of the conformal length (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Hence there remain 3 conformal parameters to be attributed genuinely to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  To find explicit formulas for them, we start with the conformal invariants of pairs  of spheres (Sm, Sn) of the same or of different dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' There is a rigorous math-  ematical treatment for arbitrary dimensions in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Applied to our case it means  that to each of the pair (S1  −, S1  +), (S1  −, S2  +), (S2  −, S1  +), (S2  −, S2  +) belongs just one con-  formal parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' To proceed, we first study these four invariants and will show  afterwards, that only three of them are independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (S1  −, S1  +)  The tangents of the osculating circles agree with those of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Therefore the related conformal invariant is  A11 = ⃗t−⃗t+ =  − cos α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (6)  Here α is the cusp angle (understood as the opening angle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' α = π in the smooth  case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (S2  −, S2  +)  Now the conformal invariant is given by the so-called inversive product,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [5,9]  A22 = R2  − + R2  + − (c− − c+)2  2R−R+  = (c− − x)(c+ − x)  R−R+  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (7)  R− and R+ are the radii of S2  − and S2  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Obviously A22 is the cosine of the angle  between the vectors pointing from x to the centers of the two osculating spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Strictly speaking, only its absolute value is invariant, since A22 changes sign under  those special conformal transformations for which the preimage of infinity is situated  inside just one of the spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' The same comment applies to A12 and A21 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (S2  −, S1  +) and (S1  −, S2  +) For a sphere and an intersecting circle the conformal invariant  2  is the scalar product of the unit vector pointing from x to the center of the sphere  with the unit tangent of the circle  A12 =  ⃗t−(c+ − x)  R+  ,  A21 = (c− − x) ⃗t+  R−  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (8)  Using (4),(5) for both legs of the cusp we get from (7) and (8)  A22 = κ−τ−κ+τ+ ⃗n−⃗n+ + κ′  −κ′  + ⃗b−⃗b+ − κ−τ−κ′  + ⃗n−⃗b+ − κ+τ+κ′  − ⃗b−⃗n+  �  κ2  −τ 2  − + κ′2  −  �  κ2  +τ 2  + + κ′2  +  ,  (9)  A12  =  κ+τ+ ⃗t−⃗n+ − κ′  + ⃗t−⃗b+  �  κ2  +τ 2  + + κ′2  +  ,  (10)  A21  =  κ−τ− ⃗t+⃗n− − κ′  − ⃗t+⃗b−  �  κ2  −τ 2  − + κ′2  −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (11)  All the scalar products in the above formulas can be expressed in terms of three Euler  angles ϕ, ϑ, ψ needed to rotate the Frenet frame {n−, b−, t−} to {n+, b+, t+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Then  we get  A11 = ⃗t−⃗t+ = cosϑ ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' ϑ = π − α ,  (12)  A22  =  1  �  κ2  −τ 2  − + κ′2  −  �  κ2  +τ 2  + + κ′2  +  �  κ−τ−κ+τ+ (cosϕcosψ − sinϕsinψcosϑ)  +κ′  −κ′  +(cosϕcosψcosϑ − sinϕsinψ) + κ−τ−κ′  + (sinϕcosψ + cosϕsinψcosϑ)  −κ+τ+κ′  − (cosϕsinψ + sinϕcosψcosϑ)  �  ,  (13)  A12  =  sinϑ  �  κ2  +τ 2  + + κ′2  +  �  κ+τ+ sinϕ − κ′  + cosϕ  �  ,  (14)  A21  =  sinϑ  �  κ2  −τ 2  − + κ′2  −  �  κ−τ− sinψ + κ′  − cosψ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (15)  The part of A22 containing the factor cosϑ is related to the product of A12 and A21  in an obvious manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' With a bit more careful inspection one gets for the whole A22  A22(ϕ, ϑ, ψ) =  1  sin2ϑ  �  A12(ϕ + π  2 ) A21(ψ + π  2 ) − cosϑ A12(ϕ)A21(ψ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (16)  Now A12 and A21 for arguments shifted by π  2 are not independent, but related by  �  A12(ϕ)  �2 +  �  A12(ϕ + π  2)  �2 = (A21(ψ)  �2 +  �  A21(ψ + π  2 )  �2 = sin2ϑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (17)  This means that a complete set of independent conformal invariants attributed to the  cusp is given by the three parameters 4  α ,  B12 = κ+τ+ sinϕ − κ′  + cosϕ  �  κ2  +τ 2  + + κ′2  +  ,  B21 = κ−τ− sinψ + κ′  − cosψ  �  κ2  −τ 2  − + κ′2  −  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  (18)  4Remember ϕ, ϑ, ψ Euler angles, α = π − ϑ opening angle of the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  3  Let us add a warning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Inserting by brute force ϕ = ψ = 0 into B12 and B21 one  could reach the wrong conclusion that  κ′  √  κ2τ 2+κ′2 could be an invariant for smooth  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' But since putting ϕ or ψ to zero is not a conformal invariant statement, this  conclusion is not allowed and also straightforwardly proven to be wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  We continue with some casual comment on the conformal invariants in 4-dimensio-  nal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' To fix all the osculating spheres from S1 to S3 at a smooth point one needs  κ, κ′, κ′′, τ1, τ ′  1, τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' This for the limits from both sides of the cusp, together with 6 Euler  angles, needed in 4D for the rotation of the Frenet frame, gives 18 metrical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  The difference of the number of parameters oft the conformal and isometry group is  now 5, hence we tentatively reach 13 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Now on both sides the  limits of the differential of the conformal length and the first conformal torsion5 are  not related to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Hence we can expect 13 − 4 = 9 conformal parameters to be  attributed genuinely to the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  On the other side, among the nine pairs of osculating spheres  (Si  −, Sj  +), i, j = 1, 2, 3, the pairs (S2  −, S2  +), (S2  −, S1  +), (S1  −, S2  +) have two invariants and  all other only one [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' This yields 12 conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' We keep it as an open  question, whether there are indeed three relations of the 3D type (16),(17) to reach  the minimal number of 9 independent parameters seen in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  We close this section with a comment on the cusp anomalous dimension for Wilson  loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' It is generally believed to depend on the cusp angle α only, and therefore  calculations have been done using straight edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' While in field theoretic perturbation  theory this can be justified by power counting in the corresponding Feynman integrals,  a rigorous proof for the generic situation at strong coupling is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' We have  presented a proof for the planar case with generically curved edges in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' For full  generality in 3D, a proof of its independence from B12 and B21 is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  3  Comments on Wilson loops on polygon-like  curves with circular edges  The polygon-like curves with circular edges, whose related Wilson loops have been  studied in our papers [1, 2], are not covered by the setting for generic curves as  presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Along their edges one has constant curvature κ  and zero torsion τ, resulting in zero conformal length (1) and undefined conformal  curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In mathematical language these edges are conformal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='6  Although such single edges carry no conformal data, their combination to a poly-  gon does7 It has still no extension in the sense of conformal length, but there are of  course the cusp angles at each cusp of the polygon and for more than 3 cusp points  the corresponding cross ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In addition, like generic curves, these polygons can  wind themselves out of a plane and exhibit torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In our paper [1] we have this  issue parameterised by the introduction of torsion angles βj, defined at a given cusp  5In 3D the conformal torsion (3) cannot be build out of the metrical parameters needed to fix all  the osculating spheres at the cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' But in 4D the osculating S3 inherits all the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  6See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' [3,5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  7In a sense one could call it a conformal vertex with internal conformal substructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  4  point xj by  βj = ∡({xj, xj+1}, ccj) ,  (19)  where {xj, xj+1} denotes the circular edge between xj and xj+1 and ccj the circle  fixed by the three cusp points xj−1, xj, xj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  However, in contrast to B12 and B21 for cusps of generic curves, these torsion  angles are not attributed to the local properties at the corresponding cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' This is  simply seen by changing in (19) the neighbouring cusp point xj+1 along the circle  of which the edge {xj, xj+1} is a part .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Then the tangent of the edge at xj remains  the same, but the circle ccj and its tangent at xj changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' By this manipulation βj  changes, although the local situation at the cusp at xj remains the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  We end with a remark on some setting for Wilson loops on piecewise smooth  curves intermediate between those of full generality considered in section 2 and the  polygons with circular edges in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In conformal invariant gauge field theories as in  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' N = 4 SYM there will be an anomalous conformal Ward identity of the type  derived in [1], imposing for the Wilson loop the structure of a conformally covariant  factor depending on the distances of the tips of the cusps times a conformally invariant  remainder factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' In the generic case this remainder factor is a function of the cross  ratios formed out of cusp points and the conformal cusp parameters identified in  section 2, but in addition also a functional of the conformal invariants as functions  along the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  For the case of polygons with circular edges the remainder is a  function of only a finite number of conformal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  To have a curve characterised by a finite number of conformal parameters but  nevertheless having nonzero conformal length, one could consider polygons with edges  which are pieces of curves with constant conformal curvature and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Such curves  have been classified in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Among them are loxodromes on rotational surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='  Acknowledgement  I thank the Quantum Field and String Theory Group at Humboldt University for  kind hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+page_content=' O’Hara and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content=' Sakata, “Space of subspheres and conformal  invariants of curves” arXiv:1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
+page_content='0344 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/G9AzT4oBgHgl3EQfjP2K/content/2301.01513v1.pdf'}
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+Astronomy & Astrophysics manuscript no. Marchandetal2023
+©ESO 2023
+January 5, 2023
+Fast methods to track grain coagulation and ionization. III.
+Protostellar collapse with non-ideal MHD
+P. Marchand1, 2, U. Lebreuilly3, M.-M. Mac Low2, V. Guillet4, 5
+1 Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, UT3-PS, CNRS, CNES, 9 av. du Colonel Roche,
+31028 Toulouse Cedex 4, France
+2 Department of Astrophysics, American Museum of Natural History, 200 Central Park West, NY, NY, 10024, USA
+3 AIM, CEA, CNRS, Université Paris-Saclay, Université Paris Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette, France
+4 Université Paris-Saclay, CNRS, Institut d’astrophysique spatiale, 91405, Orsay, France
+5 Laboratoire Univers et Particules de Montpellier, Université de Montpellier, CNRS/IN2P3, CC 72, Place Eugène Bataillon, 34095
+Montpellier Cedex 5, France
+ABSTRACT
+Dust grains influence many aspects of star formation, including planet formation, opacities for radiative transfer, chemistry, and
+the magnetic field via Ohmic, Hall, and ambipolar diffusion. The size distribution of the dust grains is the primary characteristic
+influencing all these aspects. Grain size increases by coagulation throughout the star formation process. We describe here numerical
+simulations of protostellar collapse using methods described in earlier papers of this series. We compute the evolution of the grain
+size distribution from coagulation and the non-ideal magnetohydrodynamics effects self-consistently and at low numerical cost. We
+find that the coagulation efficiency is mostly affected by the time spent in high-density regions. Starting from sub-micron radii, grain
+sizes reach more than 100 µm in an inner protoplanetary disk that is only 1000 years old. We also show that the growth of grains
+significantly affects the resistivities, and indirectly the dynamics and angular momentum of the disk.
+1. Introduction
+Grains play a major role during star formation. Firstly, they
+are the seed of planet formation. While their characteristic size
+is sub-micron in the interstellar medium (ISM) (Mathis et al.
+1977), they grow by coagulation during the collapse, and can
+reach sizes larger than 10 µm in the early stages of protostel-
+lar collapse (Guillet et al. 2020; Silsbee et al. 2020; Tsukamoto
+et al. 2021; Vorobyov et al. 2022; Bate 2022). Observations also
+suggest they reach these sizes, if not larger, in the envelopes of
+Class 0-I objects (Kwon et al. 2009; Miotello et al. 2014; Le
+Gouellec et al. 2019; Galametz et al. 2019). Their growth then
+continues in protoplanetary disks until they eventually become
+planetesimals. Variations in their size also significantly impact
+non-ideal magnetohydrodynamics (MHD) effects through their
+ionization and their chemical interactions with the gas, with a
+direct feedback on the dynamics of the gas (Marchand et al.
+2016; Zhao et al. 2016, 2018; Marchand et al. 2020; Guillet et al.
+2020). Non-ideal MHD effects have been shown to be critical to
+the regulation of magnetic field and angular momentum during
+the protostellar collapse and the protoplanetary disk evolution
+(Mouschovias & Paleologou 1979; Machida et al. 2006; Duf-
+fin & Pudritz 2008; Mellon & Li 2009; Li et al. 2011; Tomida
+et al. 2015; Wurster et al. 2016; Masson et al. 2016; Vaytet et al.
+2018; Marchand et al. 2020; Lebreuilly et al. 2021). Grains are
+also the main source of opacity in protostellar environments, af-
+fecting the cooling of the gas and the observations made of those
+systems. Their high optical depth at densities ρ > 10−13 g cm−3
+leads to the formation of the first hydrostatic core (Larson 1969).
+However, numerical simulations usually do not account for
+grain coagulation self-consistently due to the great cost of com-
+puting a coagulation algorithm on-the-fly (although new meth-
+ods are being developed, see the recent work by Lombart &
+Laibe 2021). The dust evolution used to be pre-processed or
+post-processed with no self-consistent feedback on the dynam-
+ics (Rossi et al. 1991; Dullemond & Dominik 2005; Zhao et al.
+2016; Marchand et al. 2020). Recently, more and more studies
+include the growth of grains in their hydrodynamics simulations
+(Tsukamoto et al. 2021; Vericel et al. 2021; Vorobyov et al. 2022;
+Bate 2022). In Marchand et al. (2021, hereafter Paper I), we
+presented a simple and fast method to track coagulation self-
+consistently that is particularly suited for modeling star forma-
+tion. We now apply this method to non-ideal MHD protostellar
+collapse simulations. It is coupled with the second method pre-
+sented in Paper I: a fast calculation of the ionization and grain
+charge, to obtain non-ideal MHD resistivities. These 3D simula-
+tions are the first to include a self-consistent grain growth with a
+direct feedback on the dynamics through the self-consistent cal-
+culation of MHD resistivities.
+The paper is organized as follows. In Section 2, we describe
+the methods used in our study. Section 3 presents the results of
+our calculations, both analytical in Section 3.1 and numerical in
+Section 3.2. We compare our results to other works and discuss
+the caveats in Section 4, and conclude in Section 5.
+2. Methods
+We perform non-ideal MHD simulations with the RAMSES
+code (Teyssier 2002). RAMSES is an Eulerian gas dynamics
+code with adaptive mesh refinement (AMR) and self-gravity. It
+includes a monofluid treatment of non-ideal MHD effects (Mas-
+son et al. 2012; Marchand et al. 2018). We have implemented
+the methods presented in Paper I to calculate the coagulation
+and ionization of grains on-the-fly in a self-consistent manner.
+Article number, page 1 of 11
+arXiv:2301.01510v1  [astro-ph.SR]  4 Jan 2023
+
+A&A proofs: manuscript no. Marchandetal2023
+2.1. Grain coagulation and ionization
+2.1.1. Coagulation
+In Paper I, we demonstrated that the coagulation process as
+described by the Smoluchowski (1916) equation is a one-
+dimensional process with certain types of coagulation kernels.
+Consequently, the size distribution of the coagulated grains de-
+pends only on the initial size distribution and a variable χ that
+encompasses the whole history of the physical conditions seen
+by the grains. At a given χ that is integrated along the path of
+the grains, the coagulated distribution is always the same inde-
+pendently of the actual path taken. This method works for every
+coagulation kernel for which the dependence on the gas variables
+such as density and temperature can be separated from the grain
+properties such as size and mass. In Paper I and in the present
+paper, we use the turbulent kernel derived by Ormel & Cuzzi
+(2007) in the intermediate coupling regime, which is suited for
+star formation conditions. We show that in this case χ can be
+derived from integrating
+dχ = n
+3
+4
+HT − 1
+4 dt,
+(1)
+where nH is the number density of the gas, T its temperature,
+and t the time. In three-dimensional (3D) hydrodynamics simu-
+lations, only the knowledge of χ is needed to track the coagula-
+tion of grains.
+Equation (1) is a Lagrangian derivative of χ with respect to
+time along the path of the grain. We can transform it into a partial
+(Eulerian) derivative
+∂χ
+∂t + u · ∇χ = n
+3
+4
+HT − 1
+4 ,
+(2)
+where u is the velocity of the gas. We can combine equation (2)
+with the mass conservation equation
+∂ρ
+∂t + ∇ · [ρu] = 0,
+(3)
+where ρ is the gas mass density. This yields
+∂ρχ
+∂t + ∇ · (ρχu) = ρn
+3
+4
+HT − 1
+4 ,
+(4)
+which means that the quantity ρχ can be treated in an Eulerian
+framework as a passive scalar with a source term. We exploit this
+property and implement it as such in RAMSES. The value of χ
+is therefore calculated self-consistently in all cells at each time-
+step, as a mass-weighted average, ignoring any diffusion of dust
+through the gas. We used the code Ishinisan (Marchand et al.
+2021) to pre-calculate a table containing the grain size distribu-
+tion for a large number of χ values in a large relevant interval
+(between χ = 1013 and χ = 1019 in cgs units). When the size
+distribution is needed during the hydrodynamical simulation, it
+is interpolated from the table based on the value of χ.
+Our initial distribution in this paper is a Mathis et al. (1977,
+MRN) distribution. The minimum and maximum radii are amin =
+5 nm and amax = 250 nm, and the slope of the distribution is
+−3.5, so that the number density of grains n follows the variation
+dn
+da ∝ a−3.5.
+(5)
+The total quantity of grains is determined by the dust-to-gas
+mass ratio that we assume to be 0.01. In this work, we sample the
+distribution with 60 bins of size logarithmically spaced between
+10-10
+10-9
+10-8
+10-7
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+10-3
+10-2
+10-1
+100
+101
+102
+103
+Fractional mass X (ρ/nHmp)
+Radius (µm)
+Initial MRN
+χ = 1015 cgs
+χ = 1016 cgs
+χ = 1017 cgs
+χ = 1017.5 cgs
+χ = 1018 cgs
+χ = 1018.5 cgs
+χ = 1019 cgs
+Fig. 1. State of the grain size distribution for values of χ, between 1015
+cgs and 1019 cgs. The points represent the fractional abundance of the
+size-bin as function of the effective radius of the bin.
+5 nm and 5000 µm. In Figure 1, we present the coagulated MRN
+size distribution at various χ. Below χ = 1017 cgs, the shift in
+the size distribution is negligible. For higher values, the maxi-
+mum size and the peak of the distribution are located at larger
+and larger radii, while the slope of the distribution remains sim-
+ilar. In all cases, the small grains are more abundant while the
+large grains hold more mass. The mode of the size distribution
+amax is located near the largest relevant grain size.
+As grains grow, they may experience fragmentation when the
+kinetic energy of the collision is high. Contrarily to what we de-
+rived in paper I, we find that fragmentation does not occur in
+early stages of the disk, but rather only at very high densities
+ρ > 1012 cm−3 (Lebreuilly et al. 2023) for very large grains with
+a > 0.08 cm. We demonstrate this result in Appendix C. We
+therefore neglect fragmentation in this work. We also do not ac-
+count for the grain drift with respect to the gas in this work. We
+discuss the possible consequences in Section 4.2. Drift is, how-
+ever, compatible with our coagulation model, and we detail the
+method in Appendix B. Other limitations are discussed in Sec-
+tion 4.4.
+2.1.2. Ionization and resistivities
+In Paper I, we also presented a fast method to calculate the ion-
+ization of the gas-grain mixture. For an arbitrary size distribu-
+tion, we can calculate the average electric charge of each grain
+size, the number of ions and the number of electrons, provided
+the cosmic-ray (CR) ionization rate, the density and temperature
+of the gas, the average atomic mass of ions µi and the sticking
+probability of electrons on grains se. Here, we assume µi = 28,
+which corresponds to the ion HCO+, and se = 0.6 as in Marc-
+hand et al. (2016). We also assume ζ = 5 × 10−17 s−1. The den-
+sity and the temperature, are taken from the hydrodynamic sim-
+ulation. The calculation is performed by the Newton-Raphson
+scheme described in Appendix A of Paper I. The resistivities are
+computed using a similar method to Marchand et al. (2016), with
+one difference. For each grain size, they sum over the contribu-
+tions of the whole charge distribution (between -1 to +1 in their
+case). Instead, we average the contributions using the mean elec-
+tric charge. We explicit the method in more details in Appendix
+A.
+Article number, page 2 of 11
+
+P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+2.2. Star formation simulations
+2.2.1. Model
+We perform four numerical simulations using the RAMSES
+code. We solve the following MHD equations
+∂ρ
+∂t + ∇ · �ρu� = 0,
+(6)
+∂ρu
+∂t + ∇ ·
+�
+ρuu +
+�
+P + B2
+2
+�
+I − BB
+�
+= −ρ∇Φ,
+(7)
+∂B
+∂t − ∇ ×
+�
+u × B − ηΩJ − ηH
+J × B
+B
++ ηAD
+(J × B) × B
+B2
+�
+= 0,
+(8)
+∇ · B = 0,
+(9)
+where u is the velocity of the gas, P its pressure, B the mag-
+netic field, J = ∇ × B the current, I the identity matrix, Φ the
+gravitational potential, and ηΩ, ηH and ηAD the ohmic, Hall and
+ambipolar resistivities. The temperature evolution is prescribed
+by the barotropic equation
+T = T0
+�������1 +
+�
+ρ
+10−13 g cm−3
+�γ−1������� ,
+(10)
+with T0 = 10 K and γ = 5/3 the adiabatic index. The initial
+condition is a sphere of gas of 1 M⊙. The radius of the sphere is
+controlled by the thermal over gravitational energy ratio α. We
+choose α = 0.3, which sets a radius of R = 2946 au. The density
+distribution has an m = 2 azimuthal perturbation
+ρ(ϕ) = ρ0(1 + δρ sin ϕ),
+(11)
+where ρ0 is the density of a uniform sphere of same mass and
+radius, and ϕ the azimuthal angle. We choose δρ = 0.05. The
+computational domain outside of the sphere is filled with gas of
+density ρ0/100. The sphere undergoes a solid rotation, with a
+ratio of rotational to gravitational energy of β = 0.02. The mag-
+netic field is initially uniform and parallel to the rotation axis.
+It is defined using the mass-to-flux ratio over the critical value
+(Mouschovias & Spitzer 1976)
+µB =
+M/ΦB
+(M/ΦB)crit
+,
+(12)
+with
+� M
+ΦB
+�
+crit
+= 0.53
+3π
+�
+5
+G.
+(13)
+Observations show that dense cores are slightly super-critical
+(Crutcher 1999), although recent numerical simulations indicate
+that observations may overestimate the actual strength of the
+magnetic field due to projection effects (Kuznetsova et al. 2020).
+We choose µB = 5 as a fiducial value. Those parameters are
+used in our reference case C-3. We change the value of α to 0.4,
+and the initial mass M to 5 M⊙ for two other cases called C-4
+and C-3-M5, respectively, to investigate the influence of the col-
+lapse time on the grain coagulation. The final run, named NC-3,
+is similar to C-3 without coagulation. The grain distribution and
+ionization evolve as described in sections 2.1, 2.1.2, and Paper I.
+The four cases are summarized in table 1.
+2.2.2. Grid and Algorithm
+The simulation box is a cube that is four times as large as the ra-
+dius of the sphere, with periodic boundary conditions. The initial
+grid is composed of 323 cells (level 5 of AMR) and is refined to
+ensure at least 10 points per Jeans length, strongly satisfying the
+Truelove et al. (1997) criterion. The maximum AMR refinement
+level is 13 for C-3 and NC-3 (resolution of 1.4 au), 14 for C-4
+(resolution of 0.96 au) and 16 for C-3-M5 (resolution of 0.90
+au).
+Simulations are performed with a 3D unsplit slope limiter
+to avoid overshooting of the magnetic field at shock boundaries,
+while keeping the second order convergence for the Hall effect.
+We use the HLLD Riemann solver (Miyoshi & Kusano 2005)
+for non-magnetic variables and the 2D HLL Riemann solver for
+the magnetic field and the Hall effect (Balsara 2012; Marchand
+et al. 2018). The Poisson equation is solved using the multigrid
+method of Guillet & Teyssier (2011) in our periodic domain.
+3. Application to star formation
+In this section, we present the results of our calculations. We first
+apply our coagulation method to an analytical one-zone model in
+Section 3.1, then in 3D MHD simulations in Section 3.2.
+3.1. Analytical collapse
+During spherical protostellar collapse, the time evolution of the
+contraction ratio of a gas cloud compared to its original radius
+x = R(t)/R0 can be described by (Flower et al. 2005):
+dx
+dt = − π
+2τff
+�
+1
+x − 1,
+(14)
+where τff is the free-fall time, given by
+τff =
+�
+3π
+32Gρ0
+,
+(15)
+where ρ0 is the initial density. Guillet et al. (2020) showed that
+assuming a uniform compression of the gas nicely reproduces
+the isothermal phase of the collapse, particularly when compar-
+ing the dust size distribution at the same gas density. In this case,
+the density of mass scales as ρ(t) = ρ0(R0/R[t])3. The gas density
+then evolves as
+1
+nH
+dnH
+dt = 1
+ρ
+dρ
+dt = −3
+x
+dx
+dt .
+(16)
+We use a second-order Runge-Kutta scheme to numerically
+integrate this equation from the beginning of the collapse at
+ρ = ρ0 until the formation of the first Larson core at ρ =
+10−13 g cm−3. The evolution of χ with ρ is plotted in Figure 2,
+assuming T = 10 K. The solid and dashed lines represent dif-
+ferent initial densities, ρ0 = 3.8 × 10−20 g cm−3 and ρ0 =
+3.8×10−18 g cm−3, respectively. At densities nH > 10−16 g cm−3,
+the value of χ is independent from ρ0 and increases as χ ∝ ρ1/4.
+This evolution is expected as χ ∼ ρ3/4t and t ∼ τff ∼ ρ−1/2. At
+ρ = 10−13 g cm−3, the coagulation variable reaches χ ≈ 5.6×1017
+cgs, which corresponds to a peak of the size distribution of ∼ 10
+µm, indicating a significant grain growth. That is consistent with
+the results of Guillet et al. (2020), who use the same collapse
+model, but solve coagulation on the fly.
+Article number, page 3 of 11
+
+A&A proofs: manuscript no. Marchandetal2023
+Table 1. Parameters of the protostellar collapse simulations: name of the simulation, thermal over gravitational energy ratio α, radius R, mass M,
+initial density ρ0 and initial magnetic field B of the initial cloud, formation time of the first hydrostatic core tfc, use of coagulation.
+Name
+α
+R (au)
+M (M⊙)
+ρ0 (g cm−3)
+B (µG)
+tfc (kyr)
+Coagulation
+C-3
+0.3
+2946
+1
+5.49 × 10−18
+133
+∼ 30
+Yes
+C-3-M5
+0.3
+14738
+5
+2.19 × 10−19
+27
+∼ 150
+Yes
+C-4
+0.4
+3930
+1
+2.31 × 10−18
+75
+∼ 47
+Yes
+NC-3
+0.3
+2946
+1
+5.49 × 10−18
+133
+∼ 30
+No
+Fig. 2. Evolution of the coagulation variable χ with increasing density
+nH during the isothermal collapse. The solid line represents an initial
+density of ρ0 = 3.8 × 10−20 g cm−3, and the dashed line ρ0 = 3.8 ×
+10−18 g cm−3.
+3.2. Numerical collapse
+The numerical simulations were run as described in Section 2
+until 1000 years after the density reaches 10−13 g cm−3, the for-
+mation of the first hydrostatic core at ∼ 30 kyr (Tab. 1). In refer-
+ence simulation C-3, a small circumstellar disk with a radius of ≈
+20 au forms and a disk wind is launched by magneto-centrifugal
+acceleration (Blandford & Payne 1982). Figure 3 shows face-on
+and edge-on slices of density at the final time-step of the simu-
+lation, with arrows indicating the gas velocity.
+3.2.1. Grain growth
+We show in Figure 4 the value of χ as a function of the density in
+simulation cells at the final time-step, for runs C-3, C-4 and C-
+3-M5. The increase is quasi unidimensional in the isothermally
+collapsing envelope for ρ < 1015 g cm−3. Beyond this density,
+there is a large spread of χ values of over an order of magnitude.
+This spread most likely occurs in gas falling in the pseudo-disk,
+then the disk and the first Larson core, or in the outflow over dif-
+ferent timescales, and spending unequal times in a given density
+range. The overall trend agrees well with the analytical calcula-
+tion.
+There is no significant difference in the χ values, and thus
+dust size distributions, between the three runs, despite run C-4
+needing 50% more time to collapse to the first Larson core stage,
+and C-3-M5 needing 400% more time. Coagulation happening
+in the isothermally collapsing envelope is therefore hardly im-
+pacted by the initial conditions, as growth accelerates with in-
+creasing density.
+-40
+-20
+ 0
+ 20
+ 40
+x (au)
+-40
+-20
+ 0
+ 20
+ 40
+y (au)
+-14
+-13
+-12
+-11
+log(ρ) (g cm-3)
+-40
+-20
+ 0
+ 20
+ 40
+x (au)
+-40
+-20
+ 0
+ 20
+ 40
+y (au)
+-40
+-20
+ 0
+ 20
+ 40
+-40
+-20
+ 0
+ 20
+ 40
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+-17
+-16
+-15
+-14
+-13
+-12
+log(ρ) (g cm-3)
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+-200
+-100
+ 0
+ 100
+ 200
+-200
+-100
+ 0
+ 100
+ 200
+Fig. 3. Density slices of the C-3 simulation, with grain coagulation. The
+top panel is a face-on slice of the plane z=0, and the bottom panel an
+edge-on slice of the plane y=0. White arrows represent the direction of
+the gas velocity. The snapshot is taken at the final time-step, 1000 years
+after the formation of the first Larson core.
+Figure 5 shows the mode of the size distribution as a func-
+tion of density, corresponding to the distribution of χ shown in
+Figure 4. Three regions are clearly demarcated, the first being
+the envelope, in which grain coagulation is not efficient enough
+to form large grains (ρ < 10−16 g cm−3). The second comprises
+Article number, page 4 of 11
+
+1018
+cm
+-3
+16
+X
+101s
+1014
+10-19
+10-13
+10-18
+10-17
+10~16
+10-14
+10-12
+10-11
+02-01
+p (g cmP. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+Fig. 4. Evolution of the coagulation variable χ with increasing density
+nH in the numerical collapse models C-3 (purple), C-3-M5 (light blue),
+and C-4 (green). Each point corresponds to a simulation cell at the final
+time-step. The red line is the analytical collapse solution for ρ0 = 3.8 ×
+10−20 g cm−3.
+the pseudo-disk and the early protoplanetary disk, where grains
+grow by a factor 100 from sub-micron sizes to several tens of
+µm. The third region is the first Larson core, which has even
+larger grains that reach 400 µm within a mere 103 yr years af-
+ter its formation. That value is in line with similar recent studies
+(Kawasaki et al. 2022; Lebreuilly et al. 2023). There is little dif-
+ference between runs C-3, C-4 and C-3-M5, confirming that co-
+agulation in the envelope does not impact large grains, as found
+by previous studies (for example Silsbee et al. 2022). We discuss
+observations of large grains in the envelope in Section 4.2.
+The spatial distribution of those grain sizes for run C-3 is
+displayed in Figure 6. Size distributions shifting significantly
+from the initial MRN distribution are indeed confined to the mid-
+plane, in the disk and pseudo-disk. The bottom panel also shows
+moderately larger grains in the outflow, as they only traveled
+through the upper layers of the pseudo-disk before being ejected
+(Marchand et al. 2020). If they had been in the mid-plane of the
+disk, they would have grown much more, as coagulation is irre-
+versible in our model. The upper panel shows that grains reach
+a radius larger than 1 µm in the outskirts of the disk, within 100
+au of the center. Growth then occurs rapidly as density increases
+in the inner 15 au, which is shown by the almost overlapping
+contours delimiting amax = 5 µm and amax = 10 µm.
+3.2.2. Resistivities and gas dynamics
+We describe here the impact of grain coagulation on non-ideal
+MHD resistivities and their macroscopic effects on gas dynam-
+ics. Previous studies emulated the coagulation of grains by re-
+moving the very small grains (a < 0.1 µm) and redistributing
+their mass to the larger end of the distribution (Zhao et al. 2016,
+2018; Marchand et al. 2020). This method leads to an increase
+of resistivities, in particular the ambipolar resistivity, resulting in
+weaker coupling between the magnetic field and the gas, hence
+weaker magnetic braking and larger, more unstable disks. How-
+ever, we observe here the exact opposite behavior.
+The middle panel of figure 7 presents the volume-weighted
+average resistivities as a function of density, at the final time-
+step, for simulations C-3 and NC-3. A non-evolving size distri-
+Fig. 5. Mode of the coagulated grain size distribution as a function of
+density in simulations (purple), C-3-M5 (light blue), and C-4 (green).
+The discrete values of the sizes are due to the binning of the size distri-
+bution.
+bution produces resistivities relatively similar in the envelope,
+but two to four orders of magnitude larger at disk densities, par-
+ticularly the Ohmic and ambipolar resistivities. Consequently,
+the magnetic braking is weakened, and the gas retains more an-
+gular momentum without the grain coagulation, forming a larger
+disk, as shown in Figure 8.
+This difference in regime can be quantified by the ambipolar
+Elsasser number Am = B2/(ρηADΩ). In regions where Am < 1,
+the ambipolar diffusion has a significant impact on the dynam-
+ics of the gas. The bottom panel of Figure 7 shows the radial
+profile of the ambipolar Elsasser number in runs C-3 and NC-
+3, azimuthally-averaged in the mid-plane. The higher resistivity
+in run NC-3 results in Am < 1 in the inner ∼ 12 au, indicat-
+ing active ambipolar diffusion and magnetic field dissipation,
+while Am ≳ 104 for C-3 over the same radial range, indicat-
+ing weak ambipolar diffusion. Figure 9 compares the disk size
+and angular momentum in both simulations, further confirming
+the lower magnetic braking in run NC-3. A second effect of the
+weaker magnetic forces from the stronger ambipolar diffusion in
+run NC-3 is the absence of outflow at this stage of evolution, as
+shown in the lower panel of Figure 8.
+The discrepancy of resistivity values between actual coagu-
+lation and methods simply redistributing the mass to the large-
+mass-end of the distribution originates from the lack of very
+large grains (> 10 µm) in the latter. In both cases, removing
+the small grains decreases the electron absorption by dust, and
+therefore decreases the Ohmic resistivity. However, the ambipo-
+lar resistivity is controlled by the relative abundance of ions
+and charged grains in the gas. Although the small grain removal
+method barely changes the abundance of ions, the dominant pos-
+itive charge carriers, it reduces significantly the charged grain
+population, leading to an increase of resistivity at low density. At
+high density, both with a standard MRN and a truncated-MRN
+distribution, the grains become the dominant charge carriers and
+the abundance of ions decreases (see the dashed lines in the top
+panel of Figure 7). That does not happen with coagulation be-
+cause the number of grains decreases significantly, hence leading
+to a higher number of ions and a lower resistivity than without
+proper coagulation. This kind of method without larger grain cre-
+ation is therefore inappropriate for emulating coagulation alone
+at a low cost for non-ideal MHD calculations. At later times, at
+Article number, page 5 of 11
+
+1020
+C-3
+C-4
+C-3-M5
+1019
+Analytical collapse
+1018
+C1017
+X
+1016
+1015
+10-T8
+10-T6
+10-T4
+10-20
+10°10
+p (g cm103
+C-3
+C-4
+C-3-M5
+10°
+10-78
+10-10
+-3
+p (g cmA&A proofs: manuscript no. Marchandetal2023
+-100
+-50
+ 0
+ 50
+ 100
+x (au)
+-100
+-50
+ 0
+ 50
+ 100
+y (au)
+0.1
+1
+10
+amax (µm)
+-100
+-50
+ 0
+ 50
+ 100
+x (au)
+-100
+-50
+ 0
+ 50
+ 100
+y (au)
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+amax (µm)
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+Fig. 6. Slices of the C-3 simulation with coagulation showing the mode
+of the grain size distribution amax, 1000 years after the first core forma-
+tion. The color scale is different for each panel to yield a better contrast.
+On the top panel, black contours indicate amax = 1, 5 and 10 µm, and
+amax = 0.5 and 1 µm on the bottom panel.
+densities ρ > 10−12 g cm−3, Lebreuilly et al. (2023) showed that
+the replenishment of small grains by fragmentation would lead
+to an increase in both Ohmic and ambipolar resistivities.
+4. Discussion and caveats
+4.1. Grain growth
+As explained in previous sections and displayed in Figures 4 and
+5, initial gas conditions have little to no influence on grain coag-
+ulation for later stages, and coagulation is ineffective in the en-
+velope for growing large grains. This reinforces the idea that the
+system forgets the initial conditions at the formation of the first
+hydrostatic core and the disk (Vaytet & Haugbølle 2017). Con-
+sequently, calculations such as those presented in this work may
+provide standard initial dust grain size distributions for studies of
+10-20
+10-18
+10-16
+10-14
+10-12
+10-10
+10-8
+10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10
+ne / nH, ni / nH
+ρ (g cm-3)
+Ions
+Electrons
+No coagulation
+1010
+1012
+1014
+1016
+1018
+1020
+1022
+10-20 10-19 10-18 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10
+MHD resistivities (cm2 s-1)
+ρ (g cm-3)
+Ohm
+Ambipolar
+Hall
+No coagulation
+10-2
+10-1
+100
+101
+102
+103
+104
+105
+1
+10
+100
+Elsasser number
+r (au)
+C-3 (coagulation)
+NC-3 (No coagulation)
+Fig. 7. Top panel: Abundances of ions (purple) and electrons (green)
+as a function of density in models with coagulation (C-3; solid lines) or
+without it (NC-3; dashed lines). Middle panel: Volume averaged Ohmic
+(purple), ambipolar (green), and Hall (blue) resistivities for the same
+models. Bottom panel: Average ambipolar Elsasser number Am in the
+mid-plane as a function of radius. The thin grey line represents Am = 1.
+protoplanetary disk evolution. Although other coagulation ker-
+nels affect the size distribution in different ways (see Section
+4.4), it is certain that even young protoplanetary disks contain
+grains significantly larger than the classical MRN distribution.
+Article number, page 6 of 11
+
+P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+-40
+-20
+ 0
+ 20
+ 40
+x (au)
+-40
+-20
+ 0
+ 20
+ 40
+y (au)
+-14
+-13
+-12
+-11
+log(ρ) (g cm-3)
+-40
+-20
+ 0
+ 20
+ 40
+x (au)
+-40
+-20
+ 0
+ 20
+ 40
+y (au)
+-40
+-20
+ 0
+ 20
+ 40
+-40
+-20
+ 0
+ 20
+ 40
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+-17
+-16
+-15
+-14
+-13
+-12
+log(ρ) (g cm-3)
+-200
+-100
+ 0
+ 100
+ 200
+x (au)
+-200
+-100
+ 0
+ 100
+ 200
+z (au)
+-200
+-100
+ 0
+ 100
+ 200
+-200
+-100
+ 0
+ 100
+ 200
+Fig. 8. Same as figure 3 for simulation NC-3, without grain coagulation,
+1000 years after the formation of the first Larson core.
+This has important implications for the dynamics of grains in the
+disk. Larger grains couple differently to the gas and may trigger
+the streaming instability (Youdin & Goodman 2005; Johansen
+& Youdin 2007; Yang et al. 2017), which is an early step toward
+planet formation. Although this regime is not reached in our sim-
+ulations, the fast growth of the grains in circumstellar disks could
+predict an early onset for this process.
+4.2. Large grains in envelopes and dust diffusion
+Although grain coagulation is negligible in the envelope of our
+simulations, large grain signatures in envelopes have been ob-
+served. Galametz et al. (2019), for example, report "low and
+varying dust emissivity indices" at the envelope scale for some
+Class 0 and Class I protostars. This could be due to the pres-
+ence of mm-size grains in low numbers. The time-scale to form
+such large grains in the envelope and cold ISM cores is large
+0
+5
+10
+15
+20
+25
+30
+35
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.40
+1e+51
+2e+51
+3e+51
+4e+51
+5e+51
+6e+51
+7e+51
+Disk radius (au)
+Disk angular momentum (g cm2 s-1)
+t (kyr) after First Core formation
+Disk radius
+Disk angular momentum
+No coagulation
+Fig. 9. Disk size (purple, left axis) and angular momentum (green, right
+axis) as a function of time after the formation of the first Larson core.
+Solid lines represent simulation C-3 with coagulation, and dashed lines
+are simulation NC-3 without coagulation.
+(> 100 Myr), so we exclude the possibility of early coagulation.
+Similarly, Valdivia et al. (2019) report that their synthetic polari-
+sation observations of young protostellar envelopes in RAMSES
+calculations require grains larger than 10 µm to be consistent
+with observations, which is also inconsistent with our findings
+and those of recent studies (Silsbee et al. 2022; Lebreuilly et al.
+2023, for the latest ones).
+The aerodynamic properties of the grains may cause differ-
+ential velocities between grain populations and the gas, lead-
+ing to varying dust-to-gas ratio throughout the cloud (Lebreuilly
+et al. 2020). We note however that, as they have found, dust dif-
+fusion will start to play a significant role only for very large
+grains of a few hundred microns. Generally, large grains tend
+to accumulate in higher density regions. Tsukamoto et al. (2021)
+found that this dust diffusion can lead to what they call an ash fall
+phenomenon, in which large coagulated grains (up to millimeter-
+size) from the disk are ejected by an outflow, then decouple from
+the gas and fall back in the envelope. This process may explain
+the observations of low spectral index in Class 0 envelopes, as
+the outflow fuels the envelope with large grains formed in the
+central region. Eventually, those ejected grains circle back to the
+outer edge of the disk, enriching the large-end of the size distri-
+bution. In our case, the disk wind is fueled by the upper layers of
+the pseudo-disk, in which dust only moderately grows. However,
+dust-to-gas ratio enhancement in this region due to the grain dif-
+ferential velocities may lead to an accelerated growth.
+Further studies are needed to investigate this discrepancy
+about the size of grains in envelopes between observations and
+theory.
+4.3. Coagulation in the pseudo-disk
+Grains in our simulations seem to grow faster than in Bate (2022)
+despite their use of the same turbulent kernel as in our work. The
+peaks of the distributions in that work reach a few microns at a
+maximum density of nH = 1013 cm−3 (see their Figure 7), while
+in our simulations the peak exceeds 100 µm at a lower maximum
+density of nH ≈ 3×1012 cm−3 (or ρ ≈ 10−11 g cm−3; see our Fig-
+ure 5). That discrepancy is mainly due to a different modelling
+Article number, page 7 of 11
+
+A&A proofs: manuscript no. Marchandetal2023
+of the Reynolds number to calculate the turbulent coagulation
+kernel. They assume a constant value of Re = 108, while we use
+(Ormel et al. 2009; Guillet et al. 2020)
+Re = 6.7 × 107 �
+nH
+105 cm−3
+� 1
+2 .
+(17)
+Hence, in the central regions, our Reynolds number can be larger
+by three orders of magnitude. Their grains are then stuck in the
+tightly-coupled regime where relative grain velocities are lower
+than in the intermediate coupling-regime, which is more adapted
+to this situation (as demonstrated in Section 4.1 of paper I). The
+lower relative velocities result in lower coagulation rates, and
+therefore a slower growth rate. The poor constraints on the value
+of the Reynolds number in protostellar environments therefore
+represents a source of uncertainty for grain growth by coagula-
+tion.
+An additional explanation may involve an excess of growth
+in the pseudo-disk that forms in our simulations. The pseudo-
+disk is an over-density, typically denser than ρ ≈ 10−15 g cm−3,
+created by the convergence of gas flowing along magnetic field
+lines (Galli & Shu 1993), that takes the shape of a disk perpen-
+dicular to the magnetic field, but not supported against gravity.
+It is apparent in the bottom panels of Figures 3 and 8.
+Gas in the rotationally supported disk generally comes di-
+rectly from the pseudo-disk, and the overall efficiency of coag-
+ulation is mainly affected by the time spent in high-density re-
+gions like the pseudo-disk. This is what appears in the bottom
+panel of Figure 6, in which there is a ∼ 30 au-thick layer of large
+grains in the mid-plane. A passage of grains through the pseudo-
+disk would therefore provide an acceleration of coagulation in
+the early stage of star formation, even before arrival in the disk.
+That does not happen in Bate (2022) since they do not consider
+magnetic fields, resulting in less-coagulated size distributions as
+grains enter the disk.
+4.4. Coagulation kernel
+Our coagulation methods works for every coagulation kernel
+where the environment variables and grain variables can be sep-
+arated, one example of which is the well-known turbulent ker-
+nel derived by Ormel & Cuzzi (2007) that we use. This kernel
+is appropriate to calculate the growth of large grains, but it has
+several limitations. We assume a steady-state Kolmogorov turbu-
+lence spectrum, and that the injection-scale of the turbulence is
+equal to the Jeans length corresponding to the local density (see
+section 2.2 of paper I), which may lead to an over-estimation of
+the grain collision rate. Other kernels may have different effects
+on the size distribution of grains. Guillet et al. (2020) showed
+that this turbulent kernel would increase the maximum size of
+the distribution, while a kernel derived from ambipolar diffusion,
+that creates a drift between charged and neutral grains, is effi-
+cient at removing small grains. This also happens in Bate (2022)
+and Lebreuilly et al. (2023), which combine several processes
+including Brownian motion and pressure gradients that generate
+drift velocities between grains and rapidly remove the smaller
+grains. These processes are not, however, efficient at producing
+larger grains without the help of the turbulent kernel. Contrary
+to fragmentation, the addition of those kernels would steepen the
+distribution at its lower end.
+5. Conclusions
+We present here the results of numerical simulations of protostel-
+lar collapse with dust coagulation and self-consistent calculation
+of non-ideal MHD resistivities, using the methods detailed in
+Marchand et al. (2021). We performed four simulations, includ-
+ing three with coagulation and different collapse times, and one
+without coagulation for reference. Here are our main results.
+- Coagulated size distributions retain some characteristics of
+the initial MRN distribution, in particular the dominance of
+small grains in number and large grains in mass.
+- Dust coagulation is inefficient at growing larger grains in the
+envelope, even with long free-fall times. What matters is the
+time spent in high-density regions (ρ > 10−15 g cm−3) like
+the pseudo-disk. Fragmentation can also be ignored in the
+cloud-collapse phase.
+- Grain growth is extremely rapid in the disk. The peak of the
+size distribution in mass, which is also the largest relevant
+grain size of the distribution, reaches 1 µm in the pseudo-
+disk and more than 100 µm in the inner disk only 103 yr
+after its formation.
+- Grain sizes have a significant impact on non-ideal MHD re-
+sistivities. Coagulated grains result in resistivities up to four
+orders of magnitude lower than non-coagulated grains, with
+a significant impact on the dynamics of the disk. Simple re-
+distribution approximations fail to capture this effect, as it
+occurs because of the growth of the largest grains.
+Accounting for grain coagulation is therefore necessary in
+star formation and protoplanetary disk simulations, as grains
+grow rapidly to ≥ 100 µm in radius in disks. The effects of grain
+growth on chemistry and radiative transfer will be explored in
+future papers.
+Acknowledgements. P.M. acknowledges the financial support of the Kathryn W.
+Davis Postdoctoral Fellowship of the American Museum of Natural History and
+of the European Research Council (ERC) under the European Union’s Hori-
+zon 2020 research and innovation programme (ERC Starting Grant “Chemtrip”,
+grant agreement No 949278). M.-M.M.L. was partly supported by US NSF grant
+AST18-15461. U.L. acknowledges the financial support of the European Re-
+search Council (ERC) via the ERC Synergy Grant ECOGAL (grant 855130).
+Appendix A: Calculating the resistivities
+Appendix A.1: Solving for the ionization
+Let us assume a size distribution of grains divided into N bins.
+We have the following system of equations (see Paper I)
+Zk =ψτk +
+1 − ϵ2Θ2
+1 + ϵΘαk + ϵ2Θ2 ,
+(A.1)
+⟨ ˜J(τk)⟩ =(1 − ψ) +
+2
+τk
+�
+ϵ2Θ2 + ϵΘ
+�
+1 + ϵΘαk + ϵ2Θ2 ,
+(A.2)
+ϵ =1 − ψ
+Θeψ ,
+(A.3)
+ni = −
+1
+1 − ϵ
+�
+k
+nkZk,
+(A.4)
+f(ψ) =⟨σv⟩ieϵn2
+i
+ζnH
++ nivi
+ζnH
+�
+k
+nkπa2
+k⟨ ˜J(τk)⟩ − 1 = 0.
+(A.5)
+ϵ = ne/ni < 1 is the ratio between the number of electrons and
+ions, ψ is the ratio between the electric potential of the grains
+and the kinetic energy of electrons, nk, Zk and ak represent the
+number density, the average charge and the radius of grains in
+bin k, ζ is the CR ionization rate, and vi = [8kBT/πµimH]1/2
+Article number, page 8 of 11
+
+P. Marchand et al.: Fast methods to track grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD
+is the thermal velocity of ions, with T the temperature, µi the
+average atomic mass of ions and mH the mass of the pro-
+ton. We also have Θ = se[µimH/me]1/2, with se the sticking
+probability of electrons on grains and me the electron mass.
+⟨ ˜J(τk)⟩ represents the enhancement factor for ion recombina-
+tion on grains, τk = akkBT/e2 is the reduced temperature of
+grains (Draine & Sutin 1987), and αk = [8/(πτk)]1/2. Finally,
+⟨σv⟩ie = 2 × 10−7[T/300]−1/2 is the collision rate between ions
+and electrons.
+We solve the system of equations (A.1) - (A.5) for ψ and
+find ni, ne and Zk for all bins. A Newton-Raphson algorithm is
+described in details in the Appendix A of Paper I.
+Appendix A.2: The resistivities
+The collision time-scale of species s with neutral H2, the most
+abundant species in the gas, is given by
+τsH2 =
+1
+asHe
+ms + mH2
+mH2
+1
+nH2⟨σω⟩s
+.
+(A.6)
+asHe accounts for the collisions with He and is equal to 1.14
+for ions, 1.16 for electrons and 1.28 for grains (Desch &
+Mouschovias 2001). ms and mH2 are the mass of the species s
+and the H2 molecule. nH2 represents the number density of H2
+(roughly equal to the density of the gas). ⟨σω⟩s is the collision
+rate, taken from Pinto & Galli (2009) for electrons and ions
+⟨σω⟩e = 3.16 × 10−11v1.3
+rms,e,
+(A.7)
+⟨σω⟩i = 2.4 × 10−9v0.6
+rms,i,
+(A.8)
+where
+vrms,s =
+� 8kBT
+πµs,H2
+� 1
+2
+,
+(A.9)
+is in km s−1, with µs,H2 the reduced mass of species s and H2.
+These velocities have been calculated in a three-fluid formalism
+but we use them in our monofluid framework. See some related
+concerns in the appendix of Marchand et al. (2020). For grains,
+the collision rate is given by (Draine & Sutin 1987; Kunz &
+Mouschovias 2009)
+⟨σω⟩k = πa2
+k
+�8kBT
+πmH2
+� 1
+2 ��������1 +
+� π
+2τk
+� 1
+2 �������� .
+(A.10)
+We also define the conductivity σs and cyclotron frequency ωs
+of species s
+σs = nsq2
+sτsH2
+ms
+,
+(A.11)
+ωs = qsB
+cms
+,
+(A.12)
+where qs is the electric charge of species s, B the magnetic field
+strength and c the speed of light. The parallel, perpendicular and
+Hall conductivities are expressed as
+σ|| =
+�
+s
+σs,
+(A.13)
+σ⊥ =
+�
+s
+σs
+1 + (ωsτsH2)2 ,
+(A.14)
+σH =
+�
+s
+σsωsτsH2
+1 + (ωsτsH2)2 .
+(A.15)
+(A.16)
+Finally, the Ohmic, Hall and ambipolar resisitivities are defined
+as
+ηO = 1
+σ||
+,
+(A.17)
+ηH =
+σH
+σ2
+⊥ + σ2
+H
+,
+(A.18)
+ηA =
+σ⊥
+σ2
+⊥ + σ2
+H
+− 1
+σ||
+.
+(A.19)
+(A.20)
+Appendix B: Coagulation and dust-to-gas ratio
+Our coagulation model assumes that grains are perfectly cou-
+pled with the gas and well-mixed, so that the dust-to-gas mass
+ratio is constant. However, grains of different sizes have dif-
+ferent aerodynamic properties and may experience a significant
+drift through the gas. This characteristic is usually determined
+by their Stokes number, that is the ratio between the stopping
+time of the grain (Epstein 1924) and the dynamical timescale of
+the system. In particular, Lebreuilly et al. (2020) showed strong
+variations in the dust-to-gas mass ratio during the protostellar
+collapse. The disk and first core tend to be dust-enriched while
+the envelope becomes dust-depleted. This has strong implication
+for the coagulation, as a higher grain density promotes collisions
+and speeds up their growth (conversely for a lower density). In
+this section, we briefly explore a refinement to the coagulation
+model presented in Paper I.
+The general expression of the Smoluchowski (1916) equa-
+tion, from which we derive the expression of χ, is (Paper I eq. 5)
+dX(a, t)
+dt
+= CglocalnHI(a, X, t),
+(B.1)
+where C is a constant, glocal a function of the local properties of
+the gas (density, temperature...), and
+I(a, X, t) = −
+� ∞
+0
+h(m, m′)X(m, t)X(m′, t)dm′
++ 1
+2
+� m
+0
+h(m − m′, m′)X(m − m′, t)X(m′, t)dm′.
+(B.2)
+The relative number of grains of size a at a given time t is
+X(a, t) = ngrain/nH where ngrain is the number density of grains.
+We can then write X = dgx, where dg is the dust-to-gas ratio, and
+x the normalized relative number of grains. We can then rewrite
+equation (B.1)
+dx(a, t)
+dt
+= CglocalnHdgI′(a, x, t),
+(B.3)
+and include dg in the definition of our coagulation variable, giv-
+ing it the form
+dχ′ = glocalnHdgdt.
+(B.4)
+In our case, for the Ormel & Cuzzi (2007) kernel,
+dχ′ = n
+3
+4
+HT − 1
+4 dgdt.
+(B.5)
+This expression means that the dust-to-gas ratio does not change
+the evolution path of the size distribution, only the coagulation
+Article number, page 9 of 11
+
+A&A proofs: manuscript no. Marchandetal2023
+speed. A dust-to-gas ratio N times higher requires a χ value N
+times lower to reach the same coagulated state. This new defini-
+tion of χ′ can be used in an environment with varying dust-to-
+gas ratio. In hydrodynamics simulations, it is possible to couple
+it with the drift of one grain size close to the peak of the mass
+density distribution, as allowed by the method proposed by Le-
+breuilly et al. (2019).
+Although it is not yet possible to associate this coagulation
+method with the differential drift of grains of different sizes, this
+is a first step towards a self-consistent treatment of dust grains
+in hydrodynamics simulations. This method will lead to an over-
+estimate of the small-to-larger grain ratio but this is probably a
+decent approximation to get the total dust-to-gas ratio. As shown
+in Lebreuilly et al. (2020), as long as the large grains dominate
+the mass in the distribution, they also dominate the differential
+gas and dust dynamics. In other words, most of the dust enrich-
+ment comes from the dynamics of large grains.
+Appendix C: The fragmentation barrier
+Ormel et al. (2009) derived criteria for the fragmentation of dust
+aggregates. They define the rolling energy Eroll that determines
+the energy needed to restructure the grain. In Section 4.2 of pa-
+per I, we misinterpreted their results by assuming the fragmen-
+tation would occur for a kinetic energy Ekin = 5Eroll. The actual
+criterion is
+Ekin > 5NtotEbr,
+(C.1)
+where Ekin is the kinetic energy, Ebr is the breaking energy and
+Ntot is the total numbers of monomers composing the two collid-
+ing grains. Ebr is defined by (Dominik & Tielens 1997)
+Ebr = Abrγ5/3
+grain
+(a0/2)4/3
+ε2/3
+,
+(C.2)
+with Abr = 2.8 × 103, γgrain the surface energy density of the ma-
+terial, a0 the size of the monomers composing the grains, and ε
+the reduced elastic modulus. As in Ormel et al. (2009), we adopt
+a0 = 0.1 µm. Ice mantles on grains make them more resistant
+to fragmentation. Therefore, we assume bare silicates to obtain a
+lower limit for a fragmentation criterion. In this case, γgrain = 25
+erg cm−2 and ε = 2.8×1011 dyn cm−2. The kinetic energy of two
+grains of mass m and m′ reads
+Ekin = 1
+2
+mm′
+m + m′ ∆v2,
+(C.3)
+where ∆v is the relative velocity between the two grains. This
+velocity is given by the kernel of Ormel & Cuzzi (2007) that we
+use in Paper I
+∆v =
+� 3√
+8
+z0[kBG]
+1
+2 γρs
+µmH
+� 1
+2
+n
+− 1
+4
+H T − 1
+4 a
+1
+2 ,
+(C.4)
+where z0 = 2.97, kB and G are the Boltzmann and gravitational
+constants, γ = 5/3 the adiabatic index of the gas, ρs = 2.3 g cm−3
+the bulk density of the grains, µ = 2.3 the average atomic mass
+of the gas, mH the proton mass, and a the radius of the larger of
+the two grains. Assuming two identical grains and
+m = 4
+3πρsa3
+0Ntot,
+(C.5)
+equation C.1 becomes
+a > 15Ebr
+πρsa3
+0
+� 3√
+8
+z0[kBG]
+1
+2 γρs
+µmH
+�−1
+n
+1
+2
+HT
+1
+2 .
+(C.6)
+Replacing by numeric values, we find
+a > 826 µm
+�
+nH
+1010 cm−3
+� 1
+2 � T
+10 K
+� 1
+2
+.
+(C.7)
+This limit is much higher than the one derived in paper I, and we
+do not reach it in our simulations. The value of γgrain we use is
+taken from Ormel et al. (2009), but surface energies of different
+materials can reach 10 to 100 times this value, resulting in a
+much higher estimate of the fragmentation threshold (as Ebr ∼
+γ5/3
+grain). The value derived in equation (C.7) should then be taken
+as very conservative.
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