| { | |
| "paper_id": "1991", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T07:35:47.320930Z" | |
| }, | |
| "title": "AN EFFICIENT CONNECTIONIST CONTEXT-FREE PARSER", | |
| "authors": [ | |
| { | |
| "first": "Klaas", | |
| "middle": [], | |
| "last": "Sikkel", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "University of Twente", | |
| "location": { | |
| "postBox": "PO box 217", | |
| "postCode": "7500 AE", | |
| "settlement": "Enschede", | |
| "country": "The Netherlands" | |
| } | |
| }, | |
| "email": "[email protected]" | |
| }, | |
| { | |
| "first": "Anton", | |
| "middle": [], | |
| "last": "Nijholt", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "University of Twente", | |
| "location": { | |
| "postBox": "PO box 217", | |
| "postCode": "7500 AE", | |
| "settlement": "Enschede", | |
| "country": "The Netherlands" | |
| } | |
| }, | |
| "email": "[email protected]" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "A connectionist network is defined that parses a grammar in Chomsky Normal Form in logarithmic time, based on a modification of Rytter's recognition algorithm. A similar parsing network can be defined for an arbitrary context-free grammar. Such net works can be integrated into a connectionist parsing environment for interactive distributed processing of syntactic, semantic and pragmatic information.", | |
| "pdf_parse": { | |
| "paper_id": "1991", | |
| "_pdf_hash": "", | |
| "abstract": [ | |
| { | |
| "text": "A connectionist network is defined that parses a grammar in Chomsky Normal Form in logarithmic time, based on a modification of Rytter's recognition algorithm. A similar parsing network can be defined for an arbitrary context-free grammar. Such net works can be integrated into a connectionist parsing environment for interactive distributed processing of syntactic, semantic and pragmatic information.", | |
| "cite_spans": [], | |
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| "section": "Abstract", | |
| "sec_num": null | |
| } | |
| ], | |
| "body_text": [ | |
| { | |
| "text": "Connectionist networks are strongly intercon nected groups of very simple processing units. Such networlcs are studied in natural language processing since their inherent parallelism and distributed deci sion making allows an integration of syntactic, semantic and pragmatic processing for language analysis. See, e.g., (Waltz and Pollack, 1988) , (Cotrell and Small, 1989) . By isolat_ing the syntactic component -without abandoning the connectionist paradigm -it becomes possible to study context-free parsing in environments where we can make different assumptions about types of networks, learning rules and representations of concepts. Examples of this type of research can be found in (Fanty, 1985) , a sim ple connectionist implementation of the CYK method; (Selman and Hirst, 1987) , Boltzmann machine parsing; (Howells, 1988) , a relaxation algo rithm that utilizes decay over time; (Nakagawa and Mori, 1988) , a parallel left-corner parser incorporated in a learning network; (Nijholt, 1990 ), a Fanty-like connectionist Earley parser.", | |
| "cite_spans": [ | |
| { | |
| "start": 320, | |
| "end": 345, | |
| "text": "(Waltz and Pollack, 1988)", | |
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| { | |
| "start": 348, | |
| "end": 373, | |
| "text": "(Cotrell and Small, 1989)", | |
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| }, | |
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| "start": 691, | |
| "end": 704, | |
| "text": "(Fanty, 1985)", | |
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| }, | |
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| "end": 789, | |
| "text": "(Selman and Hirst, 1987)", | |
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| }, | |
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| "start": 819, | |
| "end": 834, | |
| "text": "(Howells, 1988)", | |
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| "start": 892, | |
| "end": 917, | |
| "text": "(Nakagawa and Mori, 1988)", | |
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| }, | |
| { | |
| "start": 986, | |
| "end": 1000, | |
| "text": "(Nijholt, 1990", | |
| "ref_id": "BIBREF3" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "INTRODUCTION", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In this paper we push the speed of the parsing net work to its limits, so as to investigate how much parallellism is possjble in principle. We define a parsing network that constructs a shared forest of parse trees in O (log n) time for an input string of length n, using O (n 6 ) units. Our network is based upon Fanty 's \"dynamic programming\" approach and a type of algorithm first introduced by Rytter (1985) . The network is rather large, but not too large: no logarithmic-time parsing algorithm for arbitrary contrext-free languages is known that uses less than 0 (n 6 ) processors. Furthermore, the number of units can drastically be reduced (albeit within the same complexity bounds) by a meta-parsing algorithm that constructs a minimal network custom-tailored for a specific grammar.", | |
| "cite_spans": [ | |
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| "start": 398, | |
| "end": 411, | |
| "text": "Rytter (1985)", | |
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| "section": "INTRODUCTION", | |
| "sec_num": null | |
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| "text": "After some preliminary definitions, we construct a network for a grammar in Chomsky Normal Form. At the end of this paper we argue that a similar net work can be built for an arbitrary CFG; space limita tions do not all<?W a detailed presentation.", | |
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| "section": "INTRODUCTION", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Let G = (N, \ufffd, P, S) be a grammar in Chomsky Normal Form (CNf), i.e., production rules have the form A-+BC or A-+a. We consider input strings a 1 \u2022 \u2022 \u2022 a m with m < n, where n is an implementation dependent constant.", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A ce \u2022 ntral role in the parsing algorithm is played by items of the form (A, i,j), which are called trian gles.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A triangle (A , i,j) is called recognizable\u2022 if A \ufffd + a i+l \u2022 \u2022 \u2022 a j '. The set of triangles S is defined by def S = { (A , i,j) I A EN, 0 s i < j s n} . A triangle (A ,i,j) is called parsable if it is recog nizable and S \ufffd + a 1 \u2022 \u2022 \u2022 a i Aa j + l \u2022 \u2022 \u2022 a m .", | |
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| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
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| "text": "The collec tion of all arsable trian les is called the shared fo rest of an in ut sentence\u2022 \"forest\" because it comprises all different parse trees for that sentence, \"shared\" as common sub-trees of different parse trees are represented only once. The algorithm and net work in section 3 compute the shared forest of a sen tence.", | |
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| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
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| { | |
| "text": "We shall also need items of a different kind, called triangles with a gap, denoted ((A,i,j),(B,k, l)). A triangle with a gap ((A,i,j), ( B,k, l)) is called pro", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
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| { | |
| "text": "posable if A \ufffd + a i+I \u2022 \u2022 \u2022 a k Ba 1+1 \u2022 \u2022 \u2022 a j .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
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| "text": "During the application of the algorithm, we will propose trian gles with a gap that need further investigation. If ((A,i,j),(B,k, l)) has been proposed and (13,k,l) can pe recognized, we can fill up the gap and recognize (A , i, j) . The set of all triangles with a gap is denoted by", | |
| "cite_spans": [ | |
| { | |
| "start": 221, | |
| "end": 231, | |
| "text": "(A , i, j)", | |
| "ref_id": null | |
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| "section": "PRELIMINARIES AND DEFINITIONS", | |
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| { | |
| "text": "def r = {((A,i,j),(B,k,l)) I ( A,i,j)E s, (B, k, l)Es, i s k < l s j, i ;it k or l ;it n}", | |
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| "text": "The gap can be at the inside or at the outside of a tri-angle, as shown in Figure 1 . The size of a triangle is defined as the length of the substring a;+ 1 \u2022 \u2022 \u2022 a j : size((A,i,j)) = j -i. The size of a ' triangle with a gap is defined as the size of the triangle minus the size of the gap: size (((A,i,j) , (B,k, l) ", | |
| "cite_spans": [ | |
| { | |
| "start": 298, | |
| "end": 307, | |
| "text": "(((A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 310, | |
| "end": 318, | |
| "text": "(B,k, l)", | |
| "ref_id": null | |
| } | |
| ], | |
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| "start": 75, | |
| "end": 83, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A / \u2022 \ufffd i --k/::-. l=j", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ")) = size((A,i,j)) -size((B,k, l)) = j -i -1 + k.", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "PRELIMINARIES AND DEFINITIONS", | |
| "sec_num": null | |
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| "text": "The algorithm presented here is a (for our pur pose) improved version of Rytter\ufffds recognition algo rithm (Gibbons and Rytter, 1988) . It can be trivially extended into a parsing algorithm and has a simpler correctness proof. Remarks about the differences with the original algorithm are deferred to the end of this section, so as to keep the expose as clear as pos sible. We will describe first what is to be computed by the algorithm, and elaborate on how to compute it afterwards. The recognition algorithm uses two tables of boolean values:", | |
| "cite_spans": [ | |
| { | |
| "start": 105, | |
| "end": 131, | |
| "text": "(Gibbons and Rytter, 1988)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 recognized ((A,i,j)) for (A ,i,j)E'E, which is true once we have established that (A , i,j) is indeed recognizable, and fa lse otherwise;", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 proposed (((A,i,j),(B,k, l))) for ({A,i,j), (B,k, l))", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "E r, which is true once we have established that (B,k, l) ) is indeed proposable, and fa lse otherwise.", | |
| "cite_spans": [ | |
| { | |
| "start": 49, | |
| "end": 57, | |
| "text": "(B,k, l)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "((A,i,j),", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The algorithm will satisfy the following loop invariant properties: Acceptance or rej ection of the input string depends on the recognizability of (S, O,m), hence the number of steps that need to be performed is 1 2 1og ml (the smallest integer \ufffd 2 log m).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The well-known Cocke-Younger-Kasami algo rithm uses an upper triangular recognition table T CYK: The nonterminal A is added to table entry t;,j if {A, i,j) is recognized. The statements A E t;,j and recognized ({A,i,j)) = true are equivalent. We can illustrate the results of our algorithm (though the operations are different) with an extension of the T CYK recognition table. In this case, the recognition table is a three-dimensional structure, The third index k denotes the size of an item. When a triangle (A ,i,j) is recognized, the nonterminal A is added to t;, j ,j-i \u2022 When a triangle with a gap Figure 2 shows the surface of T R after appli cation of the algorithm.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 605, | |
| "end": 613, | |
| "text": "Figure 2", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "T R = {t;,j,k I Osi<jsn, Osks j -i} .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "( {A, i,j), (B, k, l)) is proposed, an object {A, B, k, l) is added to t;, i, h with h j -i -l + k size(({A,i,j),(B,k, l))", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Having illustrated the purpose of the recognition algorithm, we can now explain \u2022 how it works. We define the following operations on 2, r and the tables recognized and proposed; A, i,j) , (B ,k, I ) ", | |
| "cite_spans": [ | |
| { | |
| "start": 179, | |
| "end": 186, | |
| "text": "A, i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 189, | |
| "end": 199, | |
| "text": "(B ,k, I )", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "A FAST CONNECTIONIST PARSING NET WORK FOR CNF GRAMMARS A variant of Rytter's recognition algorithm", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "i E P then recognized((A, i -l,i)) := true ti od PROPOSE : for all (A \ufffd i,j), (B,i,k), (C,k,j)E 2 such that A -+BC E P do if recognized((B, i,k)) ) then proposed(((A,i,j), (C, k,j))) : = true ti ; } if recognized((C, k, j)) then proposed(((A,i,j), (B,i,k))) := true ti od A / )c . _ _ _ _ _ ---k \u2022 \u2022\u2022 \u2022 \u2022\u2022\u2022 \u2022 \u2022 \u2022 j", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "INITIA LIZE for aJI (A,i,j)E 'E. do recognized((A, i, j)) := fa lse od ; for ali ((A,i,j),(B,k, l))E r do proposed(((A,i,j),(B,k, l))) := fa lse od ; for all ( A, i -1 , i)E 'E. do if A --+a", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ")) and recognized((B,k, l)) then recognized((A, i,j)) := true ti od A \ufffd B \ufffd i-k \u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022\u2022 l-j + B /\ufffd k --1 A =/\ufffd -------j", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "RECOGNIZE : for all ((A, i,j), (B,k, l))Er , (B ,k, l)E 2 do if proposed(( (", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "and proposed (((B,k, l) ,(C,m,n))) then proposed(((A,i,j), (C, m,n ) In the sequel, we will give a proof of the correctness of the modified Rytter algorithm. But let's first look at an example.", | |
| "cite_spans": [ | |
| { | |
| "start": 13, | |
| "end": 23, | |
| "text": "(((B,k, l)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 59, | |
| "end": 68, | |
| "text": "(C, m,n )", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "COMBINE : fo r all ((A,i,j),(B,k, l)) , ((B, k, l),(C,m,n))Er \ufffd do if proposed(((A,i,j),(B,k,l)))", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ")) := true ti od B /c\ufffd k -m \u2022 \u2022 \u2022 \u2022 \u2022 n -l A i --m \u2022\u2022\u2022\u2022\u2022n --j", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "COMBINE : fo r all ((A,i,j),(B,k, l)) , ((B, k, l),(C,m,n))Er \ufffd do if proposed(((A,i,j),(B,k,l)))", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In Figure 6 one parse tree of the input sentence is shown. The algorithm obviously recognizes much more than a single parse tree, but it is sufficient to show that all items in one parse tree are recognized in order to make clear that the top item is recognized. (S, 0, 8) can be recognized in a number of different ways, but that would only clutter up the example. The nodes in the parse tree have been numbered, so \ufffde can identify the triangles by their number rather than by the more cumbersome (A,i,j) notation. We will apply the algorithm step-:-by-step on the items in this tree.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 3, | |
| "end": 11, | |
| "text": "Figure 6", | |
| "ref_id": "FIGREF6" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "COMBINE : fo r all ((A,i,j),(B,k, l)) , ((B, k, l),(C,m,n))Er \ufffd do if proposed(((A,i,j),(B,k,l)))", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Step O is shown in Figures 7-8, step 1 in Figures 9-12 and (the first half ot) step 2 in . Figures 13-14 . Circles correspond to recognized triangles, lines .correspond to proposed triangles with gaps. The example shows that the algorithm may need less than 1 2 log kl steps in some cases; we need only 2 steps although 3 are allowed. We will give a proof that is a great deal simpler than the proof of the original algorithm (Gibbons and Rytter, 1988) . For more details see (Sikkel and Nijholt, 1990) . Basis. It is easy to verify that proposable trian gles with a gap of size 1 have been proposed after the initialization step and recognizable triangles of size s 2 have been recognized after step 1.", | |
| "cite_spans": [ | |
| { | |
| "start": 427, | |
| "end": 453, | |
| "text": "(Gibbons and Rytter, 1988)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 477, | |
| "end": 503, | |
| "text": "(Sikkel and Nijholt, 1990)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 42, | |
| "end": 54, | |
| "text": "Figures 9-12", | |
| "ref_id": "FIGREF0" | |
| }, | |
| { | |
| "start": 91, | |
| "end": 105, | |
| "text": "Figures 13-14", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "COMBINE : fo r all ((A,i,j),(B,k, l)) , ((B, k, l),(C,m,n))Er \ufffd do if proposed(((A,i,j),(B,k,l)))", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Induction hypothesis. We write (I) k for the claim that (I) holds for s with size(s) s 2 k and (Il) k for the claim that (II) holds for (s, YI) with size@, Y I )) s 2 k . Hence (11) 0 and (1) 1 have been established above. From the induction hypothesis (II) k -l, (Ih we will derive (II) k , (I) k +l \u2022 (Il)k-Given (Il)t_ 1 , (l)t, we prove (II)t. Let (s, YI) be proposable, 2 kl < size ((s, YI)) s 2 k . ", | |
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| "section": "COMBINE : fo r all ((A,i,j),(B,k, l)) , ((B, k, l),(C,m,n))Er \ufffd do if proposed(((A,i,j),(B,k,l)))", | |
| "sec_num": null | |
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| "text": "We define a connectionist network in a way that resembles the parsing network of Fanty (1985) . The network consists of simple units computing AND and OR functions. The output of every unit is either 1 or 0. An AND unit is activated -i.e. its outputs have value 1 -iff all its inputs have value one; an OR unit is activated iff at least one of its inputs has value 1. In neural networks terminology: an AND unit with k inputs has a threshold value k -0.5, an OR unit has a threshold value 0.5, irrespective of its number of inputs. In order to make a distinction between the two types of units we will write OR units between parentheses \"( )\" and AND units between brackets \"[ ] \" . A, i,j) ). Likewise, proposed (((A, i,j) , (B,k, l) )) is represented by a unit (((A,i,j) , (B,k,l) A,i,j) , (C,k,j))) and a link from (R (C,k,j)) to (((A,i,j) , (B, i, k) )) are added to the network. \u2022 For an implementation of RECOGNIZE, we need additional match units [((A,i,j) , (B,k, l) ) ] for each ((A,i,j) , (B,k,l) )Ef. This is because a unit (R (A ,i,j)) can be recognized in more than one way. It should be recognized if both (((A,i,j) , (B,k,l) )) and (R (B, k, l) ) are active for some (B, k, l) E2. To this end, (R (B, k, l) ) and (((A,i,j) , (B,k,l) )) are linked to [((A,i,j) , (B,k, l) )] that ANDs their values; [((A,i, j) , (B,k, l) )] is linked to (R (A,i,j) ).", | |
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| "start": 81, | |
| "end": 93, | |
| "text": "Fanty (1985)", | |
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| "start": 694, | |
| "end": 723, | |
| "text": "Likewise, proposed (((A, i,j)", | |
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| "text": "l) E2. To this end,", | |
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| "end": 1273, | |
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| "end": 1284, | |
| "text": "(B,k, l)", | |
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| { | |
| "start": 1293, | |
| "end": 1322, | |
| "text": "ANDs their values; [((A,i, j)", | |
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| "start": 1325, | |
| "end": 1333, | |
| "text": "(B,k, l)", | |
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| { | |
| "start": 683, | |
| "end": 690, | |
| "text": "A, i,j)", | |
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| { | |
| "start": 763, | |
| "end": 772, | |
| "text": "(((A,i,j)", | |
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| }, | |
| { | |
| "start": 783, | |
| "end": 789, | |
| "text": "A,i,j)", | |
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| { | |
| "start": 833, | |
| "end": 842, | |
| "text": "(((A,i,j)", | |
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| }, | |
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| "start": 1149, | |
| "end": 1158, | |
| "text": "(B, k, l)", | |
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| "start": 1350, | |
| "end": 1360, | |
| "text": "(R (A,i,j)", | |
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| } | |
| ], | |
| "eq_spans": [], | |
| "section": "A recognizing network", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 For the COMBINE operation, we also need addi tional match units. For each ((A,i,j) , (B,k, l) ) and ((B,k, l) , (C,m,n) An example of a small fraction of the network is given in Figure 17 . It represents the units that are used for the recognition of the propositional phrase ( PP, 5, 8). ", | |
| "cite_spans": [ | |
| { | |
| "start": 76, | |
| "end": 84, | |
| "text": "((A,i,j)", | |
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| }, | |
| { | |
| "start": 87, | |
| "end": 95, | |
| "text": "(B,k, l)", | |
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| }, | |
| { | |
| "start": 102, | |
| "end": 111, | |
| "text": "((B,k, l)", | |
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| { | |
| "start": 114, | |
| "end": 121, | |
| "text": "(C,m,n)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 180, | |
| "end": 189, | |
| "text": "Figure 17", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "For each triangle (A ,i,j)E2, the network contains a unit (R (A ,i,j)) with an activation level correspond ing to the value of recognized((", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The main purpose of modifying Rytter's algo rithm is the introduction of invariant (II). It will become clear now why we need it. Tacitly we have done all the necessary preparations for the extension to a parsing algorithm, all that is left is to reap the results.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Let (A,i,j) be parsable for a particular input string a 1 \u2022\u2022\u2022a m (m sn), and assume (A ,i,j In other words, (A,i,j) is parsable if (A ,i,}) is recog nizable and ((S, 0,m), (A ,i, j ) ) is proposable. Conse quently, parsed ((A,i,j) proposed (((S, 0,m), (A,i,j) )) and recognized ( (A,i,j) ) are true. This can be done in parallel in one step! We define an additional boolean table parsed((A,i,j) ) for (A,i,j)E'E. and an operation on 'E., r and the table parsed, as follows: parsed({A,i,j) ((A,i,j) ) then parsed((A,i,j) ", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 4, | |
| "end": 11, | |
| "text": "(A,i,j)", | |
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| }, | |
| { | |
| "start": 58, | |
| "end": 91, | |
| "text": "\u2022\u2022\u2022a m (m sn), and assume (A ,i,j", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 92, | |
| "end": 139, | |
| "text": "In other words, (A,i,j) is parsable if (A ,i,})", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 161, | |
| "end": 182, | |
| "text": "((S, 0,m), (A ,i, j )", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 206, | |
| "end": 230, | |
| "text": "quently, parsed ((A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 231, | |
| "end": 259, | |
| "text": "proposed (((S, 0,m), (A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 280, | |
| "end": 287, | |
| "text": "(A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 380, | |
| "end": 394, | |
| "text": "parsed((A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 474, | |
| "end": 488, | |
| "text": "parsed({A,i,j)", | |
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| }, | |
| { | |
| "start": 489, | |
| "end": 497, | |
| "text": "((A,i,j)", | |
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| }, | |
| { | |
| "start": 505, | |
| "end": 519, | |
| "text": "parsed((A,i,j)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "PAR SE : for all (A ,i,j) E'E. . do", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ") := true fi od", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The recognition algorithm is extended to a full fledged parsing algorithm by inserting one PARSE operation after the repeat loop.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We can extend the network accordingly. The table parsed will be represented by a collection of AND units [P (A,i,j) ]. Additionally, we need a col lection of match units [Q m (A ,i,j)] for lsm sn and (A ,i,j)E'E. and a collection of OR match units (Q (A ,i,j)) for (A ,i,j)E'E.. \u2022 (Q (A ,i,j)) will be activated if the above holds for some m. To this end, each [Q m (A ,i,j)] is linked to (Q (A,i,j) ).", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 108, | |
| "end": 115, | |
| "text": "(A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 389, | |
| "end": 399, | |
| "text": "(Q (A,i,j)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 [P (A ,i,j)], obviously, receives input from (R (A ,i,j)) and (Q (A ,i,j)).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 In order to start the parsing phase, all [ accept, m] units are linked to (Q (S, 0,m)). If a string of length m is accepted, then (R (S, 0,m)) will be active, hence [P (S, 0,m)] will be activated.", | |
| "cite_spans": [ | |
| { | |
| "start": 43, | |
| "end": 55, | |
| "text": "[ accept, m]", | |
| "ref_id": null | |
| } | |
| ], | |
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| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "If (Q (A ,i,j)) is activated (via [Q 1 (A ,i,j) A,i,j) ) can be COMBINEd. Thus [ P (A ,i,j)] will be activated if and only if (A ,i,j) is parsable.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 29, | |
| "end": 47, | |
| "text": "(via [Q 1 (A ,i,j)", | |
| "ref_id": "FIGREF0" | |
| }, | |
| { | |
| "start": 48, | |
| "end": 54, | |
| "text": "A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 126, | |
| "end": 134, | |
| "text": "(A ,i,j)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Construction of the shared forest", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The number of input units ((a, i)) is m: J + 1) \u2022 (n + 1) = 0(1: \ufffd: J \u2022 n). The COMBINE match units [((A,i,j) , (B,k,l) , (C,m,n))] account for the highest order of all other types of units, 0 (!NI 3 \u2022 n 6 ), yielding a total of O(J\"f.J \u2022 n + JNJ 3 \u2022 n 6 ) units. It is easy to verify that the number of connections is also O(JLI \u2022n + JNJ 3 \u2022n 6 ).", | |
| "cite_spans": [ | |
| { | |
| "start": 100, | |
| "end": 109, | |
| "text": "[((A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 112, | |
| "end": 119, | |
| "text": "(B,k,l)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Complexity of the network", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "These numbers conform to the best known com plexity measures for logarithmic parsing algorithms: O(log n) time on a.CRCW PRAM and O(log 2 n) time on a CREW PRAM. PRAM models use O(n 6 ) pro cessors. It is not obvious that an equivalent network exists with the same order of complexity. A general method to construct a network composed of AND and OR units for an arbitrary PRAM is given by Stockmeyer and Vishkin (1984) . Applying this gen eral method, however, would yield O (n 1 3 ) units, rather than our custom-tailored network of O (n 6 ) units.", | |
| "cite_spans": [ | |
| { | |
| "start": 389, | |
| "end": 418, | |
| "text": "Stockmeyer and Vishkin (1984)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Complexity of the network", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We defined units for all (A ,i,j)E'2 and ((A,i,j), (B,k, l))E f. A large fraction of these units will never be needed. For any \u2022 particular grammar we can establish a much smaller network, by an algo rithm that closely resembles the parsing algorithm. Such analysis has been called meta-parsing (Nijholt, 1990) . Similarly, ((A,i,j) , (B,k,l) ) is called meta-proposable if there is an input string such that (A ,i,j) is proposable; (A.,i,j) is called metaparsable if there is an input string such that (A.,i,j) is parsable. These meta-properties can be computed in advance, and incorporated in the structure of the net work. The meta-recognizable and meta-parsable items for our example grammar and n = 8 are shown in tabular form in Figures 18 and 19 . The meta parsing algorithm is identical to the parsing algorithm but for two small differences:", | |
| "cite_spans": [ | |
| { | |
| "start": 295, | |
| "end": 310, | |
| "text": "(Nijholt, 1990)", | |
| "ref_id": "BIBREF3" | |
| }, | |
| { | |
| "start": 335, | |
| "end": 342, | |
| "text": "(B,k,l)", | |
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| }, | |
| { | |
| "start": 433, | |
| "end": 441, | |
| "text": "(A.,i,j)", | |
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| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 313, | |
| "end": 332, | |
| "text": "Similarly, ((A,i,j)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 735, | |
| "end": 752, | |
| "text": "Figures 18 and 19", | |
| "ref_id": "FIGREF0" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 meta-recognized ((A,i-l,i) ) is made true if A-a EP fo r any a EL,", | |
| "cite_spans": [ | |
| { | |
| "start": 18, | |
| "end": 28, | |
| "text": "((A,i-l,i)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 meta-parsed((S, O, i)) is made true fo r every (S, 0, i) that has been meta-recognized.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "It is easy to verify that after application of the meta algorithm, (A ,i,j) has been recognized if and only if (A ,i,j) is meta-recognizable; similarly for meta proposable and meta-parsable items.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "For the construction of the shared forest, we only have to consider triangles that are meta-parsable. All triangles that are not meta-parsable can be discarded:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "124 l *d *n NP VP s VP s *v *p pp NP pp NP *d *n NP VP s VP *v *p pp NP pp *d *n NP VP s VP *v *p pp NP pp *d *n NP VP s *v *p pp NP *d *n NP VP *v *p pp *d *n NP VP *v *p pp *d *n NP *v *p *d *n *v *p Figure 18. A Eti,j if (A.,i,j) is meta-recognizable I *d NP s s NP *n *v VP VP * p pp *d NP NP *n *v VP * p pp *d NP *n -- Figure 19. A Eti,j if (A ,i,j) is meta-parsable", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "even if such a triangle is recognized, it can never con tribute to a parse tree for any input. Thus we define the minimal set of triangles and triangles with gaps, as follows:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ".:::.min {(A ,i,j)E'2 I (A ,i,j) is meta-parsable} ,", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 24, | |
| "end": 32, | |
| "text": "(A ,i,j)", | |
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| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "r min {((A,i,j),(B,k,l))E f I ( (A, i,j), (B,k, l)) is meta-proposable}", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "While constructing the network, we only have to introduce units for (A.,i,j)E'2 min , ((A.,i,j),(B,k,l))", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Er min and appropriate match units. The reduced network still yields the shared forest.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Meta-parsing", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In contrast with Fanty's network, even the minimal network is rather robust. When a few units do not function, it is most likely that the proper input strings will be accepted. There is a multitude of dif ferent ways in which a triangle can be recognized; if the most direct path is broken, chances are that the triangle is recognized by an alternative path, using slightly more time. That is, unless one of the rela tively few vital units breaks down, the recognition network shows graceful degradation. The parsing part of the network has no redundancy, however. If any unit fails, a triangle in the shared forest may be lost: But this is less . dramatic than failure to recog nize a valid sentence.", | |
| "cite_spans": [], | |
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| "section": "Robustness of the network", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "It is possible to supplement the recognition net work with a robust parsing network if a top-down structure is used that is equivalent to the bottom-up structure, as in Fanty's network. Such a top-down network would yield a parse forest in logarithmic, rather than constant time. But :that :does not really matter as time complexity of the network is loga rithmic anyway.", | |
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| "section": "Robustness of the network", | |
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| "text": "The CYK alg o rithm cari be found in any textbook on formal languages, e.g. (Harrison, 1978) . A con nectionist network for the CYK algorithm has been defined by Fanty (1985) and circulated on a wider scale in (Fanty, 1986 ).", | |
| "cite_spans": [ | |
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| "end": 92, | |
| "text": "(Harrison, 1978)", | |
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| "text": "(Fanty, 1986", | |
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| "section": "Bibliographic notes", | |
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| { | |
| "text": "Rytter's recognition algorithm is presented in (Rytter, 1985) and (Gibbons and Rytter, 1988) . A similar algorithm is independently described by Brent and Goldschlager (1984r The operators PROPOSE, COMBINE and RECOGNIZE were called ACTNA TE , SQUARE and PEBBLE in the original algorithm. The word \"activate\" had to be changed so as to avoid confusion with activation of a unit. The new identifiers are chosen because we operate in a parsing context (\"recognize\") rather then a combina torial context (\"pebble\"). Rytter's algorithm per forms the following steps:", | |
| "cite_spans": [ | |
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| "start": 47, | |
| "end": 61, | |
| "text": "(Rytter, 1985)", | |
| "ref_id": null | |
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| "start": 66, | |
| "end": 92, | |
| "text": "(Gibbons and Rytter, 1988)", | |
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| "section": "Bibliographic notes", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 step 0: INITIALIZE", | |
| "cite_spans": [], | |
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| "section": "Bibliographic notes", | |
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| "text": "\u2022 step k (k > 0): ACTNA TE ; SQUARE; SQUARE; PEBBLE I which do not satisfy invariant (II)! Hence the algorithm does not allow a similar trivial extension for the computation of a shared forest. In (Gibbons and Rytter, 1988) , the correctness of the Rytter's algo rithm is derived from the correctness of a \"pebble game\" on binary trees, which has a rather compli-", | |
| "cite_spans": [ | |
| { | |
| "start": 197, | |
| "end": 223, | |
| "text": "(Gibbons and Rytter, 1988)", | |
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| ], | |
| "back_matter": [ | |
| { | |
| "text": "\u2022n is possible to define a similar parsing network for 'an arbitrary context-free grammar. \u2022 Rytter's algo rithm can be regarded as a speed-up of the CYK algo rithm, using more resources. In the same way, bottom-up versions of Eatley's algorithm (Graham, Harrison and Ruzzo, 1980) , (Chiang and Fu, 1984) can be speeded up in a similar way. Triangles have the form (A\ufffda.f3,i,j) for A\ufffdaf3EP and O\ufffdi sj \ufffd\u2022n.A \ufffda. 13 is recognizable iff a ==:Proposability can be defined accordingly. A triangle (A \ufffda. 13, i, j) is parsable if there is a yThe network for arbitrary CFGs has O(g 3\u2022 IPl 3 \u2022 n 6 ) units and O(g 3 \u2022 IPl 3 ; n 6 ) connec tions, in which g is t\ufffde average number of symbols in the right-hand side of a grammar rule. For a full treatment we refer to (Sikkel and Nijholt, 90) .A similar parsing algorithm for arbitrary CFGs on PRAM models is discussed in ( de Vreught and Honig, 1991) .", | |
| "cite_spans": [ | |
| { | |
| "start": 246, | |
| "end": 280, | |
| "text": "(Graham, Harrison and Ruzzo, 1980)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 283, | |
| "end": 304, | |
| "text": "(Chiang and Fu, 1984)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 757, | |
| "end": 777, | |
| "text": "(Sikkel and Nijholt,", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 778, | |
| "end": 781, | |
| "text": "90)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 866, | |
| "end": 890, | |
| "text": "Vreught and Honig, 1991)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "EXTENSION;TO ARBITRARY CONTEXT FREE-GRAMMARS", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A modification of Rytter's logarithmic \u2022 time recognition algorithm for CNF grammars has been introduced. This algorithm is conceptually easier than the original, and the correctness proof is a great deal simpler. Furthermore, the construction of a shared parse forest represented by a set of triangles can be added in constant time.We have defined a connectionist network that parses a CNF grammar with the above algorithm in O(log n) time using o m: i -n + INl 3 \u2022n 6 ) units. This conforms to the best known complexity bounds on a CRCW PRAM, and is a factor logn faster than the best algorithm on a CREW PRAM known to date. A Similar . network can be constructed for an arbitrary context-free grammar.A network of minimum size for a particular gram mar can be custom-tailored. The meta-parsing algo rithm . that estab. lishes the configuration of a network for the specific grammar is almost id\ufffdntical to the parsing algorithm that is implemented by the net work.The network is robust in the sense that a few bro-. ken down units will most likely cause some degrada tion in performance but still all valid sentences will be recognized.A network structure with O (n 6 ) units is too large for any serious practical implementation in natural language processing. The purpose of our investiga tions, however, has been to push the time complexity to its very limits to see how much parallelism is pos sible in principle. These results confirm that connec tionist networks can be used as a suitable abstract machine model for parallel algorithms. It is also confirmed that traditional parsing algorithms for general context-free languages can be given connec tionist implementations, allowing integration \u2022 into \u2022 more comprehensive connectionist networks for natural language analysis. Brent, Richard P. and Goldschlager, Leslie M. (1984) . \"A Parallel Algorithm for Context-Free Pars ing,\" Australian Computer Science Communications 6, 7, 7.1-7.10. Chiang, YT. and Fu, K.S. (1984) . PAMI-6, 3, 302-314. Cottrell, Garrison W. (1989) . \"A Connectionist Approach to Word Sense Disambiguation \", Research Notes in Artificial Intelligence, Morgan Kaufmann Publishers. Fanty, Mark A. (1985) . \"Context-free Parsing in Connectionist Networks \", report TR 174, Computer Science Dept., University of Rochester, Rochester, NY. Fanty, Mark A. ( 1986) . M.A. Fanty: \"Context free Parsing in Connectionist networks,\" in: J.S: Denker (Ed.), \"Neural Networks fo r Computing\", Snowbird, UT, API conference proceedings 151, American Institute of Physics, 140-145. , Gibbons, Alan and Rytter, Wojciech (1988) ; 1 1 \u2022 \"Efficient Parallel Algorithms \", Cambridge Univer sity Press. __ . -Graham, Susan L, Harrison, Michael A. and Ruzzo, Walter L. (1980) . \"An Improved Context Free Recognizer,\" ACM Transactions on Program ming Languages and Systems 2 415-462. Harrison, Michael (1978) . \"Introduction to For mal Language Theory \", Addison-Wesley, Reading, Mass. Howells, Tim (1988) . \"VITAL: a Connectionist Parser,\" Proc. 10th Conf of the Cognitive Science Society 18-25.", | |
| "cite_spans": [ | |
| { | |
| "start": 1785, | |
| "end": 1837, | |
| "text": "Brent, Richard P. and Goldschlager, Leslie M. (1984)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 1866, | |
| "end": 1980, | |
| "text": "Context-Free Pars ing,\" Australian Computer Science Communications 6, 7, 7.1-7.10. Chiang, YT. and Fu, K.S. (1984)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 1983, | |
| "end": 2031, | |
| "text": "PAMI-6, 3, 302-314. Cottrell, Garrison W. (1989)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 2121, | |
| "end": 2184, | |
| "text": "Intelligence, Morgan Kaufmann Publishers. Fanty, Mark A. (1985)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 2245, | |
| "end": 2339, | |
| "text": "TR 174, Computer Science Dept., University of Rochester, Rochester, NY. Fanty, Mark A. ( 1986)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 2471, | |
| "end": 2590, | |
| "text": "UT, API conference proceedings 151, American Institute of Physics, 140-145. , Gibbons, Alan and Rytter, Wojciech (1988)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 2655, | |
| "end": 2733, | |
| "text": "Press. __ . -Graham, Susan L, Harrison, Michael A. and Ruzzo, Walter L. (1980)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 2832, | |
| "end": 2865, | |
| "text": "415-462. Harrison, Michael (1978)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 2912, | |
| "end": 2962, | |
| "text": "Addison-Wesley, Reading, Mass. Howells, Tim (1988)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "CONCLUSIONS", | |
| "sec_num": null | |
| } | |
| ], | |
| "bib_entries": { | |
| "BIBREF0": { | |
| "ref_id": "b0", | |
| "title": "The proof of the modified algorithm as presented ;above . is a lot simpler, mainly due to the 'introduction of invariant", | |
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| "raw_text": "proof. The proof of the modified algorithm as presented ;above . is a lot simpler, mainly due to the 'introduction of invariant (II).", | |
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| "BIBREF2": { | |
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| "links": null | |
| } | |
| }, | |
| "ref_entries": { | |
| "FIGREF0": { | |
| "num": null, | |
| "uris": null, | |
| "text": "Triangles with a gap", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF1": { | |
| "num": null, | |
| "uris": null, | |
| "text": "I) if size((A,i,j)) s 2 k and (A ,i,j) is recognizable then recognized((A ,i,j)) = true after k steps, (II) if size(((A,i,j), (B,k, l))) s 2 k and ({A,i,j),(B,k, l)) is proposable then proposed(((A,i,j),(B,k, l))) = true after k steps.", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF2": { | |
| "num": null, | |
| "uris": null, | |
| "text": "The surface of T R as it should be computed by the algorithm", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF3": { | |
| "num": null, | |
| "uris": null, | |
| "text": "Figure 3. PROPOSE", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF4": { | |
| "num": null, | |
| "uris": null, | |
| "text": "RECOGNIZE119", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF5": { | |
| "num": null, | |
| "uris": null, | |
| "text": "COMBINEThe functioning of the operators PR OPOSE, RECOGNIZE and COMBINE is illustrated in Figures 3 -5. Everything in a for all statement can be corn-\ufffd puted in parallel. The recognition algorithm, using these operators, can be given as:if recognized((S, 0,m)) then accep t else reject fi", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF6": { | |
| "num": null, | |
| "uris": null, | |
| "text": "A parse tree .", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF7": { | |
| "num": null, | |
| "uris": null, | |
| "text": "After step O(b ): After step l(a): RECOGNIZE After step l(b ): PROPOSE After step l(c) : COMBINE \ufffd (\u00bd\u00beGo \ufffd lq J.vi\ufffd /\ufffd' \ufffd After step 1( d): COMBINE Figure 13. After step 2(a): RECOGNIZE After step 2(b): PROPOSE Correctness of the algorithm Theorem. After application of the above algo rithm, a triangle will have been recognized if and only if it is recognizable; a triangle with a gap will have been proposed if and only if it is proposable.", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF8": { | |
| "num": null, | |
| "uris": null, | |
| "text": "We denote triangles with greek letters s, Y I , t etc. The triangles YI, t are called a pair of sons of s if s= (A ,i,j), YI = (B,i,k), t = (C, k,j) for some A,B, CEN with A-\">BCEG and Osi < k <j sn. For technical reasons we allow empty triangles with a gap (s, s)-For such an empty triangle, pro posed ((s, s)) = true by definition.", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF9": { | |
| "num": null, | |
| "uris": null, | |
| "text": "Claim A. There is a <P with sons ,p, ,p', such that <P, ,p, ,p' are recognizable, (s, <P),(,p, ri) are pro posable, size ((s, <P)) s 2 kl , size ((,p, ri)) s 2 kl . SeeFigure 15. Proof If_ (s, Y1) is proposable, there is a sequence to, \u2022 \u2022 \u2022 , t p with \ufffd = s, \ufffd P = 'Y) such that each (S i , Si+i ) is proposed by a PROPOSE operation; these \"atomic\" triangles with a gap are subse quently COMBINEd into(;, 11).Choose( <I>, 1jJ) = (S i , Si+ 1 ) with the largest i such \u2022 that size ((;, sJ) s 2 k -l . From size ((S i + ] , 11)) > 2 k -l it follows that size ((;, S i +l )) s 2 k -l ,hence a larger i could have been chosen. ,cp)) s 2 k -l size ((lJJ,11)) s 2 k -l Figure 15. Claim A From the induction hypothesis _we find that (s, cp),. (1jJ, 11) have been proposed after step k-l; 1jJ 1 has\u2022 been recognized after the . RECOGNIZE in step k. Then (<I>, 1JJ) is PROPOSEd in step' k \u2022and two COM BINE operations yield proposed @, 11)). (lh + I \u2022 Given (II) k -1, (I)k, we prove (I) k +l \u2022 Let s be recognizable, 2 k < size(;) s 2 k +l . \u2022 Claim B. There is an 11 with a pair of sons 0, s such that size((;, 11)) s 2\\ size(0) s 2 k , size(s) s 2 k and 11, 0, s are recognizable. Proof let cp 1 be the largest son of s, cp 2 the largest son of q> 1 , etc. Let <j> j tie the first one with size s 2 k . Then 11 = <l>j -1 \u2022If l'J = s, (I) k+ I follows trivially. Otherwise, Claim A holds and we find a situation as shown inFigure 16.", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF10": { | |
| "num": null, | |
| "uris": null, | |
| "text": "Claims Band A From the induction hypothesis we find that(s, cp),(1JJ, 11) have been proposed after step kl; 1jJ' , 0, \ufffd have been recognized after the RECOGNIZE in step k. Then (<I>, 1JJ),(11, t) are PROPOSEd in step k and (\ufffd. 1.v), (1J\u2022. n arc COMBINEd. The second COM BINE in sk;l k proposed(s, t). Hence s will be recognized in step k + 1.", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF11": { | |
| "num": null, | |
| "uris": null, | |
| "text": ") E r, an AND unit [((A,i,j), (B,k, l), (C,m,n))] is added. It receives input from (((B, k, l),(C,m,n))) (((A,i,j),(C, m,n ))). ((( A ,i,j),(B,k, l))) and and sends output to \u2022 The (accept) unit receives input from match units [ accept, i] that will be activated if a sentence of length i could be recognized. This is accom plished by linking (($, i+l)) and (R (S, 0, i)) to [accept, i].", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF12": { | |
| "num": null, | |
| "uris": null, | |
| "text": "A fraction of the recognizing network", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF13": { | |
| "num": null, | |
| "uris": null, | |
| "text": ") \ufffd (S, 0,m). Then the following two conditions hold: (i) A==-+ a; + 1 \u2022 \u2022 \u2022 a j , (ii) S =:> + a 1 \u2022 \u2022 \u2022 a;Aa j + I \u2022 \u2022\u2022a m .", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF14": { | |
| "num": null, | |
| "uris": null, | |
| "text": ") := fa lse od ; if recognized((S, 0,m)) then parsed((S, 0,m)) := true ; for all (A,i,j)E'E. fi do ifproposed(((S, 0,m), (A,i,j))) and recognized", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF15": { | |
| "num": null, | |
| "uris": null, | |
| "text": "[Q m (A ,i,j)] will be activated if parsed ((S, 0,m)) = true and proposed (((S, 0,m), (A ,i,j))) = true. That is, for all possible values of m, [P (S, 0,m)] and (((S, 0,m), (A,i,j))) are linked to [Q m (A ,i,j)].", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF16": { | |
| "num": null, | |
| "uris": null, | |
| "text": "]) by any [P (S, 0, /)] with lsl sm, it is also activated by [P (S, O,m )], because ((S, O,m ), (S, 0,l)) and ((S, 0, /), (", | |
| "type_str": "figure" | |
| }, | |
| "FIGREF17": { | |
| "num": null, | |
| "uris": null, | |
| "text": "triangle (A ,i,j) is called meta-recognizable if (A ,i,j) is recognizable for s\ufffdme input string a 1 \u2022 \u2022 \u2022 a m EL m , (m sn).", | |
| "type_str": "figure" | |
| }, | |
| "TABREF0": { | |
| "text": "). Hence the surface of T R is equal to T CYK, representations of triangles with a gap are contained in entries inside the table. Invariants (I) and (II) guarantee that a table entry with height k will be completed within i 2 logkl steps. As a simple example, consider the grammar", | |
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| "TABREF2": { | |
| "text": ") must be \u2022 made true if both", | |
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| "html": null, | |
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| } | |
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| } |