| { | |
| "paper_id": "1993", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T07:36:04.515926Z" | |
| }, | |
| "title": "A Proof-Theoretic Reconstruction of HPSG *", | |
| "authors": [ | |
| { | |
| "first": "Stephan", | |
| "middle": [], | |
| "last": "Raaijmakers", | |
| "suffix": "", | |
| "affiliation": {}, | |
| "email": "" | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "A reinterpretation of Head-Driven Phrase Structure Grammar {HPSG) in a proof-theoretic context is presented. This approach yields a decision procedure which can be used to establish whether certain strings are generated by a given HPSG grammar. It is possible to view HPSG as a fragment of linear logic (Girard, 1987), subject to partiality and side conditions on inference rules. This relates HPSG to several categorial logics (Morrill, 1990). Specifically, HPSG signs are mapped onto quantified formulae, which can be interpreted as second-order types given the Curry-Howard is. omorphism. The logic behind type inference will, aside from the usual quanti fier introduction and elimination rules, consist of a partial logic for the undirected implication connective. It will be shown how this logical perspective can be turned into a parsing perspective. The enterprise takes the standard HPSG of Pollard-Sag {1987) as a starting point, since this version of HPSG is well-documented and has been around long enough to have displayed both. merits and shortcomings; the approach is directly applicable to more recent versions of HPSG, however. In order to make the proof-theoretic recasting smooth, standard HPSG is reformulated in a binary format.", | |
| "pdf_parse": { | |
| "paper_id": "1993", | |
| "_pdf_hash": "", | |
| "abstract": [ | |
| { | |
| "text": "A reinterpretation of Head-Driven Phrase Structure Grammar {HPSG) in a proof-theoretic context is presented. This approach yields a decision procedure which can be used to establish whether certain strings are generated by a given HPSG grammar. It is possible to view HPSG as a fragment of linear logic (Girard, 1987), subject to partiality and side conditions on inference rules. This relates HPSG to several categorial logics (Morrill, 1990). Specifically, HPSG signs are mapped onto quantified formulae, which can be interpreted as second-order types given the Curry-Howard is. omorphism. The logic behind type inference will, aside from the usual quanti fier introduction and elimination rules, consist of a partial logic for the undirected implication connective. It will be shown how this logical perspective can be turned into a parsing perspective. The enterprise takes the standard HPSG of Pollard-Sag {1987) as a starting point, since this version of HPSG is well-documented and has been around long enough to have displayed both. merits and shortcomings; the approach is directly applicable to more recent versions of HPSG, however. In order to make the proof-theoretic recasting smooth, standard HPSG is reformulated in a binary format.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Abstract", | |
| "sec_num": null | |
| } | |
| ], | |
| "body_text": [ | |
| { | |
| "text": "\u2022 The parser should have reasonable time/space complexity.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Existing parsers for HPSG do not obey these demands; e.g., the Popowich/Vogel parser (Popowich -Vogel, 1990) \u2022 The parser should separate grammatical theory from parsing issues.", | |
| "cite_spans": [ | |
| { | |
| "start": 85, | |
| "end": 108, | |
| "text": "(Popowich -Vogel, 1990)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The main concern of this paper lies in building a parser for HPSG. The result of the enterprise should meet the following desiderata:", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 The parser should make an operationalisa tion of the grammatical theory explicit, as declaratively as possible.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The main concern of this paper lies in building a parser for HPSG. The result of the enterprise should meet the following desiderata:", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 It should be easy to alter the grammatical theory.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The main concern of this paper lies in building a parser for HPSG. The result of the enterprise should meet the following desiderata:", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In the parsing-as-deduction field, several pars ing routines have arisen from proof-theoretic investigations (Moortgat, 1988; Konig, 1989) . While these routines are not all among the most efficient, once a proof-theoretic formulation of HPSG has been made, one can benefit from these results.", | |
| "cite_spans": [ | |
| { | |
| "start": 109, | |
| "end": 125, | |
| "text": "(Moortgat, 1988;", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 126, | |
| "end": 138, | |
| "text": "Konig, 1989)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The main concern of this paper lies in building a parser for HPSG. The result of the enterprise should meet the following desiderata:", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "*This research was carried out within the framework of the research programme 'Human-Computer Communica tion using natural language' (MMC). The MMC programme is sponsored by Senter, Digital Equipment B.V., SUN Microsystems Nederland B.V. and AND Software.", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "The main concern of this paper lies in building a parser for HPSG. The result of the enterprise should meet the following desiderata:", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Some terminological remarks: we refer to HPSG of Pollard -Sag (1987) with '( classi cal) HPSG', and to its type-theoretic (deductive) equivalent with 'V-HPSG'.", | |
| "cite_spans": [ | |
| { | |
| "start": 49, | |
| "end": 68, | |
| "text": "Pollard -Sag (1987)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The main concern of this paper lies in building a parser for HPSG. The result of the enterprise should meet the following desiderata:", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "HPSG is a lexicalist, feature-based formalism ror syntactic and semantic analysis of natural lan guage. HPSG puts all relevant linguistic infor mation in the lexicon, and has general rules and principles governing the construction of phrases from subphrases.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An overview of HPSG", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "As a syntactic formalism, HPSG divides the labour of tree construction into separate processes of mobile construction and mobile ordering. A mobile is a tree-like structure with unordered trees; actually, a mobile can be interpreted as a description of a set of trees.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An overview of HPSG", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Socalled immediate dominance (ID) rules build these mobiles, which are then turned into trees by linear precedence (LP) principles. HPSG is a feature-based formalism, employing various feature mechanisms transporting feature informa tion through feature structures. In HPSG, lex emes are bundles of so-called attribute-value pairs where A j is a certain linguistic (phonologi cal/syntactic/semantic) property taking its speci fication from a set of values containing l7i. These bundles are called signs. The reader is referred to Pollard -Sag (1987 ,1992 where the phon, syn, sem and dtrs values de scribe respectively the phonological, syntactic, se mantic and configurational properties of the sign.", | |
| "cite_spans": [ | |
| { | |
| "start": 530, | |
| "end": 548, | |
| "text": "Pollard -Sag (1987", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 549, | |
| "end": 554, | |
| "text": ",1992", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An overview of HPSG", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Attributes take either atomic or complex val ues; an attribute like person ranges over the set {first, second, third}, whereas an attribute like dtrs ( describing daughters of phrases) takes RAAIJMAKERS full signs as values. The notion of head is a central concept in HPSG. Basically, a head of a phrase is a subphrase determining the relevant combinato rial properties of the sign . . Heads can be phrasal or lexical; lexical heads are simply signs having no daughters. For instance, the head of a VP sees Mary is the verb sees; The grammatical proper ties of sees determine the properties ( viz. agree ment) \u2022 of the VP as a whole, and not those of the direct object Mary. The head of the sentence John sees Mary is the VP sees Mary.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An overview of HPSG", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "HPSG heavily leans on the notion of unifica tion (Shieber, 1986) . Simplifying matters some what, two signs S 1 and S 2 are unifiable with each other, written S 1 U S 2 , if for any attribute they are both specified for, they bear non-conflicting val ues. Further, any fully disjunct parts of two signs (consisting of different attribute-value pairs) of the two signs can be combined directly. ", | |
| "cite_spans": [ | |
| { | |
| "start": 49, | |
| "end": 64, | |
| "text": "(Shieber, 1986)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An overview of HPSG", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "Is, which subcategorises for both an infinitival VP and a nominative NP. The operation order-constituents gives the disjunction of all permutations of the phonology of the daughters.\u2022 At least one of these permu tations will have to be consistent with the con straints of order expressed by the LP principles, which are specific for English (and related languages):", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "phrasal) head daughter and being fully saturated, that is, signs having a subcat(egorisation) list of length zero ( written as ()). There is one com plement daughter (indicated with the variable '_d'); an example would be an S having a VP as head daughter and a subject as only complement daughter. The loc attribute is used to describe \"local\" properties of a sign, such as lexicality and subcategorisation demands; this contrasts with the bind attribute, describing anaphoric links . over signs. Rule 2 caters for instance for VP 's, which have a lexical head daughter (the verb), and are one short of becoming saturated: they subcategorise for a subject. Rule 3 admits of saturated signs with a lexical, inverted head daughter, like in Is John sleeping ?, the head daughter of which is the finite auxiliary", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "\u2022 \u2022", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Constituent Order Principle", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "instance, are in Germanic languages less oblique than direct objects, which means they \u2022are more obligatory: they cannot be omitted, in general.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP l says that lexical heads precede all their sisters (the empty sign O acts _like a ''\ufffdildc\ufffdrd\" symbol here, unifying with every sig\ufffd). _ LP2 says that less oblique complements precede more oblique phrasal sisters; < < is precedence between oblique elements, where obliqueness corresponds \u2022 inversely to degree of obligatoriness. Subjects, for", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "sJteats an apple.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "John eats.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "by the order of complements on the subcat list of signs: the less oblique elements follow the more oblique elements. The HFP enforces identity between the head fea tures of the head daughter and the mother sign. ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "The degree of obliqueness is mirrored (in reverse)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The derivation in figure 1 shows the operation of the various principles and rules. Notice that HPSG as presented in Pollard -Sag (1987) ", | |
| "cite_spans": [ | |
| { | |
| "start": 117, | |
| "end": 136, | |
| "text": "Pollard -Sag (1987)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": ".4 Sample derivation", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "We start off by presenting a binary version of HPSG which removes some of the unattractive features of classical HPSG. Most significantly, this version makes no use of vacuous application of rules to signs, and thus allows for signs to monotonously evolve from lexical to non-lexical status. The theory remains very close to classical HPSG in all other aspects. The binarity is mainly motivated from practical reasons; it facilitates the linking of HPSG to a logical type calculus. Bina rity is by no means a, strong c9mm. itment, how ever. Focus is on the desire to analyse a fragment of Dutch declarative main-clauses, although some examples illustrate the applicability of the binary apparatus on fragments of English as well.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "First, we . qefine lexic, ajity, in terms of daugh ters, using; common predkate notation.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "\u2022 lexical(Sign) \u2022 if dtrs(Sign)= (), parafrased as: Sign is lexical if Sign has zero daughters.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "We then define:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "\u2022 argsn(Sign) if length(subcat(Sign))= n, n \ufffd l, parafrased as: Sign wants n argu ments if the subcat list of Sign has length n (an empty list has length zero) .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "We refer to a functor :F with argsn as :F n .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "The crucial observation for languages like Dutch and English is that the aniourit of satu ration together with the lexicality of a functor ( a sign with non-empty subcat list) determines the position of the functor with respect to its argu ment. A post-modifier like with pictures, mod ifying a noun like book, follows the noun: it is non-lexical, and has args 1 . Similarly, intran sitive verbs -assuming they are lexicalised as VP's, i.e. non-lexical, verbal args 1 functorsfollow their subjects. Semi-saturated verbal func tors like gives John precede their objects: they are args 2 functors. We can capture the order determiner-noun by assuming that determiners subcategorise for non-maximal noun projections (like book, little bo_ok with black cover) , so they are args 1 ; they are lexical, and precede their ar gument. This contrasts with the view of Pollard -Sag (1987) , which analyses nouns as subcate gorising for determiners. 1 So, the generalisation seems to be that:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "1. Ordering effects triggered by the lexicality of functors come into play only for :F 1 func tors: a lexical :F 1 is ordered before its argu ment; a non-lexical :F 1 is ordered after its argument.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "2. A functor :F n where n > l is ordered before its argument.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "The following LP principles mak\ufffd this precise: 2", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "(BLPl) [ lex + l < a args 1 r \u2022 \ufffd ., , \u2022 (BLP2) a < [ lex a\ufffdgs 1 -l (BLP3) [ args n ] '< a", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "To i,ee how these principles work, consider the derivation in figure 2.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": ". ,.,\u2022 ..", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "1 We shall neglect the question on how to encode (non)-maximality of phrases' here; a bar-'level along the lines C\ufffdoper (1990) suggests may be necessary here.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "2 The signs in these principles are only partly specified. We also need the regular LP principle for inverted phrases:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "(BLP4) [ inv + ] < er I", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "This all works fine for concatenative phenom ena, i.e. the combination of two phrases under adjacency. Certain adjuncts appear to be non concatenative, however. In Dutch, one observes:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "\u2022 Jan geeft met plezier Marie een boek.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "John gi'lies eagerly Mary a book.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "\u2022 Jan geeft Marie met plezier een boek.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Binary-Branching HPSG", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "This suggests-that the phonological operation as sociated with certain adverbial modifiers sh\ufffduld not be concatenation but jn\ufffdf. #ion. ' };\ufffdpe (1990) has made similar remarks concer\ufffding-'semi free word order phenoinena. We then arrive a:t the following LP principf\ufffd ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "\u2022 Jan ge\ufffdft Marie een boek met plezier.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "phonology-values be -nested lists (lists of lists) rather than fiat lists, for instance [Jan,[[geeft,[de,man] ], [een,boek] ", | |
| "cite_spans": [ | |
| { | |
| "start": 88, | |
| "end": 109, | |
| "text": "[Jan,[[geeft,[de,man]", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 113, | |
| "end": 123, | |
| "text": "[een,boek]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "241", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "]]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "241", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The infixation of graag into the VP _ phonology [[geeft,[de,man] ], [een,boek] The ID rules of original HPSG must be adapted as well; while Rules 1 and 3 can be kept, Rule 2 must now be altered to cater for gener alised incompleteness: a sign having more than one item on its subcategorisation list is a well formed sign as well.", | |
| "cite_spans": [ | |
| { | |
| "start": 48, | |
| "end": 64, | |
| "text": "[[geeft,[de,man]", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 68, | |
| "end": 78, | |
| "text": "[een,boek]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "241", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "With the binary version of HPSG we are set to give HPSG a deductive basis. First, we show that it is possible to reinterpret signs as types. Then we introduce a deductive apparatus performing type-deduction with these derived types. This calculus builds binary proof trees (proof terms) , which are orthogonal to (binary-) \u2022 HPSG deriva tion trees.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Deduction fo r HPSG", | |
| "sec_num": "4" | |
| }, | |
| { | |
| "text": "We propose to view signs as types, or, with the Curry-Howard isomorphism in mind, . as fo rmu l\ufffde of a certain logic. Ideas in this spirit can al ready be found in work of Blackburn (interpret ing signs as modal formulae), Morrill and others. The concept of types has many interpretations, but one particularly apt for linguistics is that a type is a set of expressions, or, in more tradi tional terms, a category. Together with a set of combinatorial principles, types form an alge bra of expressions over a certain domain: a type system. Essentially, these combinatorial princi ples constitute a derivabzlity relation between se quences of types '-+': A -+ B saying that from the type sequence A the type sequence B can be derived. An example of a type system would be any syntactic algebra consisting of a set of type formation rules ( e.g. the prod uctio\ufffd' rules in a rewrite system) and a set of syntactic -categories (types) (Moortgat, 1988) .", | |
| "cite_spans": [ | |
| { | |
| "start": 923, | |
| "end": 930, | |
| "text": "(types)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 931, | |
| "end": 947, | |
| "text": "(Moortgat, 1988)", | |
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| } | |
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| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Signs as Formulae", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "The intuition that signs can be interpreted as types arises from the functionality expressed by the subcat feature: essentially, this feature e\ufffda\ufffdesses fha:.t a certain sign is functionally {in Jt9\ufffd p l : ete for one or more other signs. This im medfu;t' el\ufffd suggests a functional type equivalent ( a functor) for these signs. Saturated signs then can be interpreted to correspond to saturated func tors, or atomic types, i.e. types not being made up from a type-forming connective and one or more subtypes. HPSG's Subcat principle, which allows for the combination of a non-saturated sign with a subset o( the signs it subcategorises for should then correspond to a combinatorial rule of type formation, i.e. an inference rule in a type calculus.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Signs as Formulae", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "When we want to make a correspondence be tween the signs of HPSG and types of a certain kind, we immediately notice that HPSG signs en code much more information than the monadic categories of simple type systems like produc tion grammars. A category like S, for instance, is represented\u2022 in lIPSG as a fine-grained spec ification of a verbal projection having various properties among which is an empty subcate gorisation frame. Clearly, we need a more so phisticated type language than can be offered by monadic where each Pi is a predicate symbol.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Signs as Formulae", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "We now turn to the translation from signs to types, where we let t(S) yield the formula (type) e .9-uivalent of the sign S . The crucial thing to note is that the subcat information of a sign is reformulated as the func tional demands of a functor type: a subcat list of length n yields a functor with functional degree n,", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Signs as Formulae", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "where n now indicates the number of arguments the functor is incomplete for.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Signs as Formulae", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "The following example illustrates the mapping from signs to formulae. Variables are prefixed with a don't care '_ , .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Signs as Formulae", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "[syn , [ [loc, [[head , [ [maj , n] [n(syn(loc(head(case(_c) ", | |
| "cite_spans": [ | |
| { | |
| "start": 7, | |
| "end": 35, | |
| "text": "[ [loc, [[head , [ [maj , n]", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 36, | |
| "end": 60, | |
| "text": "[n(syn(loc(head(case(_c)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Signs as Formulae", | |
| "sec_num": "4.1" | |
| }, | |
| { | |
| "text": "Now that we have types, the question arises: what do we do with these types? In this section we show how we can interpret the HPSG apparatus of ID rules and various principles as an inference mechanism for type deduction. Before we do so, a few words on type deduction are necessary.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Type deduction", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "As mentioned in section 4.1, types have a truth-conditional interpretation: they correspond to propositions (formulae). This logical point of view makes it possible to identify type derivability relations with logical derivability relations from proof theory. A statement A \ufffd B expressing the derivability of type sequence B from type se quence A is then called a sequent (Gallier, 1986) .", | |
| "cite_spans": [ | |
| { | |
| "start": 372, | |
| "end": 387, | |
| "text": "(Gallier, 1986)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Type deduction", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "A sequent A 1 , ... , A n \ufffd B can be interpreted_ as: the validity of the formulae A1 , ... , A n implies the validity of B; i.e., there is no model for the formu lae A 1 , ... , A n that is not also a model for B. The sequence A 1 , ... , A n is called the antecedent of the sequent; the sequence B (in the present case of length 1) is called the succedent of the sequent.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Type deduction", | |
| "sec_num": "4.2" | |
| }, | |
| { | |
| "text": "fi .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Th", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Pi . . . P n .d -h e con gurat10n C 1s rea as: t e conclusion sequent C is valid iff the premise se quents P1 , ... , P n are valid. As an example, here is a fragment of so-called linear non-commutative propositional logic. 'Linear' (Girard, 1987) Av.(t(v)(v) ). We then end up with the following rules: >-----------------------. .", | |
| "cite_spans": [ | |
| { | |
| "start": 224, | |
| "end": 247, | |
| "text": "'Linear' (Girard, 1987)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 248, | |
| "end": 259, | |
| "text": "Av.(t(v)(v)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 304, | |
| "end": 329, | |
| "text": ">-----------------------.", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Th", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "I-t -: X _ -+ _ t_: _ X _ n => r, v : X -+ t: Y r -+ Av.t : x => Y n => v: x, r -+ t = Y r -+ Av.t : x => Y", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "RAAIJMAKERS occurs exactly once in t; so we do not have terms Av.w where v does not occur in \u2022 w, nor terms like", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A PROOF-THEORETIC RECONSTRUCTION OF HPSG", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": ".:,-,:__ ___;_----,---r 1 , t: \ufffd => Y, A, r2 -+ A A-+ t' : X r1 ,t(t') : Y, r 2 -+ A \u00a3, =>= -------------------r1 , A, t: X => Y, r 2 -+ A The term for \ufffdhe proof above would be AP.P(t) giving the term t as a proof for A. Once one adds the so-called Cut rule to the calculus: C r2 -+ A r1 ,A, r3-+ A ut \u2022 \u2022 r1 , r2 , r3-+ A A-terms in non-normal -form occur as proof terms. The Cut rule expresses the transitivity of the derivability relation -+. Cut-free sequent calculus for the linear frag ment of propositional logic has the so-called sub formula property: premise sequents contain all and only subformulae of the conclusion sequent. Premise sequents have lower degree ih -terms of type-forming connectives: they contain one c_ on nective less than the conclusion \u2022sequents. From a top-down theorem proving regime, this means a steady reduction of complexity . during deduc tion: one starts with a 'complex' sequent con taining a lot of connectives, breaking this sequent down into sequents of smaller degree, until one reaches the axiom sequents of type A -+ A, thus settling the conjecture of the conclusion sequent. In calculi with Cut, the subformula property no longer holds, since A can be any type, possi bly increasing the degree of the premise sequent r1 , A, r 3 -+ A. Fortunately, the Cut elimina tion theorem (Gentzen's Hauptsatz (Gentzen, 1934)) says that Cut is a derivable rule: every proof with Cut can \u2022 be transformed into a Cut free proof. Cut-elimination leads to normal-form proof terms.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Here are the sequent rules for second-order quantifier types (Morrill, 1990) . \u00a3\\:/ r 1 , t(t') : A[t' / ...x] , r2 \ufffda: X r 1 , t: \\:/ (...x) .A, r2 \ufffda: t: 3(... x) r, r 1 , r2 ; t[a/ ,B] is the substitution of a for ,Bin t and 1ri(t) is the i-th projection of the pair term t: 1r1 ((a, ,B) a; 1r2 ((a, ,B) We shall be silent about proof terms from now on, as they do not play an evident role in parsing HPSG. They could be of use in proving meta results about HPSG parsing, however.", | |
| "cite_spans": [ | |
| { | |
| "start": 61, | |
| "end": 76, | |
| "text": "(Morrill, 1990)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 153, | |
| "end": 164, | |
| "text": "t: 3(... x)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [ | |
| { | |
| "start": 135, | |
| "end": 141, | |
| "text": "(...x)", | |
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| }, | |
| { | |
| "start": 165, | |
| "end": 187, | |
| "text": "r, r 1 , r2 ; t[a/ ,B]", | |
| "ref_id": "FIGREF2" | |
| }, | |
| { | |
| "start": 278, | |
| "end": 290, | |
| "text": "1r1 ((a, ,B)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 291, | |
| "end": 306, | |
| "text": "a; 1r2 ((a, ,B)", | |
| "ref_id": "FIGREF2" | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "245", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "X \u00a33 r 1, 1r2 (t) : A[1r1 (t)/ ...x] , r2 \ufffd a : X r 1 ,", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "245", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": ") =", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "245", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Given the calculus presented above, let us es tablish the fragment needed to perform deduction for HPSG.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "245", | |
| "sec_num": null | |
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| { | |
| "text": "HPSG rules describe admissible, i.e. well-formed signs. In a type-theoretic setting, they can be interpreted as type definitions, since here, , signs become types. A simple way to implement these definitions, is to formulate them as axiom schemata in a type calculus. That is, every rule R defining the sign :E becomes an axiom scheme: R t ( :E ) \ufffd t ( :E ) Every axiom sequent thus becomes an in stance of an ID rule. This assures that, whenever an axiom schema is used during deduction, the type check is effective.", | |
| "cite_spans": [ | |
| { | |
| "start": 351, | |
| "end": 357, | |
| "text": "( :E )", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "ID rules as axiom schemata.", | |
| "sec_num": "4.3" | |
| }, | |
| { | |
| "text": "For binary HPSG, this results in the following axiom schemata, where \ufffd schematises over types Q....x i => \u2022 \u2022 \u2022 => ....x n , 1 \ufffd i \ufffd (_phan, _syn, _sem, _dtrs)] \\:/(_cat)\\:/(_phan)\\:/(_syn)\\:/(_sem)V(_dtrs) -\ufffd . [_cat(_phan, _syn, _sem, _dtrs) [_cat(_phan, _syn, _sem, _dtrs)] It is not too hard to recognise equivalents of the HPSG rules Rl and R2 in these axiom schemata, once one remembers that functors now combine with their arguments one at a time: in classical HPSG, there were only two kinds of func tional configurations: a functor having consumed all of its arguments ( treated by Rl) and a func tor having consumed all of its arguments but one (R2). In 'D-HPSG, many more configurations arise, generally speaking: n -1 for any n-placed functor. So, where the first axiom schema restores Rl, the second can be seen to be a generalisation of R2 to cover any kind of functional, incomplete ness. The lexicality demand on the head daughter Rule 2 makes vanishes here; functors consume one argument at a time, and once they have consumed one, they are no longer lexical. 3", | |
| "cite_spans": [ | |
| { | |
| "start": 94, | |
| "end": 132, | |
| "text": "Q....x i => \u2022 \u2022 \u2022 => ....x n , 1 \ufffd i \ufffd", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 133, | |
| "end": 160, | |
| "text": "(_phan, _syn, _sem, _dtrs)]", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 212, | |
| "end": 243, | |
| "text": "[_cat(_phan, _syn, _sem, _dtrs)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 244, | |
| "end": 276, | |
| "text": "[_cat(_phan, _syn, _sem, _dtrs)]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "ID rules as axiom schemata.", | |
| "sec_num": "4.3" | |
| }, | |
| { | |
| "text": "n. \\:/( _cat)\\:/( _phan )V( _syn )V( _sem )\\:/( _dtr s) [_cat(_ph qn , _syn, _sem, _dtrs)] \ufffd \\:/( _cat)\\:/(_phan )\\:/(_syn )\\:/( _sem )V( _dtrs) [_cat", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "ID rules as axiom schemata.", | |
| "sec_num": "4.3" | |
| }, | |
| { | |
| "text": "] \ufffd \\:/(_cat)\\:/(_phan)\\:/(_syn)\\:/(_sem)V(_dtrs) \ufffd .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "ID rules as axiom schemata.", | |
| "sec_num": "4.3" | |
| }, | |
| { | |
| "text": "There is another option here: the ID-rules could be compiled away by ensuring they are con sequently applied to every sign and its phrasal subsigns when the lexicon is created. Although this idea entirely hides the important concept of ID rules in the process of lexicon creation, it al lows for using the regular axiom scheme I-t -: X _ ---+ _t _: X -", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "ID rules as axiom schemata.", | |
| "sec_num": "4.3" | |
| }, | |
| { | |
| "text": "The various principles of HPSG appear to be eas ily reconciliable with the logical setting proposed. In HPSG, they do not form a homogenous class; some principles govern the flow of information in a feature structure, others create new information (like the LP principles). This is reflected in their proof-theoretic reconstruction. The Subcat principle covers concatenative functors only, i.e. functors which either follow or precede their arguments. For non-concatenative functors, such as the adverbial modifiers of sec tion 3, we cannot use the concatenative connec tive \u21d2.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "4.4\u2022 Principles as inference rules and conditions", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Borrowing the connective ! from categorial grammar (Moortgat, 1988) , then, A ! B is a ex pression wanting to penetrate in an expression of type B to form an A. The adverbial adjuncts are typed vp ! vp, where vp is an abbreviation of a formula n( ... ) \u21d2 v( ... ). It turns out to be tech nically impossible to establish a full logic for this connective under the perspective of antecedents as lists; only the rule \u00a3 ! can be formulated. 4 See Moortgat (1988, 1990) for discussion resp. a so lution. For our purposes, this is enough, however: HPSG displays partial logics (left rules only) for functional connectives. The rule becomes: r2 ,r3 ---+ t' : B r1 ,t(t') : A, r4 ---+ A r1 ,r2 ,t: A ! B,r3,r4 --- ", | |
| "cite_spans": [ | |
| { | |
| "start": 51, | |
| "end": 67, | |
| "text": "(Moortgat, 1988)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 453, | |
| "end": 459, | |
| "text": "(1988,", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 460, | |
| "end": 465, | |
| "text": "1990)", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 636, | |
| "end": 664, | |
| "text": "r2 ,r3 ---+ t' : B r1 ,t(t')", | |
| "ref_id": null | |
| }, | |
| { | |
| "start": 667, | |
| "end": 706, | |
| "text": "A, r4 ---+ A r1 ,r2 ,t: A ! B,r3,r4 ---", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Head Feature Principle", | |
| "sec_num": "4.4.1" | |
| }, | |
| { | |
| "text": "\u00a3 !", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Head Feature Principle", | |
| "sec_num": "4.4.1" | |
| }, | |
| { | |
| "text": "where r i is a type sequence of length \ufffd 0, with the exception that at least one of r 2 , r 3 is non em pty. Notice that this rule generalizes over in completeness in the following way: if r2 is empty, ! is an instance 9f /; if r3 is empty, ! is an in stance of \\.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "247", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The Semantics Principle can be 'compiled away' as well, by putting in the lexicon the semantics of a sign as a product of the semantics of its daugh ter signs. This makes it possible to incorporate various kinds of semantics into lexical signs, for instance, a simple application semantics: phon, ... ] [syn , .. . [subcat , [[ ... [sem,X] ", | |
| "cite_spans": [ | |
| { | |
| "start": 291, | |
| "end": 339, | |
| "text": "phon, ... ] [syn , .. . [subcat , [[ ... [sem,X]", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Semantics Principle", | |
| "sec_num": "4.4.3" | |
| }, | |
| { | |
| "text": "[ [", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Semantics Principle", | |
| "sec_num": "4.4.3" | |
| }, | |
| { | |
| "text": "]]] ... ] [sem,f(X)] , [dtrs, ... ]]", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Semantics Principle", | |
| "sec_num": "4.4.3" | |
| }, | |
| { | |
| "text": "Once a functor combines with one of its argu ments to form a mobile the LP principles apply to order the functor and argument branches by ordering the respective phon values to arrive at the phon value of the mother node. LP principles can address both aspects of argument and func tor, . so they must be functions of a pair of types T to sets of types:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "T x T \ufffd POW(T)", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "In case a concatenative functor combines with its arguments, the string ordering functions \u2022 yield a singleton set of result types; for non concatenative functors, this result set often has an arity greater than one, since there is gener ally more than one string position for a non concatenative functor, and each separate string position determines a new sign.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "The operationalisation of LP principles in V HPSG is as follows. Once a functor has applied to its arguments, both functor and argument types are fed to the LP principles, which figure out the phon value of the range subtype of the functor. This entails that LP principles in V-HPSG should operate as side-conditions on inference rules: a, b) . This is done to optimise the expression of list construction: concatenation of two lists can now be expressed via variable shar ing with one unit clause:", | |
| "cite_spans": [ | |
| { | |
| "start": 337, | |
| "end": 342, | |
| "text": "a, b)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "5 Pi F. ___ C _____ n _if LP i V ... V", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "conc_dl(A-B,B-C,A-C). For example, conc_dl([a, blC) -C, [c) -0, [a, b, c) -\u25a1). (BLPl) \ufffd X n.Y(phon(S2 -S3), -, -, -) \u00ae \ufffd X 1 .Z(phon(S1 -S2), syn(loc(..h, lex( + )), _b ), _u, _w) = \ufffd X .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": ".", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "1 .Z(phon(S1 -S3), syn(loc(..h, lex( + )), _b ), _u, _w) [r [-[-[-[ ---> z", | |
| "cite_spans": [ | |
| { | |
| "start": 60, | |
| "end": 67, | |
| "text": "[-[-[-[", | |
| "ref_id": null | |
| } | |
| ], | |
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| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "(BLP2) \ufffd X n.Y(phon(S1 -S2), -, -, -) \u00ae \ufffd X 1 .Z(phon(S2 -S3), syn(loc(..h, lex(-)), _b), _u , _w) Ii) COHSOLE", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": ") .", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "Notice how the lexicality value is changed to 0 here, once the functor has combined with its argument. Above we argued that the lexicality feature is a derived feature, arising from the ab sence or presence of daughters in a tree. Since lex ical signs already have (variable) daughters in V HPSG, checking for lexicality could (and should) be implemented here by inspection of the daugh ter specifications on the functor type: if the first daughter entry in the compdtrs attribute list has, say, a variable phon value, the sign as a whole can be concluded to be lexical. For reasons of ef ficiency, we implement this view on lexicality by switching to non-lexicality the moment a functor combines with an argument. The variable L ex presses the irrelevance of the (non-)lexicality of the functor symbol: no matter what value the functor has for lex, the range type will have the value -for the attribute lex.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "LP principles", | |
| "sec_num": "4.4.4" | |
| }, | |
| { | |
| "text": "Various theorem proving techniques can be imple mented quite easily in Prolog. As theorem prover, we use a simple sequent-proving device, imple mented as follows: the prover is a set of clauses for a predicate prove (+Goal , -Rules) , where Goal is either a sequence of sequents of length mini mally 1, or a structure {G1 , ... ,G n } with each G i (1 ::; i ::; n) a non-sequent goal; Rules is a list encoding the inference rules and axioms used for proving Goal. Initially, Goal is the sequent to be proved. The predicate prove/2 calls a rou tine matching the sequent against the database of inference rules, i.e. if Goal is of the form X 1 , \u2022\u2022\u2022 ,X n -+ Y, it tries to match (resolve) the sequent against the rules and axioms of the cal culus, which take the shape of A -+ B (if C).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Theorem prover", | |
| "sec_num": "5.4" | |
| }, | |
| { | |
| "text": "Once the match of X 1 , ... , X n against A and Y against B has been made, the eventual premises C are attempted to prove. The linear precedence principles are (as illustrated in section 5.3) encoded as goals {lpi , ... , lp n }, to be called before entering the eventual premise sequents.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Theorem prover", | |
| "sec_num": "5.4" | |
| }, | |
| { | |
| "text": "Here are some linear precedence principles. They are written as lp (Name ,Arg , Funct , NewFunct),", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Principles", | |
| "sec_num": "5.5" | |
| }, | |
| { | |
| "text": "with Name the name of the principle, Arg , Funct , NewFunct types such that NewFunct has as its phonology value the ordered phonology val ues of Arg and Funct. Uninteresting variables are written as underscores. with WORD , SIGN resp. a lexeme and its sign rep resentation, and the optional <-VAR_CONDITIONS encoding instantiations of variables mentioned in SIGN ( this is just done to avoid having to type very corn plex signs).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Principles", | |
| "sec_num": "5.5" | |
| }, | |
| { | |
| "text": "Some (partially specified) sample lexical en tries are:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Principles", | |
| "sec_num": "5.5" | |
| }, | |
| { | |
| "text": "The demand that Rule 2 makes on the non-invertedness of the head daughter is left unexpressed here.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "A full logic for this connective would make the structural rule of Permutation: r _ f:l. permut ation = derivable. This means that antecedents now become treated as multisets (sets with repetition) rather than lists, w,hich is not desirable for linguistic purposes.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "This relates the current enterprise to Gabbay's (Gabbay, 1991) labelled deductive systems, where side-conditions on inference rules occur aa well.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "", | |
| "sec_num": null | |
| } | |
| ], | |
| "back_matter": [ | |
| { | |
| "text": ",r4 __. tJ. (Wallen, 1990) . This effect does not take place here, since we only have universal quantification.", | |
| "cite_spans": [ | |
| { | |
| "start": 12, | |
| "end": 26, | |
| "text": "(Wallen, 1990)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "t : V(.:x ).A \u00a3 ! r 2 ,r3 __. t' : B r1,t(t') : A, r4 __. A r1,r 2 ,t: A ! B, r3", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "In Prolog format, the axioms and rules of the cal culus have the following shape:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": ".3 Calculus in Prolog format", | |
| "sec_num": "5" | |
| }, | |
| { | |
| "text": "Given a functor type A1 \u21d2 ... A n \u21d2 B, which is right-associative, i.e. A1 \u21d2 ( ... (A n \u21d2 B) ", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 83, | |
| "end": 92, | |
| "text": "(A n \u21d2 B)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "axiom(Name , Antecedent--->Succedent rule (Name , Antecedent --->Succedent (if C)", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "the man walks.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "de man loopt.", | |
| "sec_num": "2." | |
| }, | |
| { | |
| "text": "the man walks gladly.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "de man loopt graag.", | |
| "sec_num": "3." | |
| }, | |
| { | |
| "text": "john has fast walked.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "jan heeft hard gelopen.", | |
| "sec_num": "4." | |
| }, | |
| { | |
| "text": "john hits the man.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "jan slaat de man.", | |
| "sec_num": "5." | |
| }, | |
| { | |
| "text": "john hits gladly the dog.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "john slaat graag de hond.", | |
| "sec_num": "6." | |
| }, | |
| { | |
| "text": "8. jan geeft marie de hond. john gives mary the dog.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "de man koopt een boek met plaatjes. the man buys a book with pictures.", | |
| "sec_num": "7." | |
| }, | |
| { | |
| "text": "11. dat jan de hond slaat. that john the dog hits. ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "jan geeft marie graag een boek. john gives mary gladly a book.", | |
| "sec_num": "10." | |
| }, | |
| { | |
| "text": "where each Xi is distinct, the analysis them, 1991) into the theorem prover. Proof in variants are structural validities for antecedent succedent pairs, which serve to prune irrelevant options from the search space. The attractive feature of the current setting is that any opti malisation coming from proof theory can be used to optimise the parser.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 40, | |
| "end": 51, | |
| "text": "them, 1991)", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "As can be concluded from the output pre sented in the previous section, perfp:rma_ :\u00b5ce is rel atively good. Sentences taking a lot qf time (say, over 1 second), invariably contain at least one ad verbial modifier, or involve an NP closure prob lem. For instance, in \u2022 John gives Mary a book with pictures the phrase 'John gives Mary a book' can be er roneously analysed as a sentence before the PP 'with pictures' is attached to 'a book'. Once the parser detects the remaining phrase 'with pic tures', it will have to backtrack and redo a lot of work. The bad performance is a consequence of the sequent formalism: for any configuration X1 \u21d2 X2, X1, X1 \u21d2 X1", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We have shown that it is possible to give HPSG a deductive basis. The binary version of HPSG we have proposed, has been demonstrated to corre spond to a fragment of second-order linear logic. The binarity of this HPSG dialect, which is faith ful to classical HPSG in all other respects, is mo-. tivated from practical rather than theoretical rea sons; in fact, the current approach is open to any version of HPSG. The parser we developed is, al though relatively fast, in need of further optimal isation; the use of proof invariants may help to reduce the search space. Also, recently developed RAAIJMAKERS low-complexity theorem proving techniques such as proof nets (Roorda, 1991) , may be of use here. Returning to the five desiderata of section 1, then, the last item, \"The parser should have reasonable time/space complexity\" has not fully been met yet.", | |
| "cite_spans": [ | |
| { | |
| "start": 668, | |
| "end": 682, | |
| "text": "(Roorda, 1991)", | |
| "ref_id": null | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "5 4 8 Concluding remarks", | |
| "sec_num": "2" | |
| } | |
| ], | |
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| "ref_entries": { | |
| "FIGREF0": { | |
| "num": null, | |
| "text": "violates the second, fourth and fifth demand; the LiLog STUF en vironment (Dorre -Raasch, 1991) violates-the first, third, and fifth. For a full comparison, see Raaijmakers (forthcoming). \u2022 The parser should interpret the original grammatical theory, or as close a dialect as possible.", | |
| "uris": null, | |
| "type_str": "figure" | |
| }, | |
| "FIGREF2": { | |
| "num": null, | |
| "text": "Sample derivation for 'John gives Macy a book with pictures'.", | |
| "uris": null, | |
| "type_str": "figure" | |
| }, | |
| "FIGREF3": { | |
| "num": null, | |
| "text": "( the phonology of) B. The non concatenative connective ! was introduced in cat egorial grammar by Moortgat (Moortgat, 1988) for similar purposes; an expression of type A ! B infixes into expressions of type B to form an expression of type A (see section 4.4.2 below). ADVHOD describes the \u2022sign for a VP-level adver bial modifier, which is a sign subcategorising for a VP to yield a VP: it inherits the NP argument {the subject) its argument VP is still incomplete for. There is a little snag here: mere infixation of the adverbial phonology into the VP phonology :w,gq. ld result in ill-formed strings where the ad v\ufffdr\ufffd 1 penetrates into one of the verbal arguments. Fo\ufffd \ufffdn\ufffdtance, *Jan geeft de graag man een boek John gives the with-pleasure man a book \u2022 This problem cannot be fixed \u2022by letting A PROOF-THEORETIC RECONSTRUCTION OF HPSG", | |
| "uris": null, | |
| "type_str": "figure" | |
| }, | |
| "FIGREF4": { | |
| "num": null, | |
| "text": "\u2022 cat\ufffdgpxi\ufffds alone. Suppose then we switch from mop_ a<;l k\ufffd_types to types with inter nal structure: predic\ufffdtional types in stead of propositional types. The value of the category determining maj (or) ?,ttribute should become the top-level predicat\ufffd con. stant. As in predicate logic, types (propo\ufffditions)\"are made up from such . a \u2022predicate constant and terms as arguments of t ' he :p*\ufffdaica, te. , \ufffd\ufffdi\ufffdi\\'. \ufffdavjng variable values, i.e. being under$peci\u00b5e\ufffd J9 , ' l\ufffdcertain attributes, correspond to (univer's\ufffdly) \u2022 quantified formulae. E.gx ). (n(gender(neuter ), persan(_x) )] .The choice between universal and existential quantification is mainly motivated from consid erations regarding the proof terms for quantified formulae, which will be discussed in the next sec tion. A related motivaton is the fact that uni versally quantified types have a straightforward connection with Prolog literals, facilitating imple mentation. It is important to notice that there is RAAIJMAKERS no deep, 'predicate-lik\ufffd' meaning \u2022behind such a formula: it is just a description of a certain kind of category, in the case above having a variable spot for the person value. Sign-valued attributes, i.e. attributes taking a full sign as value, or a list of signs, are treated the same: whenever such an attribute takes a variable sign as value, univer sal quantification over this variable occurs. This is responsible for the second-order nature of the . type language we use. Under the logical interpretation of types as formulae, types have proof terms associated with them; these proof terms are the justification for assuming the formula is true: they correspond to proofs for the propositions the types express. These proofs are constructed in a calculus of in ference rules, the inference rules constituting a derivability relation over type sequences (like the combinatorial rules of production systems), where this derivability relation now gets a logical inter pretation as well. An alternative, quite common point of view is that proof terms are a kind of procedures (or programs) and types are the spec ification of what these programs do. For instance, the formula \\/(_x).(n(gender(masc), number(_x.)-)). would be a specification of the program recognis ing singular and plural ;masculine. \u00b5oun phrases ( this basically is what parsing is aboJJ,t}: .; A concept like reentrancy can easily be en coded by means of variab!e sha \ufffd ing, for example 'v'(_x).[P 1 (P i (..x)\\ .. _:,., Pn: (.:t)}]", | |
| "uris": null, | |
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| }, | |
| "FIGREF5": { | |
| "num": null, | |
| "text": "sign E with the sign deleted from it. Furthermore, var(X), atom(X), number(X) express respectively that X is a vari able, an atom or a number.\u2022 t( X) : = Q o.X if var(X) or atom(X) or number(X) or X = 0 \u2022 t (E [ :r: 0 l) == Q .M(F) if t (E _ [ ::r: 0 l ) == Q A1 => ... A n => M(F)if t (E -[ ::r;; (X , , ... , X n ) l) := Q , .F and t((X1 , ... , Xn) ) := Q2 . ( A1 , ... ,A n )\u2022 t( [ A V ] ) := \\l(V).A(V) if var(V) \u2022 t([ AV ]) := Q.A(X1 , ... , X n ) if t (V) := Q. {X1 , . . . , X n ) \u2022 t ( (X1 , \u2022\u2022\u2022 , Xn) ) := Q1 Q2 . (F1 , F2, ... , Fn ) if t (X1 ) := Q1 .Fl and t ( (X2 , ... , Xn) ) := Q2 . (F2 , \u2022 \u2022 \u2022 , Fn )", | |
| "uris": null, | |
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| }, | |
| "FIGREF6": { | |
| "num": null, | |
| "text": ", nform(_n), aux(nil), inv(nil), prd(nil) ), lex(l) ) , bind(..b )))]", | |
| "uris": null, | |
| "type_str": "figure" | |
| }, | |
| "FIGREF7": { | |
| "num": null, | |
| "text": "the meaning of an utterance depends\u2022 on the lin ear order of its words. Every r i is a (possibly empty) type sequence; A is a non-empty type se quence, and X, Y, A are types. The comma ',' denotes non-commutative concatenation: I'1 , r2 is the concatenation of the type sequences r 1 and r 2 \u2022 This entails that antecedents are essentially lists of types. I X -+ X The \u00a3, rules are referred to as the left rules; the n rules as the right rules of the calculus. Here is a proof of the theorem A-+ (A => B) => B.", | |
| "uris": null, | |
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| }, | |
| "FIGREF8": { | |
| "num": null, | |
| "text": ") = ,B. The func tional proof terms for \\I-types reflect the intuition istic idea that a proof for a proposition \\:/(...x).A consists of a method for proving the proposition expressed by A. This gives a more plausible in terpretation of proofs for universal quantification once this quantification ranges over infinite do mains: a mere truth-value then seems impossible to arrive at. The pair terms for 3-types say that a proof for such a formula consists of an individual (a witness) and a term in which this individual is substituted for the bound variable. As noted earlier. \u2022 the intuitionistic quantifier terms have a nice interpretation in our syntactic type calculus: a type V(...x) .II then becomes a specification of a method (proof) recognising all expressions of type II on the basis of any (instantiation of) ...x.", | |
| "uris": null, | |
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| }, | |
| "FIGREF9": { | |
| "num": null, | |
| "text": "Mt \ufffd = \ufffd-J J]JJ JJ c-cdo c--rn dtra (!,oaodt ni l] \u00bb Sentence : de 111an koopt een boek met plaatJes. Cde.man.koopt .een.boek.met.plaatJesl-Cl > .syn< loc < head(case<nil > ,vform(fln) ,aux(O), lnv(O) .prd(O)), lex(O)) ,bind (ni l)) ,se111(kopen (sem(een(se111(met(sem( plaatJesl .sem(boek> > > > > .se111(de(se111<111an> > > > > ,dtrs(headdtr <nil > .co111pdtrs<n!! <phon< tee n,boek.met.plaatJesl-Cl >. s11n< loc(head(case(ni l) .nf'or111(f'ull > .aux<nil >. inv(ni l > .prd( 11 > >. lex<O> > ,bind(nil > > ,se111(een<sem<111et(se111<plaatJes l.se111(boek) > > > > ,dtrs(headdtr(nl l) ,co111 trs(n!! ( hon( Cboek,met, laat esl -Cl >.s n( loc(head(case(nil ) ,nfor111(norm) ,aux", | |
| "uris": null, | |
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| }, | |
| "FIGREF11": { | |
| "num": null, | |
| "text": "(S2-S3) ,_,_ ,_) , [X\u00a9(_,_, _ ,_) I T] = >B\u00a9 (phon(S1-S2) , P , Q ,D) , [X\u00a9(_,_ , _ ,_) I T] = >B\u00a9 (phon (S1-S3) , P , Q ,D)) inverted(B\u00a9 (_, P ,_,_)) . 6 Sample lexical entries Lexical entries are of the form WORD : = SIGN (+-VAR_CONDITIONS)", | |
| "uris": null, | |
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| }, | |
| "TABREF0": { | |
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| "text": "for a full overview of the various attributes and their values. The generic structure of main signs in HPSG is [ phon .. \u2022 i syn .. . sem .. . dtrs .. .", | |
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| "content": "<table/>" | |
| }, | |
| "TABREF3": { | |
| "html": null, | |
| "num": null, | |
| "text": "", | |
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| "content": "<table><tr><td>[ (</td><td>, syn I loc I subcat [!] dtrs [ headdtr I syn I loc I subcat IT]+[!] l compdtrs IT] \u2022</td><td>]</td></tr></table>" | |
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| "TABREF4": { | |
| "html": null, | |
| "num": null, | |
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| "content": "<table><tr><td>for. Conceptually, the lexicality feature seems to be derivable, since, oniy those signs are non lexical (phrasal) which carry at _least . one daugh ter. The HPSG theory as originally put forward by Pollard -Sag (1987) does not lend itself directly to a proof-theoretic reconstruction. The theory, being declarative in a strong sense, has obscure operational aspects. Also, mainly for practical (but possibly also for theoretical) reasons, it ap pears to be desirable to have a version of HPSG building binary branching syntax trees. So, as a first step we present a binary version of HPSG.</td></tr></table>" | |
| }, | |
| "TABREF7": { | |
| "html": null, | |
| "num": null, | |
| "text": "is a nice interpretation . of lin\ufffd guistic types as propositional formul\ufffd ill a lo . gic: atomic types T correspond to formulae T; com plex types like A\\B correspond to A =}l B, with =?l a left-oriented version of the implication ar row =? of propositional logic. The combination of a type A with a type A =} B to a type B then becomes an instance of Modus Ponens, of which we now have two versions: A, A =}1 B -+ B and A =}r B, A -+ B. This, in fact, is an operational isation of the slogan parsing as deduction, and is basically the central theme of categorial deduc tion as in Lambek calculus", | |
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| "content": "<table><tr><td>containing expressions over some alpha bet of strings. More fine-grained type systems \u2022 . make a distinction\u2022 between atomic and complex types: atomic types being . monadic 09jects and corn plex types being made up from . (atomic or complex) subtypes with the rise of so-ca:Ifod type forming connectives which serve to expre\ufffds com binatorial properties. Typ\ufffd-forming connectives are relations over the set of type_ symbols; a famil iar example are the slashes from categorial gram mar /, \\: a fu nctor type X/ Y combines with a type Y to its right to form a \ufffdyp e X; a fandor</td></tr><tr><td>There</td></tr></table>" | |
| }, | |
| "TABREF13": { | |
| "html": null, | |
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| "text": "LP n LP principles operate on an argument type and a functor type. Here are the type-theoretic equiv alents of the LP principles of binary HPSG. The notation (BLP n )A \u00ae B = C says that the result of applying the LP principle n to argument A and", | |
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| "content": "<table><tr><td>\ufffd</td></tr><tr><td>functor B is C. As before, \ufffd Y schematises over</td></tr><tr><td>types</td></tr><tr><td>and</td></tr><tr><td>\ufffd X lZ=X=>Z.</td></tr><tr><td>Further, inverted(X) says that X is inverted,</td></tr><tr><td>ADVM0D(X) that X is an adverbial modifier, and infix( S2, SI) = S3 that S3 is the infixation of S2 into SL Uninteresting variables are sup</td></tr><tr><td>pressed with an underscore, and quantifiers \u2022 are</td></tr><tr><td>omitted. Anticipating on the implementation, we use (Prolog) difference list notation for list con</td></tr><tr><td>struction: the difference list [a, b, c)-[c) is equiva lent to the list [</td></tr></table>" | |
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