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| "paper_id": "W98-0119", | |
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| "date_generated": "2023-01-19T06:04:33.103111Z" | |
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| "title": "Partial Proof Trees and Structural Modalities\u2022", | |
| "authors": [ | |
| { | |
| "first": "Aravind", | |
| "middle": [ | |
| "K" | |
| ], | |
| "last": "Josbi", | |
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| "institution": "Cognitive Science University of Pennsylvania", | |
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| "addrLine": "Suite 400A, 3401 Walnut Street", | |
| "postCode": "19104-6228", | |
| "settlement": "Philadelphia", | |
| "region": "PA" | |
| } | |
| }, | |
| "email": "" | |
| }, | |
| { | |
| "first": "Seth", | |
| "middle": [], | |
| "last": "Kulick", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "Cognitive Science University of Pennsylvania", | |
| "location": { | |
| "addrLine": "Suite 400A, 3401 Walnut Street", | |
| "postCode": "19104-6228", | |
| "settlement": "Philadelphia", | |
| "region": "PA" | |
| } | |
| }, | |
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| { | |
| "first": "Natasha", | |
| "middle": [], | |
| "last": "Kurtonina", | |
| "suffix": "", | |
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| "institution": "Cognitive Science University of Pennsylvania", | |
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| "addrLine": "Suite 400A, 3401 Walnut Street", | |
| "postCode": "19104-6228", | |
| "settlement": "Philadelphia", | |
| "region": "PA" | |
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| "abstract": "", | |
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| "text": "An important theme in current categorial research is the shift of emphasis from individual category logics to couununicating families of such systems. The reason for this shift is that the individual logics are not expressive enough for realistic grammar development; the grammar writer needs access to the combined inferential capacities of family of logics. Categorial systems with structural modalities (see Moortgat 1997 , Kurtonina and Moortgat 1997 , Morrill 1994 for details) can incorporate not only limited relaxation of the rigid structure to provide more generative capacity, but also impose additional constraints to block undesired derivations 1 \u2022 Although they provide a powerful extension of capacities of categorial inference, their use can be linked to overgeneration in some cases. In this paper we will show how this problem can be handled if categorial systems based on partial proof trees are used as building blocks of the system. The key idea is that the use of PPTs allow us to 'localize' the management of resources, thereby freeing us from this management as the PPTs are combined.", | |
| "cite_spans": [ | |
| { | |
| "start": 411, | |
| "end": 424, | |
| "text": "Moortgat 1997", | |
| "ref_id": "BIBREF2" | |
| }, | |
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| "start": 425, | |
| "end": 454, | |
| "text": ", Kurtonina and Moortgat 1997", | |
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| }, | |
| { | |
| "start": 455, | |
| "end": 469, | |
| "text": ", Morrill 1994", | |
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| "text": "Here we provide a very brief overview of the PPT system. See Joshi and Kulick (1997) for details. The basic idea is to associate with each lexical item one or more PPTs, obtained by unfolding the arguments of the type that would be associated with that lexical item in a simple categorial grammar, such as the Ajdukiewicz and Bar-Hillel grammar. The basic PPTs then serve as the building blocks of the grammar, and complex proof trees are obtained by 'combining' these PPTs by various inference rules, that basically allow the linking of conclusion nodes", | |
| "cite_spans": [ | |
| { | |
| "start": 61, | |
| "end": 84, | |
| "text": "Joshi and Kulick (1997)", | |
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| "text": "\u2022we would like to tha.nk Gerhard Jaeger, AJain l&:omte, Owen Rambow, Mark Steedman, K. Vijay-Shan.ker, and two anonymous reviewers for their valuable comments. This work was partially supported by NSF Grant SBR96-20230.", | |
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| "text": "1 In this paper we focus an categorial systems that use structural modalities. Another branch of categorial grammar is that represented by Combinatory Categorial Grammar (CCG) (Steedman 1996) . Work is currently in progress to investigate the relation.ship between CCG and the partial proof tree system described here.", | |
| "cite_spans": [ | |
| { | |
| "start": 176, | |
| "end": 191, | |
| "text": "(Steedman 1996)", | |
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| "text": "to assumption nodes, and the stretching of a node in a proof. The main motivation of this approach is to incorporate into the categorial framework the key insights from LTAG, namely the notion of an extended domain of loca.lity and the consequent factoring of recursion from the domain of dependencies.", | |
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| "text": "In CG the engine of grammatical inference is, of course, a multiplicative fragment of intuitionistic linear Iogic (Lambek Calculus) and logical derivability of some distinguished types from a sequence of types is crucial for determination of grammaticality of linguistic expressions. On a deductive level the logical architecture of categorial inference is reflected in the rules of a calculus (for instance, sequent calculus). In contrast to CG, the PPTs system is a tree rewriting system. However, we can make explicit the underlining logic of the system to provide a logical explanation of the resource management. In fact, two kinds of logiC8 are involved in PPTs system. Construction of basic trees is guided by the logic of a CG, while operations of combining trees are monitored by a single rule -Cut. We now consider the use of two kinds of struct ural modalities, following Moortgat (1997) , Kurtonina and Moortgat (1997) , Morrill (1994) .", | |
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| "text": "(1997)", | |
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| "text": "Kurtonina and Moortgat (1997)", | |
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| "end": 947, | |
| "text": "Morrill (1994)", | |
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| "text": "Structural Relaxation: Consider the relative clauses in (la) and (2a):", | |
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| "text": "(1) (2) a. . b. a.", | |
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| "text": "b. The two sentences correspond to the sequent derivations in (lb) and (2b). The form er is a valid derivation, but 'th~ latter'is ~ot derivable. The problem is that the hypothetical np assumption is not in the required position adjacent to the verb. Here the so-called Permutation modality comes into the picture. We refine the assignment to the relative pronoun to the type r/(s/np\u2022), where the decoration with J indicates an access to restricted Permutation. 3a, with the corresponding sequent (3b), is derivable with X instantiated to s/np. However, the ungrammatical (4a), corresponding to the sequent (4b), can be derived with X instantiated tos. This problem can be fixed by refining the type assignment to 'and' to be (X\\D-1-X)/ X and by closing off the coordinate structure with the dual structural modality 0. The resulting sequent corresponding to (3) is now (5), with its validity proved by (6):", | |
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| "text": "The island violation (4) fails, because the hypothetical np assumption finde itself in the scope of modal operator. Thus, the idea of the approach is to freeze complete coordination into an island configuration. The introduction of this other type of structural modality imposes structural cnnstraints rather than structural relaxation, as with the permutation modality.", | |
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| "text": "Conflict: However, if the two types of modalities appear in the same sentence, then they require a simultaneous relaxation and constraining of the interaction between the types. Consider the derivation of (7a), with the corresponding sequent (7b).", | |
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| "text": "To derive this sequent, X must be instantiated as (s/npB), due to the presence of yesterday and today. And, as we just saw, the tne for and should have the type\u2022 assignment (X\\D X)/X, and so the type for and in this example becomes ((s/npl)\\D-1-(s/np\u2022))/(s/npl). lt is unfortunate that such a complex type for and is required simply because of the way that adverbs interact with extraction in the inference system. Using PPTs offers an interesting way to resolve the confiict, because ofthe way that it employs two different logics.", | |
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| "text": "We cannot show the relevant PPTs here for space reasons. However, the basic idea is that, as discuased in Joshi and Kulick (1997) , permutation ie not needed for an adverb with a relative clause as in (2a) since the adverb is simple inserted via \"stretching\" a node in the object relative clause tree. Refinement of the system to account for coordination allows the derivation of (3a), while (4a) is ruled because, of course, the two conjuncts need tobe of the same type, and they cannot coordinate if one iB s whlle the other is s/np. Crucia.lly, a.llowing (7a) is not a problem, since the adverbs simply come in via \"stretching\" , and have no effect w hatsoever on the type constraints for coordination. Therefore, there is no n' eed for any modification of the basic type for coordination.", | |
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| "text": "We conclude that by using PPTs, the linguistic phenomena motivating the introduction of structural modalities in categorial grammar can be handled by either eliminating them (such as for an a.dverb in a relative clause) or by retaining them but localizing them within basic PPTs (e.g., topicalization by permutation, as described in Joshi and Kulick 1997) , thus avoiding the problem of overgeneration which requires constraints on modalities. This is due to the existence of of two types of logic in the PPTs, a consequence of combining trees rather than just strings, and is a very desirable consequence of localizing the management of resources in the PPT syste.m.", | |
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