M. Moortgat. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Foris, Dordrecht, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and to build its semantic interpretation. A CG is composed of a lexicon and a small set of rules. Many versions of CGs have been defined, and they di#er in terms of the set of categories defined and or in terms of the set of rules used (e.g. Lambek 1958] van Benthem 1988] Morrill 1987] [Moortgat 1988], Carpenter 1998] and [Steedman 2000] This discussion starts with a description of AB CG, the basic CG, and is followed by a description of other configurations that extend either the set of categories, or the set of rules, or both. 4.1.1 AB Categorial Grammar The basic version of CG, known ....

....A sign is composed of three attributes (figure 4.16) orth encodes the orthographic description of words, cat encodes syntactic aspects related to the categories, and . sem encodes the semantics associated with a particular word. For an extensive description of the Lambek Calculus see [Moortgat 1988]. The descriptive power of a very similar system is investigated by Ho#man [1995] and she demonstrates that such system is able to successfully handle long distance scrambling, generating some mildly context sensitive languages. # # # # # CAT : cat # # # # # Figure 4.16: Sign ....

Moortgat, M. Categorial Investigations - Logical and Linguistic Aspects of the Lambek Calculus. Foris Publications, 1988.

....of discontinuity, modi ers and quanti er scope. However, the introduction of a type discipline and more general treatment of recursive lexical rules ( BvN94] must be considered, in future work. On the other hand, most recent developments of categorial grammars are based on the Lambek calculus [Lam58, Moo88, Mor94] (an intuitionist fragment of Linear Logic) Some implementations for the propositional fragment are based on chart parsers [K on94, Hep92] and we conjecture that FD calculus can be successfully used in such a systems, for process semantic representations. From an implementational Fig. 4. ....

Michael Moortgat. Categorial Investigations: Logical and Linguistic Aspects of Lambek Calculus. Foris, Dordrecht, 1988.

....The development of this material over a period of time has benefited from a wide range of influences; for extensive comments and suggestions I thank most of all Guy Barry, Mark Hepple, and Neil Leslie. 1 For discussion of the rule to rule grammatical architecture see Oehrle (1988) 2 See Moortgat (1989). 1 the validities according to the interpretation of types. The extension of Lambek categorial grammar given is related to linear logic (see e.g. Girard 1989) which drops structural rules from intuitionistic logic; in the present application this corresponds to the fact that grammar is ....

.... law of fi reduction (unary operators such as x will be assumed to bind more tightly than binary operators such as ) xff fi = ff[fi=x] 8) This Curry Howard formulas as types interpretation of categorial deductions is introduced in van Benthem (1983) and implemented in a sequent context in Moortgat (1989). The simplification in (8) means that introducing and then eliminating implication constitutes a detour in derivation, producing a reducible, non normal form, proof (cf. Prawitz 1965) By way of illustration of conditionalisation, the non canonical constituent John likes can be derived as of ....

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Moortgat, Michael: 1989, Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht.

....hypotheses are not allowed in the sequents of the original Lambek calculus. We do not insist on this restriction since the results presented in this paper hold in both system (i.e. the Lambek calculus with or without the empty sequence) 2 3 Moortgat s phonological algebra Following Moortgat [12], one denes the phonological algebra associated to a set of tokens V as the structure freely generated from V by a binary operation i j, and one adds to this structure an identity element such that for any a 2 V, a ) a and ( a) a. Notice that, apart from these identity laws, the ....

....= a and ( a) a. Notice that, apart from these identity laws, the phonological algebra does not obey any law. In particular, i j is not associative. The purpose of the phonological algebra is to associate with any Gentzen derivation a phonological term that would reAEect some prosodic phrasing [10, 12]. This may be achieved by decorating sequents with phonological terms: a : A Gamma a : A (Id) Gamma Gamma t : A Delta; a : A; Theta Gamma u : B (Cut) Delta; Gamma; Theta Gamma u[a: t] B Gamma Gamma t : A Delta; b : B; Theta Gamma u : C (nL) Delta; Gamma; a : A n B; Theta ....

M. Moortgat. Categorial Investigations: logical and linguistic aspects of the lambek calculus. Foris Publications, 1988.

....(also shared by TAG) as is evident in Steedman s comment in [37] p. 44: Although the (weakly context free) Lambek calculus provides an interesting start point. the addition of any of the non order preserving rules to its axioms immediately makes it collapse into permutation closure, as [23] has shown. For CTL researchers, issues of generative capacity are generally not considered to be of prime theoretical importance as Morrill for example notes in [26] p.257: There is no absolute measure as to what constitutes a restricted class of languages, so that while observations of ....

....Logics, or CTL for short, extend categorial grammar with a resourcesensitive perspective a la linear logic, in order to introduce controlled use of more powerful structural operations like commutativity. 4 The development of CTL started back in the mid 1980 s with Van Benthem [39] and Moortgat [23], and, independently, work by Oehrle and Zhang [27, 30] CTL builds o the logical approach to CG provided by the Lambek calculus [22] The main idea behind CTL is to distinguish various modes of composition, rather than just the singular one with its associated residuals n; as in applicative ....

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Michael Moortgat. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Foris, Dordrecht, The Netherlands, 1988.

....build interpretations of the appropriate functional type. However, getting the right syntactic and semantic coverage with such systems has proven di#cult, possibly because of the e#ects of their strict coupling of syntactic and semantic constraints. This has motivated researchers such as Moortgat [34] to add to categorial grammars explicit abstraction mechanisms. In pure categorial systems, interpretation operations are derived operators 5 in a semantic algebra capturing the basic properties of functional objects. 2 The goal of this paper is to show how the semantic algebra can be enriched ....

....(gap x) is given the interpretation x by the second int clause, while the universal quantification in the first clause defines the scope of the abstraction. Instead of using specialized rules for relative clauses, we could have used assumptions for that purpose, either in the semantic calculus [34, 40], or in combined syntacticsemantic rules in which the syntactic gap is supplied as an assumption in the syntactic analysis, and the corresponding bound variable is universally quantified [38] The third clause for int interprets lexical items by consulting a translation mapping trans from lexical ....

M. Moortgat. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. PhD thesis, University of Amsterdam, Amsterdam, Holland, October 1988.

....of a systematic correspondence between combined algebraic and relational interpretation of categorial logic, and the form of proof structures and paths in proof nets, illustrating with reference to medial extraction, in situ binding and discontinuity. Type logic for linguistic description (e.g. Moortgat 1988, 1997; Morrill 1994; Carpenter 1997) is based on what we may refer to as a Lambek van Benthem correspondence: logical) formulas as (linguistic) categories. Lexical signs are classified by category formulas, and the language model projected by a lexicon is determined by the consequence relation ....

....the relative pronoun binds a position which is medial in the relative clause. the dog) that i John gave t i to Mary (5) Defining the relative pronoun as R (NnS) or R (S N) where R is CNnCN) allows it to bind only left or right peripheral positions: 5) is not generated. To deal with such cases, Moortgat (1988) defines as follows a binary operator which we write e : B e A] fs 1 s 2 j8s 2 [ A] s 1 s s 2 2 [ B] g (6) Assigning the relative pronoun to category R (S e N) allows both medial and peripheral extraction, via the rule of proof (7) Gamma 1 ; A; Gamma 2 ) B e R Gamma 1 ; Gamma ....

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Moortgat, M.: 1988, Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht.

....phrases to their interpretations, and the process of semantic interpretation is the derivation of interpretation judgments for a phrase from premises giving the interpretations 2 Fernando C. N. Pereira of the constituents of the phrase. A very important special case is that of categorial semantics [2, 21], in which the basic judgments do not relate phrases directly to their interpretations, but rather to the types of their interpretations, and the actual interpretations are extracted from derivations by virtue of some version of the Curry Howard isomorphism between propositions and types [6, 11, ....

....in the directed Lambek calculus [19] the single semantic function type # # # is split into the left looking and right looking function types # # and # #. Additional specializations have been proposed as more complex grammatical phenomena, such a long distance dependencies, have been addressed [13, 21]. At the same time, rules of inference have to be far more restrictive than would be required just for soundness of type assignment, to the point of being made sensitive to the specific categories involved rather than just their forms [31] The logical elegance of early categorial grammar is thus ....

M. Moortgat. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. PhD thesis, University of Amsterdam, Amsterdam, The Netherlands, October 1988.

....possibility, having Permutation, but not association, is denoted NLP. For convenience, I will adopt distinct notations for the connectives of three of these systems, as follows: L:fffl,n, g, NL:ffi,ffin,ffi g, LP:f Omega , Gammaffi,ffi Gammag. 1 Such discontinuity connectives are proposed in [12]. Regarding the formalisation and use of multimodal systems including such connectives, see [8] 9] 13] 18] 19] 2 A sequent Gamma ) A indicates that the succedent formula A can be derived from the structured configuration (i.e. non empty bracketed sequence) of antecedent formulas ....

Moortgat, M. 1988. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht.

....the analyses that can be proposed in linearization HPSG and LFG, although certain differences are noted. The modern linguistic frameworks Generalized Phrase Structure Grammar (GPSG: Gazdar et al. 1985) Head driven Phrase Structure Grammar (HPSG: Pollard and Sag 1994) and Categorial Grammar (CG: Moortgat 1988) grew out of a structuralist tradition that assumed a rather close relationship between units of functor argument structure and units of surface phrase structure indeed they hewed to this line closely, since they denied the need for transformational operations, which much of modern linguistics ....

Moortgat, M. 1988. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Dordrecht: Foris.

....these limitations is to extend the logic. The initial kernel of the Lambek Calculus is left unchanged but it is extended with new connectives which are used to express linguistic phenomena that defy the power of the pure Lambek Calculus. This approach has been fruitfully explored by Moortgat [Moo88, Moo96], Morrill [Mor90, Mor94] Barry, Hepple, Leslie and Morrill [BHLM91] Moortgat and Morrill[MM91] who have designed new connectives, such as unary modalities and binary operators for extraction and wrapping, and new logical systems such as multimodal systems to extend the power of the Lambek ....

M. Moortgat. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Foris, Dordrecht, 1988.

.... and its sub and supersystems are closely related to several issues of current interest in logic, as e.g. linear logics [17] action logic [25] gaggle theory [14] labelled deductive systems [16] and in natural language semantics and computational linguistics (see van Benthem [2] Moortgat [22]) A thorough logical discussion of the domain can be found in [10, 5] and many linguistic applications in [23] In terms of type theory, B is a purely applicative system, while Lambek systems employ lambda abstraction. A problem which has quite early appeared in this discipline is whether ....

M. Moortgat, Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht, 1988.

....is shown in (14) For simplicity, we have assumed flat structure at the top even though most linguistic analyses assume a binary branching structure. 30 One alternative approach which avoids the use traces altogether is suggested by current research in extended categorial grammar (Steedman 1985; Moortgat 1988, 1990) These researchers account for unbounded dependencies by including non Lambek versions of functional composition and type raising (Steedman) or by adding discontinuous type constructors (Moortgat) While we believe that this line of research is a very promising one to pursue, very little ....

Moortgat, M. 1988. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Dordrecht: Foris Publications.

....of its type forming operators or the lack of a mechanism which would allow change to the order of types in antecedents of formulas of L. Different measures have been proposed to make Lambek formalism more flexible: to enrich the calculus by a new typeforming operator of extraction, see Moortgat 1988, to add a modal operator of permutation, see Morrill et al. 1990, with the polymorphic Lambek calculus also being considered, see Emms 1994. However, a method of changing the order of types can be introduced into the Lambek calculus in a straightforward way, by adopting in the system the ....

Moortgat, M. 1988. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Dordrecht: Foris.

....cheese that the rat that the cat that the dog chased saw ate stank. 9) We argue that a single simple metric of categorial processing complexity explains these and other performance phenomena. 1 Lambek calculus We shall assume some familiarity with Lambek categorial grammar as presented in e.g. Moortgat (1988), Morrill (1994) Moortgat (1997) or Carpenter (1998) and limit ourselves here to reviewing some technical and computational aspects. The types, or (category) formulas, of Lambek calculus are freely generated from a set of primitives by binary infix connectives ( over ) n ( under ) ....

Moortgat, Michael: 1988, Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht.

.... we can identify as a generalisation residuation calculi in which the Lambek connectives (corresponding to linear logic multiplicatives) are defined in a number of potentially interactive modes (Moortgat and Morrill 1991) In Morrill (1993) an improvement of the logic of discontinuity of Moortgat (1988) is developed in this way. Given Cut elimination decidability is directly demonstrable from sequent formulations, but in applications to natural language processing our further objective is efficiency. There are two main approaches in existence: sequent proof normalisation, and proof nets. The ....

....2. John j: N 3. Mary m: N 4. rang ; up)WJohn) phone j) NnS 1, 2 E 5. rang John up (phone j) NnS = 4 6. Mary rang John up ( phone j) m) S 3, 5 En (24) Discussion of semantics would take us outside the direct concerns of the present article; the reader is referred to e.g. Moortgat (1988, 1991) and Morrill (1992a, 1993) for explication. The effect of quantifier raising, whereby quantifiers are to take sentential scope, is achieved by assignment of a quantifier phrase such as everyone to a quantifying in type (S N)#S. A simple instance of quantifier raising is shown in (25) ....

Moortgat, Michael: 1988, Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht.

....A] B i R Gamma ) B= i A Multimodal calculi of discontinuity for discontinuous functors, quantifier phrases, gapping, subject and object oriented reflexives etc. are developed in Solias (1992) Morrill and Solias (1993) and Morrill (1994b, 1995b) surmounting technical difficulties with Moortgat (1988, 1990b, 1991) The treatment of 11 Morrill (1994b, 1995b) is a multimodal calculus with three families of connectives: in addition to the implications, f= ng of L, there are implications forming split strings , f ; g, and implications for interpolation f ; #g, each defined with respect to ....

Moortgat, M.: 1988, Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht.

....Furthermore, in the paraphrase (1b) there is, in addition, ellipsis of the second transitive verb. In (1c) and (1d) other components are elided, with consequent semantic effects. The present work continues in the line of others seeking to develop categorial type calculus of discontinuity (Moortgat 1988, 1990, 1996b, 1996c; Solias 1992; Morrill and Solias 1993; Morrill 1994a, chs. 4 5, 1995a, Calcagno 1995, Moortgat and Oehrle 1995) In particular, it generalises the sorted discontinuity calculus outlined in the appendix of Morrill (1995a) We begin by defining a general framework for sorted ....

....unary split and bridge operators mediating between strings and split strings, and binary operators for generalised concatenation, juxtaposition and interpolation adjunctions. 1 MULTIMODALITY AND DISCONTINUITY The first attempts to formulate categorial logic for discontinuity appear in Moortgat (1988), although as documented in that work there are a number allusions to nonconcatenative operators in earlier, non type logical, literature. Moortgat (1988) defines four binary discontinuous type constructors: infixors and extractors, each with universal and existential varieties. But as he notes, ....

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MOORTGAT, M. (1988): Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht.

....above. These considerations explain in part the recent growing interest in lexicalized grammatical frameworks such as dependency grammar (Mel cuk, 1988; Hudson, 1990; Sleator Temperley, 1991) slot grammar (McCord, 1980; McCord, 1989) categorial grammar (Lambek, 1958; Ades Steedman, 1982; Moortgat, 1988), Head Driven Phrase Structure Grammar (HPSG) Pollard Sag, 1987) and lexicalized tree adjoining grammar (Schabes, 1990) all of which lead to configurations made up of lexical items and direct relationships between them. Computational Requirements: The best formal explanation of a particular ....

Moortgat, M. (1988). Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. PhD thesis, University of Amsterdam, The Netherlands.

....to place any desired principal submatrix in the upper left corner. Incorporating permutation has also been considered in Categorial Grammar. In fact, permutation is sometimes incorporated by adding the equivalence (X = Y ) j (Y n X) Refinements of this simple approach have been proposed [10, 14], motivated by better modeling of natural language than is possible with Categorial Grammar without permutation. ffl Minors and Matroids Generalizations of both the preceding ideas have been studied in the framework of matroids. Specifically, Oxley ( 18] ch.3) reviews related results of Tutte ....

M. Moortgat, Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Providence, RI: Foris Publications, 1988.

....grammar to logic without non logical axioms. In this case there is a nostratal architecture with no essential level of derivational representation: all properties are projected from the model theoretic specification of what the formalism signifies. We take this to be the essential nature of TLG (Moortgat 1988, 1997; van Benthem 1991 95; Morrill 1994; Carpenter 1997) A language, on the Saussurian view, is a collection of signs, where each sign associates a signifier and a signified. A grammar, as a description of language, is to specify a set of pairings of representations of signifiers and signifieds ....

Moortgat, Michael (1988), Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus, Foris, Dordrecht.

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M. Moortgat. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Foris, Dordrecht, 1988.

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M. Moortgat. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Foris, Dordrecht, 1988.

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Michael Moortgat. 1988. Categorial Investigations: logical and linguistic aspects of the Lambek Calculus. Ph.D. thesis, Rijksuniversiteit, Groningen, Dordrecht.

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Michael Moortgat. 1988. Categorial Investigations: Logical and Linguistic Aspects of the Lambek Calculus. Foris.

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