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Categorial Type Logics
 Handbook of Logic and Language
, 1997
"... Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource mana ..."
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Cited by 238 (5 self)
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Contents 1 Introduction: grammatical reasoning 1 2 Linguistic inference: the Lambek systems 5 2.1 Modelinggrammaticalcomposition ............................ 5 2.2 Gentzen calculus, cut elimination and decidability . . . . . . . . . . . . . . . . . . . . 9 2.3 Discussion: options for resource management . . . . . . . . . . . . . . . . . . . . . . 13 3 The syntaxsemantics interface: proofs and readings 16 3.1 Term assignment for categorial deductions . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Natural language interpretation: the deductive view . . . . . . . . . . . . . . . . . . . 21 4 Grammatical composition: multimodal systems 26 4.1 Mixedinference:themodesofcomposition........................ 26 4.2 Grammaticalcomposition:unaryoperations ....................... 30 4.2.1 Unary connectives: logic and structure . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 Applications: imposing constraints, structural relaxation
Multimodal Linguistic Inference
, 1995
"... In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resourc ..."
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Cited by 40 (6 self)
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In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives =; ffl; n, together with a package of structural postulates characterizing the resource management properties of the ffl connective. Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system. The simple systems each have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems. The first type is obtained by combining a number of unimodal systems into one multimodal logic. The...
Structural Control
 SPECIFYING SYNTACTIC STRUCTURES, PATRICK BLACKBURN, MAARTEN DE RIJKE (EDS.)
, 1988
"... In this paper we study Lambek systems as grammar logics: logics for reasoning about structured linguistic resources. The structural parameters of precedence, dominance and dependency generate a cube of resourcesensitive categorial type logics. From the pure logic of residuation NL, one obtains L, N ..."
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Cited by 38 (8 self)
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In this paper we study Lambek systems as grammar logics: logics for reasoning about structured linguistic resources. The structural parameters of precedence, dominance and dependency generate a cube of resourcesensitive categorial type logics. From the pure logic of residuation NL, one obtains L, NLP and LP in terms of Associativity, Commutativity, and their combination. Each of these systems has a dependency variant, where the product is split up into a leftheaded and a rightheaded version. We develop a theory of systematic communication between these systems. The communication is twoway: we show how one can fully recover the structural discrimination of a weaker logic from within a system with a more liberal resource management regime, and how one can reintroduce the structural flexibility of a stronger logic within a system with a more articulate notion of structuresensitivity. In executing this programme we follow the standard logical agenda: the categorial formula language is enriched with extra control operators, socalled structural modalities, and on the basis of these control operators, we prove embedding theorems for the two directions of substructural communication. But our results differ from the Linear Logic style of embedding with S4like modalities in that we realize the communication in both directions in terms of a
A CompilationChart Method for Linear Categorial Deduction
, 1996
"... Recent work in categorial grammar has seen proposals for a wide range of systems, differing in their `resource sensitivity'... ..."
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Cited by 18 (5 self)
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Recent work in categorial grammar has seen proposals for a wide range of systems, differing in their `resource sensitivity'...
Pomset Logic as an Alternative Categorial Grammar
 IN FORMAL GRAMMAR
, 1995
"... Lambek calculus may be viewed as a fragment of linear logic, namely intuitionistic noncommutative multiplicative linear logic. As it is too restrictive to describe numerous usual linguistic phenomena, instead of extending it we extend MLL with a noncommutative connective, thus dealing with partia ..."
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Cited by 17 (2 self)
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Lambek calculus may be viewed as a fragment of linear logic, namely intuitionistic noncommutative multiplicative linear logic. As it is too restrictive to describe numerous usual linguistic phenomena, instead of extending it we extend MLL with a noncommutative connective, thus dealing with partially ordered multisets of formulae. Relying on proof net technique, our study associates words with parts of proofs, modules, and parsing is described as proving by plugging modules. Apart from avoiding spurious ambiguities, our method succeeds in obtaining a logical description of relatively free word order, headwrapping, clitics, and extraposition (these latest two constructions are unfortunately not included, for lack of space).
HigherOrder Linear Logic Programming of Categorial Deduction’, Report de Recerca LSI–94–42–R, Departament de Llenguatges i
 Sistemes Informàtics, Universitat Politècnica de Catalunya Morrill, Glyn: 1994b, Type Logical Grammar: Categorial Logic of Signs
"... We show how categorial deduction can be implemented in higherorder (linear) logic programming, thereby realising parsing as deduction for the associative and nonassociative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variet ..."
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Cited by 13 (4 self)
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We show how categorial deduction can be implemented in higherorder (linear) logic programming, thereby realising parsing as deduction for the associative and nonassociative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions. The present work deals with the parsing problem for Lambek calculus and its extensions as developed
The TuringCompleteness of Multimodal Categorial Grammars
 JFAK: Essays dedicated to Johan van Benthem on the occasion of his 50th birthday. Institute for Logic, Language, and Computation, University of Amsterdam. Available on CDROM at http://turing.wins.uva.nl
, 1996
"... this paper, we demonstrate that the multimodal categorial grammars are in fact Turingcomplete in their weak generative capacity. The result follows from a straightforward reduction of generalized rewriting systems to a mixed associative and modal categorial calculus. It turns out that we should not ..."
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Cited by 12 (0 self)
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this paper, we demonstrate that the multimodal categorial grammars are in fact Turingcomplete in their weak generative capacity. The result follows from a straightforward reduction of generalized rewriting systems to a mixed associative and modal categorial calculus. It turns out that we should not be surprised that seemingly arbitrary kinds of operations can be coded in multimodal categorial grammars. In this paper, we show that any computable grammar can be coded as a multimodal categorial grammar. From the standpoint of formal linguistics, this opening of the computational floodgates might even appear to be inevitable. Simply compare the introduction of general transformations in transformational grammars [Peters and Ritchie 1973], metarules in phrase structure grammars [Uszkoreit and Peters 1986], and lexical rules in categorial and phrase structure systems [Carpenter 1991], all of which have been shown to be Turingcomplete. Although steps may be taken to restrict the power of these systems to ensure decidability, such moves appear rather ad hoc because of their lack of linguistic motivation. For instance, consider the restrictions against metarule self application [Gazdar et al. 1985], or the finite bound placed on unary phrase structure rules (and by association, empty categories) by [Kaplan and Bresnan 1982]. Natural language syntax is a difficult matter, and no formalism has even come close to providing a universal system in which all and only natural language grammars can be expressed. Perhaps even more discouraging is the fact that no grammars for particular languages have ever been developed that even come close to covering a naturally occurring range of data in a theoretically clean fashion. On the other hand, the grammar fragments that are typically propo...
Memoisation for Glue Language Deduction and Categorial Parsing
, 1998
"... The multiplicative fragment of linear logic has found a number of applications in computational linguistics: in the "glue language" approach to LFG semantics, and in the formulation and parsing of various categorial grammars. These applications call for efficient deduction methods. Although a number ..."
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Cited by 8 (1 self)
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The multiplicative fragment of linear logic has found a number of applications in computational linguistics: in the "glue language" approach to LFG semantics, and in the formulation and parsing of various categorial grammars. These applications call for efficient deduction methods. Although a number of deduction methods for multiplicative linear logic are known, none of them are tabular meth ods, which bring a substantial efficiency gain by avoiding redundant computation (c.f. chart methods in CFG parsing): this paper presents such a method, and discusses its use in relation to the above applications.
In Situ Binding: A Modal Analysis
, 1996
"... In this paper we compare two multimodal deconstructions of the in situ binder q(A; B; C), proposed in [9] for the scoping of generalized quantifiers. The wrapping analysis of [13] is shown to be of limited generality: it restricts the occurrence of generalized quantifier expressions to associative ..."
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Cited by 7 (1 self)
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In this paper we compare two multimodal deconstructions of the in situ binder q(A; B; C), proposed in [9] for the scoping of generalized quantifiers. The wrapping analysis of [13] is shown to be of limited generality: it restricts the occurrence of generalized quantifier expressions to associative environments  environments where sensitivity for constituent structure is lacking. We propose an alternative deconstruction where the wrapping operation is independent of resource management assumptions about the structural context. The analysis is based on the general theory of structural control proposed in [8]: the interaction principles for the wrapping operation are finetuned in terms of unary modal control devices.
Mixing Modes of Linguistic Description in Categorial Grammar
, 1995
"... Recent work within the field of Categorial Grammar has seen the development of approaches that allow different modes of logical behaviour to be displayed within a single system, something corresponding to making available differing modes of linguistic description. Earlier attempts to achieve this go ..."
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Cited by 6 (0 self)
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Recent work within the field of Categorial Grammar has seen the development of approaches that allow different modes of logical behaviour to be displayed within a single system, something corresponding to making available differing modes of linguistic description. Earlier attempts to achieve this goal have employed modal op erators called structural modalities, whose use presents a number of problems. I propose an alternative approach, involving co existence and interrelation of different sublogics, that eliminates the need for structural modalities, whilst maintaining the de scriptive power they provide.