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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 299 (6 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
Lambek Grammars Based on Pregroups
 Logical Aspects of Computational Linguistics, LNAI 2099
, 2001
"... Lambek [13] introduces pregroups as a new framework for syntactic structure. In this paper we prove some new theorems on pregroups and study grammars based on the calculus of free pregroups. We prove that these grammars are equivalent to contextfree grammars. We also discuss the relation of pregrou ..."
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Cited by 34 (5 self)
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Lambek [13] introduces pregroups as a new framework for syntactic structure. In this paper we prove some new theorems on pregroups and study grammars based on the calculus of free pregroups. We prove that these grammars are equivalent to contextfree grammars. We also discuss the relation of pregroups to the Lambek calculus. 1 Introduction and
Incremental processing and acceptability
 Computational Linguistics
, 2000
"... We describe a lefttoright incremental procedure for the processing of Lambek categorial grammar by proof net construction. A simple metric of complexity, the profile in time of the number of unresolved valencies, correctly predicts a wide variety of performance phenomena including garden pathing, ..."
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Cited by 30 (4 self)
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We describe a lefttoright incremental procedure for the processing of Lambek categorial grammar by proof net construction. A simple metric of complexity, the profile in time of the number of unresolved valencies, correctly predicts a wide variety of performance phenomena including garden pathing, the unacceptability of center embedding, preference for lower attachment, lefttoright quantifier scope preference, and heavy noun phrase shift.
Categorial Formalisation of Relativisation: Pied Piping, Islands, and Extraction Sites
, 1992
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Lambek Calculus with Nonlogical Axioms
 Language and Grammar, Studies in Mathematical Linguistics and Natural Language
, 2002
"... We study Nonassociative Lambek Calculus and Associative Lambek Calculus enriched with nitely many nonlogical axioms. We prove that the nonassociative systems are decidable in polynomial time and generate contextfree languages. In [1] it has been shown that nite axiomatic extensions of Associa ..."
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Cited by 14 (10 self)
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We study Nonassociative Lambek Calculus and Associative Lambek Calculus enriched with nitely many nonlogical axioms. We prove that the nonassociative systems are decidable in polynomial time and generate contextfree languages. In [1] it has been shown that nite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages; here we give a new proof of this fact. We also obtain similar results for systems with permutation and n ary operations.
Fibred Semantics for FeatureBased Grammar Logic
, 1994
"... This paper gives a simple method for providing categorial brands of featurebased unification grammars with a modeltheoretic semantics. The key idea is to apply the paradigm of fibred semantics (or layered logics, see [15]) in order to combine the two components of a featurebased grammar logic. We ..."
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Cited by 13 (5 self)
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This paper gives a simple method for providing categorial brands of featurebased unification grammars with a modeltheoretic semantics. The key idea is to apply the paradigm of fibred semantics (or layered logics, see [15]) in order to combine the two components of a featurebased grammar logic. We demonstrate the method for the augmentation of Lambek categorial grammar with Kasper/Roundsstyle feature logic. These are combined by replacing (or annotating) atomic formulas of the first logic, i.e. the basic syntactic types, by formulas of the second. Modelling such a combined logic is less trivial than one might expect. The direct application of the fibred semantics method where a combined atomic formula like np(num:sg & pers:3rd) denotes those strings which have the indicated property and the categorial operators denote the usual left and rightresiduals of these string sets, does not match the intuitive, unificationbased proof theory. Unification implements a global bookkeeping w...
Clausal Proofs and Discontinuity
, 1995
"... We consider the task of theorem proving in Lambek calculi and their generalisation to "multimodal residuation calculi". These form an integral part of categorial logic, a logic of signs stemming from categorial grammar, on the basis of which language processing is essentially theorem provi ..."
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Cited by 11 (3 self)
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We consider the task of theorem proving in Lambek calculi and their generalisation to "multimodal residuation calculi". These form an integral part of categorial logic, a logic of signs stemming from categorial grammar, on the basis of which language processing is essentially theorem proving. The demand of this application is not just for efficient processing of some or other specific calculus, but for methods that will be generally applicable to categorial logics. It is proposed that multimodal cases be treated by dealing with the highest common factor of all the connectives as linear (propositional) validity. The prosodic (sublinear) aspects are encoded in labels, in effect the termstructure of quantified linear logic. The correctness condition on proof nets ("long trip condition") can be implemented by SLD resolution in linear logic with unification on labels/terms limited to one way matching. A suitable unification strategy is obtained for calculi of discontinuity by normalisation...
A Dialectica Model of the Lambek Calculus
 AMSTRERDAM COLLOQUIUM, 1991
, 1991
"... this paper. But it must be said from the start that the `answer' is only provided in semantical terms. The Proof Theory of the systems considered should be investigated in future work. Another warning is that the perspective of this note is basically from Category Theory as a branch of Mathemat ..."
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Cited by 7 (1 self)
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this paper. But it must be said from the start that the `answer' is only provided in semantical terms. The Proof Theory of the systems considered should be investigated in future work. Another warning is that the perspective of this note is basically from Category Theory as a branch of Mathematics, so words like categories and functors are always meant in their mathematical, rather than linguistical or philosophical sense. We first recall Linear Logic and provide the transformations to show that the Lambek Calculus L really is the multiplicative fragment of (noncommutative) Intuitionistic Linear Logic. In the second section we describe the usual String Semantics for the Lambek Calculus L and generalise it, using a categorical perspective. In the third section we describe our Dialectica model for the Lambek Calculus. In the last section we discuss modalities and some `untidiness' of the CurryHoward correspondence for the fragments of Linear Logic in question. I would like to thank Jan van Eijck for inviting me to give the talk that became this note, thereby gently `forcing' me to think about the subject, as well as, for his generous hospitality. I also would like to thank Martin Hyland, Harold Schellinx, Dirk Roorda, Mark Hepple, Glyn Morrill and Michael Moortgat for several useful discussions. Many of the ideas in this paper have been shaped by these discussions, but of course the mistakes are all mine. Finally I want to thank Jim Lambek for `putting me right' about how completeness has nothing to do with the existence of two disjunctions, in the most friendly possible way. 1 From Linear Logic to the Lambek Calculus