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Continuation semantics for the Lambek–Grishin calculus
 INFORMATION AND COMPUTATION
, 2010
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AnnotationFree Sequent Calculi for Full Intuitionistic Linear Logic ∗
"... Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic linear logic extended with par. Its proof theory has been notoriously difficult to get right, and existing sequent calculi all involve inference rules with complex annotations to guarantee soundness and cutelimination. We give ..."
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Full Intuitionistic Linear Logic (FILL) is multiplicative intuitionistic linear logic extended with par. Its proof theory has been notoriously difficult to get right, and existing sequent calculi all involve inference rules with complex annotations to guarantee soundness and cutelimination. We give a simple and annotationfree display calculus for FILL which satisfies Belnap’s generic cutelimination theorem. To do so, our display calculus actually handles an extension of FILL, called BiIntuitionistic Linear Logic (BiILL), with an ‘exclusion ’ connective defined via an adjunction with par. We refine our display calculus for BiILL into a cutfree nested sequent calculus with deep inference in which the explicit structural rules of the display calculus become admissible. A separation property guarantees that proofs of FILL formulae in the deep inference calculus contain no trace of exclusion. Each such rule is sound for the semantics of FILL, thus our deep inference calculus and display calculus are conservative over FILL. The deep inference calculus also enjoys the subformula property and terminating backward proof search, which gives the NPcompleteness of BiILL and FILL.
Linguistic, Philosophical, and Pragmatic Aspects of TypeDirected Natural Language Parsing
"... We describe how type information can be used to infer grammatical structure. This is in contrast to conventional type inference in programming languages where the roles are reversed, structure determining type. Our work is based on Applicative Universal Grammar (AUG), a linguistic theory that vi ..."
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We describe how type information can be used to infer grammatical structure. This is in contrast to conventional type inference in programming languages where the roles are reversed, structure determining type. Our work is based on Applicative Universal Grammar (AUG), a linguistic theory that views the formation of phrase in a form that is analogous to function application in a programming language. We descibe our overall methodology including its linguistic and philosophical underpinnings.
Cyclic Pregroups and Natural Language: a Computational Algebraic Analysis
"... Abstract. The calculus of pregroups is introduced by Lambek [1999] as an algebraic computational system for the grammatical analysis of natural languages. Pregroups are non commutative structures, but the syntax of natural languages shows a diffuse presence of cyclic patterns exhibited in different ..."
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Abstract. The calculus of pregroups is introduced by Lambek [1999] as an algebraic computational system for the grammatical analysis of natural languages. Pregroups are non commutative structures, but the syntax of natural languages shows a diffuse presence of cyclic patterns exhibited in different kinds of word order changes. The need of cyclic operations or transformations was envisaged both by Z. Harris and N. Chomsky, in the framework of generative transformational grammar. In this paper we propose an extension of the calculus of pregroups by introducing appropriate cyclic rules that will allow the grammar to formally analyze and compute word order and movement phenomena in different languages such as Persian, French, Italian, Dutch and Hungarian. This crosslinguistic analysis, although necessarily limited and not at all exhaustive, will allow the reader to grasp the essentials of a pregroup grammar, with particular reference to its straightforward way of computing linguistic information. 1
Generalized Discontinuity
"... Abstract. We define and study a calculus of discontinuity, a version of displacement calculus, which is a logic of segmented strings in exactly the same sense that the Lambek calculus is a logic of strings. Like the Lambek calculus, the displacement calculus is a sequence logic free of structural ru ..."
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Abstract. We define and study a calculus of discontinuity, a version of displacement calculus, which is a logic of segmented strings in exactly the same sense that the Lambek calculus is a logic of strings. Like the Lambek calculus, the displacement calculus is a sequence logic free of structural rules, and enjoys Cutelimination and its corollaries: the subformula property, decidability, and the finite reading property. The foci of this paper are a formulation with a finite number of connectives, and consideration of how to extend the calculus with defined connectives while preserving its good properties. 1 Introduction: architecture of logical grammar An argument in logic comprises some premises and a conclusion; for example: 1 (1) a. All men are mortal. Socrates is a man. Socrates is mortal.