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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 254 (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
Proofnets: The parallel syntax for prooftheory
 Logic and Algebra
, 1996
"... The paper is mainly concerned with the extension of proofnets to additives, for which the best known solution is presented. It proposes two cutelimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponential ..."
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Cited by 98 (1 self)
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The paper is mainly concerned with the extension of proofnets to additives, for which the best known solution is presented. It proposes two cutelimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponentials are also accommodated. Traditional prooftheory deals with cutelimination; these results are usually obtained by means of sequent calculi, with the consequence that 75 % of a cutelimination proof is devoted to endless commutations of rules. It is hard to be happy with this, mainly because: ◮ the structure of the proof is blurred by all these cases; ◮ whole forests have been destroyed in order to print the same routine lemmas; ◮ this is not extremely elegant. However oldfashioned prooftheory, which is concerned with the ritual question: “isthattheoryconsistent? ” never really cared. The situation changed when subtle algorithmic aspects of cutelimination became prominent: typically
Applications of Linear Logic to Computation: An Overview
, 1993
"... This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, li ..."
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Cited by 42 (3 self)
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This paper is an overview of existing applications of Linear Logic (LL) to issues of computation. After a substantial introduction to LL, it discusses the implications of LL to functional programming, logic programming, concurrent and objectoriented programming and some other applications of LL, like semantics of negation in LP, nonmonotonic issues in AI planning, etc. Although the overview covers pretty much the stateoftheart in this area, by necessity many of the works are only mentioned and referenced, but not discussed in any considerable detail. The paper does not presuppose any previous exposition to LL, and is addressed more to computer scientists (probably with a theoretical inclination) than to logicians. The paper contains over 140 references, of which some 80 are about applications of LL. 1 Linear Logic Linear Logic (LL) was introduced in 1987 by Girard [62]. From the very beginning it was recognized as relevant to issues of computation (especially concurrency and stat...
Proofnets and Context Semantics for the Additives
"... We provide a context semantics for MultiplicativeAdditive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cutelimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lvy, who provided a " ..."
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We provide a context semantics for MultiplicativeAdditive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cutelimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lvy, who provided a "geometry of optimal hreduction" (context semantics) for hcalculus and MultiplicativeExponential Linear Logic (MELL). We integrate three features: a semantics that uses buses to implement slicing; a proofnet technology that allows multidimensional boxes and generalized garbage, preserving the linearity of additive reduction; and finally, a readback procedure that computes a cutfree proof from the semantics, which is closely related to full abstraction theorems.
Looking From the Inside and From the Outside
, 1998
"... Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for ..."
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Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for actual points to move around inside them, perhaps to parameterize them in some way. The concept of formal proofs has the interesting feature that it provides opportunities for both perspectives. The inner perspective has been largely overlooked, but in fact lengths of proofs lead to new ways to measure the information content of mathematical objects. The disparity between minimal lengths of proofs with and without "lemmas" provides an indication of internal symmetry of mathematical objects and their descriptions.
Lorenzen's Games and Linear Logic
, 1999
"... This paper presents some basic notions concerning Lorenzen's game tradition and its relation with Linear Logic. It is shown that Lorenzen's idea was applied in Blass' work in order to define a game semantics for Linear Logic. Originally Blass cameup with these ideas on 70's, ..."
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This paper presents some basic notions concerning Lorenzen's game tradition and its relation with Linear Logic. It is shown that Lorenzen's idea was applied in Blass' work in order to define a game semantics for Linear Logic. Originally Blass cameup with these ideas on 70's, however they were fully developed only in 1992. Moreover, by identifying the source of the problems in Blass' approach, Abramsky developed a system using the same pattern of games for which Linear Logic is sound and complete. These systems are both exposed in the following text. Finally, some further issues are discussed and an alternative (and intuitive) approach is built in order to achieve the desired semantics taking into account finite games instead of infinite. 1 1 Lorenzen's Game Semantics Each logic has a syntactical aspect (which defines the wellformed constructions) and a semantical aspect, in which the meaning of each construction is set in an unique way. In classical logic the latter is g...
INTRODUCTION TO THE COMBINATORICS AND COMPLEXITY OF CUT ELIMINATION
"... Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted ..."
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Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted
Generation in the Lambek Calculus Framework: an Approach with Semantic Proof Nets
"... This paper wants to focus on the generation capabilities of proof nets thanks to their semantic readings as expressed in (de Groote and Reto]'d, 1996). The main features of our proposal consist in the use of proof nets for Lambek calculus, of the CurryHoward isomorphism (Howard, 1980; Girard e ..."
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This paper wants to focus on the generation capabilities of proof nets thanks to their semantic readings as expressed in (de Groote and Reto]'d, 1996). The main features of our proposal consist in the use of proof nets for Lambek calculus, of the CurryHoward isomorphism (Howard, 1980; Girard et al., 1988), of semantic p]'oo[' nets with semantic expressions 2 la Montague (Montague, 1974; Dowry et al., 1981), and in an algorithm for proof search with a target proof net. Unlike a previous proposal for generation in the Lam bek calculus framework (Merenciano and Mort'ill, 1997), this point of view avoids the use of the Xterm unification to lead the generation process, And the algorithmic undecidability of this latter mechanism (fi'om second order unification) does not occur any more. In this work, we do not consider the choice of lexi cal items fi'om a given semantic expression the syntactic realization of which we want to generate, but rather tile way we can associate given lexical entries to fit the given semantic expression and generate a syntactically correct expression. For this purpose, we express our problem as a proof search one in (multiplicative) linear logic which is decidable. Moreover, we characterize the semantic recipes of lexical items that provide a polynomial solution for the generation process
Generation, Lambek Calculus, Montague's Semantics and Semantic Proof Nets
, 2000
"... Most of the studies in the framework of Lambek calculus have considered the parsing process aud ignored the generation process. This paper wants to rely on the close link between Lambek calculus and linear logic to present a method for the generation process with semantic proof nets. We express the ..."
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Most of the studies in the framework of Lambek calculus have considered the parsing process aud ignored the generation process. This paper wants to rely on the close link between Lambek calculus and linear logic to present a method for the generation process with semantic proof nets. We express the process as a proof search procedure based on a graph calculus and the solutions appear as a matrix computation preserving the decidability properties, and we characterize a polynomial time case.
Generation with Semantic Proof Nets
"... Categorial grammars and Lambek calculus found a nice embedding in Linear Logic, and a lot of work have presented proof nets uses for linguistic purposes, with a special look at proof nets for Lambek calculus. But they have mainly explored the syntactic capabilities of proof nets, describing parsing ..."
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Categorial grammars and Lambek calculus found a nice embedding in Linear Logic, and a lot of work have presented proof nets uses for linguistic purposes, with a special look at proof nets for Lambek calculus. But they have mainly explored the syntactic capabilities of proof nets, describing parsing processes. We present here our vision of the generation process based on semantic proof nets. The main features of this proposal consist in the use of proof nets for lambek calculus but also for semantic recipes of lexical entries, so that no term unification occur with their limitations (undecidability from second order). Based on a graph calculus, the solutions of this process can be expressed as a matrix computation. Copyright c flXerox Corporation 1999. All rights reserved. XEROX R fl , The Document Company R fl Contents