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27
From ProofNets to Interaction Nets
 Advances in Linear Logic
, 1994
"... Introduction If we consider the interpretation of proofs as programs, say in intuitionistic logic, the question of equality between proofs becomes crucial: The syntax introduces meaningless distinctions whereas the (denotational) semantics makes excessive identifications. This question does not hav ..."
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Cited by 59 (1 self)
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Introduction If we consider the interpretation of proofs as programs, say in intuitionistic logic, the question of equality between proofs becomes crucial: The syntax introduces meaningless distinctions whereas the (denotational) semantics makes excessive identifications. This question does not have a simple answer in general, but it leads to the notion of proofnet, which is one of the main novelties of linear logic. This has been already explained in [Gir87] and [GLT89]. The notion of interaction net introduced in [Laf90] comes from an attempt to implement the reduction of these proofnets. It happens to be a simple model of parallel computation, and so it can be presented independently of linear logic, as in [Laf94]. However, we think that it is also useful to relate the exact origin of interaction nets, especially for readers with some knowledge in linear logic. We take this opportunity to give a survey of the theory of proofnets, including a new proof of the sequentializ
A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 53 (8 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
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 41 (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...
Proof Search in Firstorder Linear Logic and Other Cutfree Sequent Calculi
 In LICS
, 1994
"... We present a general framework for proof search in firstorder cutfree sequent calculi and apply it to the specific case of linear logic. In this framework, Herbrand functions are used to encode universal quantification, and unification is used to instantiate existential quantifiers so that the eig ..."
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Cited by 27 (2 self)
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We present a general framework for proof search in firstorder cutfree sequent calculi and apply it to the specific case of linear logic. In this framework, Herbrand functions are used to encode universal quantification, and unification is used to instantiate existential quantifiers so that the eigenvariable conditions are respected. We present an optimization of this procedure that exploits the permutabilities of the subject logic. We prove the soundness and completeness of several related proof search procedures. This proof search framework is used to show that provability for firstorder MALL is in nexptime, and firstorder MLL is in np. Performance comparisons based on Prolog implementations of the procedures are also given. The optimization of the quantifier steps in proof search can be combined effectively with a number of other optimizations that are also based on permutability. 1 Introduction Since proofs contain more information than the theorems they prove, the main challen...
Linear Logic
, 1992
"... this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may ..."
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Cited by 24 (1 self)
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this paper we will restrict attention to propositional linear logic. The sequent calculus notation, due to Gentzen [10], uses roman letters for propositions, and greek letters for sequences of formulas. A sequent is composed of two sequences of formulas separated by a `, or turnstile symbol. One may read the sequent \Delta ` \Gamma as asserting that the multiplicative conjunction of the formulas in \Delta together imply the multiplicative disjunction of the formulas in \Gamma. A sequent calculus proof rule consists of a set of hypothesis sequents, displayed above a horizontal line, and a single conclusion sequent, displayed below the line, as below: Hypothesis1 Hypothesis2 Conclusion 4 Connections to Other Logics
Lambek calculus is npcomplete
 Theoretical Computer Science
, 2003
"... We prove that for both the Lambek calculus L and the Lambek calculus allowing empty premises L ∗ the derivability problem is NPcomplete. It follows that also for the multiplicative fragments of cyclic linear logic and noncommutative linear logic the derivability problem is NPcomplete. ..."
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Cited by 23 (0 self)
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We prove that for both the Lambek calculus L and the Lambek calculus allowing empty premises L ∗ the derivability problem is NPcomplete. It follows that also for the multiplicative fragments of cyclic linear logic and noncommutative linear logic the derivability problem is NPcomplete.
Deciding Provability of Linear Logic Formulas
 Advances in Linear Logic
, 1994
"... Introduction There are many interesting fragments of linear logic worthy of study in their own right, most described by the connectives which they employ. Full linear logic includes all the logical connectives, which come in three dual pairs: the exponentials ! and ?, the additives & and \Phi, and ..."
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Cited by 21 (0 self)
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Introduction There are many interesting fragments of linear logic worthy of study in their own right, most described by the connectives which they employ. Full linear logic includes all the logical connectives, which come in three dual pairs: the exponentials ! and ?, the additives & and \Phi, and the multiplicatives\Omega and . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ . SRI International Computer Science Laboratory, Menlo Park CA 94025 USA. Work supported under NSF Grant CCR9224858. lincoln@csl.sri.com http://www.csl.sri.com/lincoln/lincoln.html Patrick Lincoln For the most part we will consider fragments of linear logic built up using these connectives in any combination. For example, full linear logic formulas may employ any connective, while multiplic
Linear Logic and Computation: A Survey
 Proof and Computation, Proceedings Marktoberdorf Summer School
, 1993
"... . This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth ..."
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Cited by 16 (6 self)
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. This is a survey of computational aspects of linear logic related to proof search. Keywords. Linear logic, cut free proof search, logic programming, complexity. 1 Introduction Linear logic, introduced by Girard [14, 36, 32], is a refinement of classical logic. While the central notions of truth (emphasized in classical logic) and proof construction (emphasized in intuitionistic logic) remain important in linear logic, it might be said that the emphasis in linear logic is on state. Linear logic is sometimes described as being resource sensitive because it provides an intrinsic and natural accounting of process states, events, and resources. Linear logic also sheds new light on classical logic and its relationship to intuitionistic logic, see Girard [15, 16] and Danos et al. [11]. An evocative semantic paradigm for linear logic by means of games is proposed by Blass [7] and by Abramsky and Jagadeesan [2]. As an intuitive motivation, let us consider reading logical deductions so tha...
linTAP: A Tableau Prover for Linear Logic
 International Conference TABLEAUX’99
, 1999
"... linTAP is a tableau prover for the multiplicative and exponential fragment M?LL of Girards linear logic. It proves the validity of a given formula by constructing an analytic tableau and ensures the linear validity using prex unication. We present the tableau calculus used by linTAP, an algorithm fo ..."
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Cited by 16 (5 self)
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linTAP is a tableau prover for the multiplicative and exponential fragment M?LL of Girards linear logic. It proves the validity of a given formula by constructing an analytic tableau and ensures the linear validity using prex unication. We present the tableau calculus used by linTAP, an algorithm for prex unication in linear logic, the linTAP implementation, and some experimental results obtained with linTAP. 1
First Order Linear Logic without Modalities Is NEXPTIMEHard
 Theoretical Computer Science
, 1994
"... The decision problem is studied for the nonmodal or multiplicativeadditive fragment of first order linear logic. This fragment is shown to be nexptime hard. The hardness proof combines Shapiro's logic programming simulation of nondeterministic Turing machines with the standard proof of the pspace ..."
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Cited by 15 (11 self)
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The decision problem is studied for the nonmodal or multiplicativeadditive fragment of first order linear logic. This fragment is shown to be nexptime hard. The hardness proof combines Shapiro's logic programming simulation of nondeterministic Turing machines with the standard proof of the pspace hardness of quantified boolean formula validity, utilizing some of the surprisingly powerful and expressive machinery of linear logic. 1 Introduction Linear logic, introduced by Girard, is a resourcesensitive refinement of classical logic [10, 29]. Linear logic gains its expressive power by restricting the "structural" proof rules of contraction (copying) and weakening (erasing). The contraction rule makes it possible to reuse any stated assumption as often as desired. The weakening rule makes it possible to use dummy assumptions, i.e., it allows a deduction to be carried out without using all of the hypotheses. Because contraction and weakening together make it possible to use an assu...