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19
A Judgmental Analysis of Linear Logic
, 2003
"... We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives ..."
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Cited by 50 (27 self)
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We reexamine the foundations of linear logic, developing a system of natural deduction following MartinL of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, extending dual intuitionistic linear logic but differing from both classical linear logic and Hyland and de Paiva's full intuitionistic linear logic. We also provide a corresponding sequent calculus that admits a simple proof of the admissibility of cut by a single structural induction. Finally, we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of doublenegation translation.
A formulaeastypes interpretation of subtractive logic
 Journal of Logic and Computation
, 2004
"... We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural ..."
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Cited by 24 (1 self)
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We present a formulaeastypes interpretation of Subtractive Logic (i.e. biintuitionistic logic). This presentation is twofold: we first define a very natural restriction of the λµcalculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for firstclass coroutines (a restricted form of firstclass continuations). Keywords: CurryHoward isomorphism, Subtractive Logic, control operators, coroutines. 1
A short proof of the Strong Normalization of Classical Natural Deduction with Disjunction
 Journal of symbolic Logic
, 2003
"... We give a direct, purely arithmetical and elementary proof of the strong normalization of the cutelimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction. 1 ..."
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Cited by 23 (14 self)
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We give a direct, purely arithmetical and elementary proof of the strong normalization of the cutelimination procedure for full (i.e. in presence of all the usual connectives) classical natural deduction. 1
ProofTerms for Classical and Intuitionistic Resolution (Extended Abstract)
, 1996
"... We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resol ..."
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Cited by 13 (3 self)
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We exploit a system of realizers for classical logic, and a translation from resolution into the sequent calculus, to assess the intuitionistic force of classical resolution for a fragment of intuitionistic logic. This approach is in contrast to formulating locally intuitionistically sound resolution rules. The techniques use the fflcalculus, a development of Parigot's calculus.
Confluency property of the callbyvalue λµ ∧∨  calculus
 Computational Logic and Applications CLA’05. Discrete Mathematics and Theoretical Computer Science proc
, 2006
"... LAMA Équipe de logique, Université de Savoie, F73376 Le Bourget du Lac, France In this paper, we introduce the λµ ∧ ∨ callbyvalue calculus and we give a proof of the ChurchRosser property of this system. This proof is an adaptation of that of Andou (2003) which uses an extended parallel reduct ..."
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Cited by 3 (0 self)
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LAMA Équipe de logique, Université de Savoie, F73376 Le Bourget du Lac, France In this paper, we introduce the λµ ∧ ∨ callbyvalue calculus and we give a proof of the ChurchRosser property of this system. This proof is an adaptation of that of Andou (2003) which uses an extended parallel reduction method and complete development. Keywords: Callbyvalue, ChurchRosser, Propositional classical logic, Parallel reduction, Complete development 1
A games semantics for reductive logic and proofsearch
 GaLoP 2005: Games for Logic and Programming Languages
, 2005
"... Abstract. Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive l ..."
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Abstract. Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive logic and show how it may be used to model two important examples of control, namely backtracking and uniform proof. 1 Introduction to reductive logic and proofsearch Theorem proving, or algorithmic proofsearch, is an essential enabling technology throughout the computational sciences. We explain the mathematical basis of proofsearch as the combination of reductive logic together with a control régime. Then we present a games semantics for reductive logic and show how it may be used to model two important
A Parigotstyle linear λcalculus for Full intuitionistic Linear Logic
, 2005
"... This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical l ..."
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Cited by 3 (0 self)
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This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical logic in that its sequentcalculus formulation has intrinsic multiple conclusions. From the intrinsic multiple conclusions comes the inspiration to modify Parigot’s natural deduction systems for classical logic, to produce a natural deduction formulation and a term assignment system for FILL. keywords: linear logic, λµcalculus, CurryHoward isomorphism 1
Some Pitfalls of LKtoLJ Translations and How to Avoid Them
 Proc CADE14, LNCS 1249
, 1997
"... . In this paper, we investigate translations from a classical cutfree sequent calculus LK into an intuitionistic cutfree sequent calculus LJ. Translations known from the literature rest on permutations of inferences in classical proofs. We show that, for some classes of firstorder formulae, all m ..."
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Cited by 2 (0 self)
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. In this paper, we investigate translations from a classical cutfree sequent calculus LK into an intuitionistic cutfree sequent calculus LJ. Translations known from the literature rest on permutations of inferences in classical proofs. We show that, for some classes of firstorder formulae, all minimal LJproofs are nonelementary but there exist short LKproofs of the same formula. Similar results are obtained even if some fragments of intuitionistic firstorder logic are considered. The size of the corresponding minimal search spaces for LK and LJ are also nonelementarily related. We show that we can overcome these difficulties by extending LJ with an analytic cut rule. 1 Introduction Characterizing classes of formulae for which classical derivability implies intuitionistic derivability was one topic in the Leningrad group around Maslov in the sixties. Such classes are called (complete) Glivenko classes which were extensively characterized by Orevkov [7]. More recently, people ar...
GoalDirected Proof Search in MultipleConclusioned Intuitionistic Logic
 In Proceedings of the First International Conference on Computational Logic, volume LNAI 1861
, 2000
"... . A key property in the definition of logic programming languages is the completeness of goaldirected proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (singleconclusioned) sequent calculus LJ, but has subsequently been adapted to multip ..."
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. A key property in the definition of logic programming languages is the completeness of goaldirected proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (singleconclusioned) sequent calculus LJ, but has subsequently been adapted to multipleconclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goaldirected proofs for a multipleconclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both singleconclusioned and multipleconclusioned systems (although the latter are less well known). In this paper we show that the language obtained for the multipleconclusioned system differs from that for the singleconclusioned case, show how hereditary Harrop formulae can be recovered, and investigate contractionfree fragments of the logic. 1 Introduction Logic programming is based upon the observation that if ...
A Parigotstyle Linear lambdaCalculus for Full Intuitionistic Linear Logic
, 2003
"... This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classic ..."
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Cited by 1 (0 self)
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This paper describes a natural deduction formulation for Full Intuitionistic Linear Logic (FILL), an intriguing variation of multiplicative linear logic, due to Hyland and de Paiva. The system FILL resembles intuitionistic logic, in that all its connectives are independent, but resembles classical logic in that its sequentcalculus formulation has intrinsic multiple conclusions. From the intrinsic multiple conclusions comes the inspiration to modify Parigot's natural deduction systems for classical logic, to produce a natural deduction formulation and a term assignment system for FILL.