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67
A New Deconstructive Logic: Linear Logic
, 1995
"... The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different a ..."
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Cited by 102 (11 self)
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The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programmingwithproofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for nonadditive proof nets, to be precise) using appropriate embeddings (socalled decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...
A CurryHoward foundation for functional computation with control
 In Proceedings of ACM SIGPLANSIGACT Symposium on Principle of Programming Languages
, 1997
"... We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatib ..."
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Cited by 77 (3 self)
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We introduce the type theory ¯ v , a callbyvalue variant of Parigot's ¯calculus, as a CurryHoward representation theory of classical propositional proofs. The associated rewrite system is ChurchRosser and strongly normalizing, and definitional equality of the type theory is consistent, compatible with cut, congruent and decidable. The attendant callbyvalue programming language ¯pcf v is obtained from ¯ v by augmenting it by basic arithmetic, conditionals and fixpoints. We study the behavioural properties of ¯pcf v and show that, though simple, it is a very general language for functional computation with control: it can express all the main control constructs such as exceptions and firstclass continuations. Prooftheoretically the dual ¯ v constructs of naming and ¯abstraction witness the introduction and elimination rules of absurdity respectively. Computationally they give succinct expression to a kind of generic (forward) "jump" operator, which may be regarded as a unif...
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 23 (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
Recursive Polymorphic Types and Parametricity in an Operational Framework
, 2005
"... We construct a realizability model of recursive polymorphic types, starting from an untyped language of terms and contexts. An orthogonality relation e # indicates when a term e and a context # may be safely combined in the language. Types are interpreted as sets of terms closed by biorthogonalit ..."
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Cited by 22 (1 self)
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We construct a realizability model of recursive polymorphic types, starting from an untyped language of terms and contexts. An orthogonality relation e # indicates when a term e and a context # may be safely combined in the language. Types are interpreted as sets of terms closed by biorthogonality. Our main result states that recursive types are approximated by converging sequences of interval types. Our proof is based on a "typedirected" approximation technique, which departs from the "languagedirected" approximation technique developed by MacQueen, Plotkin and Sethi in the ideal model. We thus keep the language elementary (a callbyname #calculus) and unstratified (no typecase, no reduction labels). We also include a short account of parametricity, based on an orthogonality relation between quadruples of terms and contexts.
On the Intuitionistic Force of Classical Search
 THEORETICAL COMPUTER SCIENCE
, 1996
"... The combinatorics of classical propositional logic lies at the heart of both local and global methods of proofsearch enabling the achievement of leastcommitment search. Extension of such methods to the predicate calculus, or to nonclassical systems, presents us with the problem of recovering ..."
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Cited by 19 (5 self)
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The combinatorics of classical propositional logic lies at the heart of both local and global methods of proofsearch enabling the achievement of leastcommitment search. Extension of such methods to the predicate calculus, or to nonclassical systems, presents us with the problem of recovering this leastcommitment principle in the context of noninvertible rules. One successful approach is to view the nonclassical logic as a perturbation on search in classical logic and characterize when a leastcommitment (classical) search yields sufficient evidence for provability in the (nonclassical) logic. This technique has been successfully applied to both local and global methods at the cost of subsidiary searches and is the analogue of the standard treatment of quantifiers via skolemization and unification. In this paper, we take a typetheoretic view of this approach for the case in which the nonclassical logic is intuitionistic. We develop a system of realizers (proofobje...
Arithmetical proofs of strong normalization results for symmetric λcalculi
"... symmetric λµcalculus ..."
A CPSTranslation of the λµCalculus
, 1994
"... We present a translation of Parigot's λµcalculus [10] into the usual λcalculus. This translation, which is based on the socalled continuation passing style, is correct with respect to equality and with respect to evaluation. At the type level, it induces a logical interpretation of classical log ..."
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Cited by 15 (1 self)
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We present a translation of Parigot's λµcalculus [10] into the usual λcalculus. This translation, which is based on the socalled continuation passing style, is correct with respect to equality and with respect to evaluation. At the type level, it induces a logical interpretation of classical logic into intuitionistic one, akin to Kolmogorov's negative translation. As a byproduct, we get the normalization of second order typed λµcalculus.
La valeur d’un entier classique en λµcalcul
 Archive for Mathematical Logic
, 1997
"... de mathématiques, équipe de logique, ..."
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 12 (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.