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48
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 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
Arithmetical proofs of strong normalization results for symmetric λcalculi
"... symmetric λµcalculus ..."
Computation with classical sequents
 MATHEMATICAL STRUCTURES OF COMPUTER SCIENCE
, 2008
"... X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X ..."
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Cited by 16 (16 self)
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X is an untyped continuationstyle formal language with a typed subset which provides a CurryHoward isomorphism for a sequent calculus for implicative classical logic. X can also be viewed as a language for describing nets by composition of basic components connected by wires. These features make X an expressive platform on which algebraic objects and many different (applicative) programming paradigms can be mapped. In this paper we will present the syntax and reduction rules for X and in order to demonstrate the expressive power of X, we will show how elaborate calculi can be embedded, like the λcalculus, Bloo and Rose’s calculus of explicit substitutions λx, Parigot’s λµ and Curien and Herbelin’s λµ ˜µ.
Sequentiality vs. Concurrency in Games and Logic
 Math. Structures Comput. Sci
, 2001
"... Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic. ..."
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Cited by 13 (0 self)
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Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.
Callbyvalue is dual to callbyname, reloaded
 In Rewriting Technics and Application, RTA’05, volume 3467 of LNCS
, 2005
"... Abstract. We consider the relation of the dual calculus of Wadler (2003) to the λµcalculus of Parigot (1992). We give translations from the λµcalculus into the dual calculus and back again. The translations form an equational correspondence as defined by Sabry and Felleisen (1993). In particular, ..."
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Cited by 11 (0 self)
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Abstract. We consider the relation of the dual calculus of Wadler (2003) to the λµcalculus of Parigot (1992). We give translations from the λµcalculus into the dual calculus and back again. The translations form an equational correspondence as defined by Sabry and Felleisen (1993). In particular, translating from λµ to dual and then ‘reloading ’ from dual back into λµ yields a term equal to the original term. Composing the translations with duality on the dual calculus yields an involutive notion of duality on the λµcalculus. A previous notion of duality on the λµcalculus has been suggested by Selinger (2001), but it is not involutive. Note This paper uses color to clarify the relation of types and terms, and of source and target calculi. If the URL below is not in blue please download the color version from
Static Typing for Dynamic Messages
 IN PROCEEDINGS OF THE 25 TH ACM SYMPOSIUM ON PRINCIPLES OF PROGRAMMING LANGUAGES
, 1998
"... Dynamic messages are firstclass messages dynamically bound to program variables. By dynamic messages, the methods to be invoked can be varied dynamically at runtime, which provides a powerful abstraction mechanism for objectoriented languages. Dynamic messages are critically needed for some progr ..."
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Cited by 10 (1 self)
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Dynamic messages are firstclass messages dynamically bound to program variables. By dynamic messages, the methods to be invoked can be varied dynamically at runtime, which provides a powerful abstraction mechanism for objectoriented languages. Dynamic messages are critically needed for some programs, but it seems that there has been no proposal of static type systems for dynamic messages. This paper presents a static typing discipline for dynamic messages and formalizes it into a second order polymorphic type system. The type system satisfies the type soundness property and has a principal type inference algorithm. The type system therefore provides a foundation for a statically typed objectoriented language enriched with polymorphic dynamic messages.
Search algorithms in type theory
, 2000
"... In this paper, we take an abstract view of search by describing search procedures via particular kinds of proofs in type theory. We rely on the proofsasprograms interpretation to extract programs from our proofs. Using these techniques we explore, in depth, a large family of search problems by par ..."
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Cited by 8 (2 self)
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In this paper, we take an abstract view of search by describing search procedures via particular kinds of proofs in type theory. We rely on the proofsasprograms interpretation to extract programs from our proofs. Using these techniques we explore, in depth, a large family of search problems by parameterizing the speci cation of the problem. A constructive proof is presented which has as its computational content a correct search procedure for these problems. We show how a classical extension to an otherwise constructive system can be used to describe a typical use of the nonlocal control operator call/cc. Using the classical typing of nonlocal control we extend our purely constructive proof to incorporate a sophisticated backtracking technique known as ‘con ictdirected backjumping’ (CBJ). A variant of this proof is formalized in Nuprl yielding a correctbyconstruction implementation of CBJ. The extracted program has been translated into Scheme and serves as the basis for an implementation of a new solution to the Hamiltonian circuit problem. This paper demonstrates a nontrivial application of the proofsasprograms paradigm by applying the technique to the derivation of a sophisticated search algorithm; also, it shows the generality of the resulting implementation by demonstrating its application in a new problem
Non deterministic classical logic: the λµ ++calculus
"... In this paper, we present an extension of λµcalculus called λµ ++calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallelor. 1 ..."
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Cited by 8 (8 self)
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In this paper, we present an extension of λµcalculus called λµ ++calculus which has the following properties: subject reduction, strong normalization, unicity of the representation of data and thus confluence only on data types. This calculus allows also to program the parallelor. 1
Explicit Substitutions and Reducibility
 Journal of Logic and Computation
, 2001
"... . We consider reducibility sets dened not by induction on types but by induction on sequents as a tool to prove strong normalization of systems with explicit substitution. To illustrate this point, we give a proof of strong normalization (SN) for simplytyped callbyname ~calculus enriched with op ..."
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Cited by 7 (1 self)
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. We consider reducibility sets dened not by induction on types but by induction on sequents as a tool to prove strong normalization of systems with explicit substitution. To illustrate this point, we give a proof of strong normalization (SN) for simplytyped callbyname ~calculus enriched with operators of explicit unary substitutions. The ~calculus, dened by Curien & Herbelin, is a variant of calculus with a let operator that exhibits symmetries such as terms/contexts and callbyname /callbyvalue reduction. The ~calculus embeds various standard calculi (and Gentzen's style sequent calculi too) and as an application we derive the strong normalization of Parigot's simplytyped calculus with explicit substitution. Introduction Explicit substitution in calculus The traditional theory of calculus relies on reduction, that is the capture by a function of its argument followed by the process of substituting this argument to the places where it is used. The ...