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82
Disjunctive Tautologies as Synchronisation Schemes
 In Computer Science Logic’00
, 2000
"... In the ambient logic of classical second order propositional calculus, we solve the specification problem for a family of excluded middle like tautologies. These are shown to be realized by sequential simulations of specific communication schemes for which they provide a safe typing mechanism. 1 ..."
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Cited by 19 (2 self)
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In the ambient logic of classical second order propositional calculus, we solve the specification problem for a family of excluded middle like tautologies. These are shown to be realized by sequential simulations of specific communication schemes for which they provide a safe typing mechanism. 1
From proofs to focused proofs: a modular proof of focalization in linear logic
 CSL 2007: Computer Science Logic, volume 4646 of LNCS
, 2007
"... dale.miller at inria.fr saurin at lix.polytechnique.fr Abstract. Probably the most significant result concerning cutfree sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications — e.g. in game semantics, Lu ..."
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Cited by 17 (7 self)
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dale.miller at inria.fr saurin at lix.polytechnique.fr Abstract. Probably the most significant result concerning cutfree sequent calculus proofs in linear logic is the completeness of focused proofs. This completeness theorem has a number of proof theoretic applications — e.g. in game semantics, Ludics, and proof search — and more computer science applications — e.g. logic programming, callbyname/value evaluation. Andreoli proved this theorem for firstorder linear logic 15 years ago. In the present paper, we give a new proof of the completeness of focused proofs in terms of proof transformation. The proof of this theorem is simple and modular: it is first proved for MALL and then is extended to full linear logic. Given its modular structure, we show how the proof can be extended to larger systems, such as logics with induction. Our analysis of focused proofs will employ a proof transformation method that leads us to study how focusing and cut elimination interact. A key component of our proof is the construction of a focalization graph which provides an abstraction over how focusing can be organized within a given cutfree proof. Using this graph abstraction allows us to provide a detailed study of atomic bias assignment in a way more refined that is given in Andreoli’s original proof. Permitting more flexible assignment of bias will allow this completeness theorem to help establish the completeness of a number of other automated deduction procedures. Focalization graphs can be used to justify the introduction of an inference rule for multifocus derivation: a rule that should help us better understand the relations between sequentiality and concurrency in linear logic. 1
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.
A unified sequent calculus for focused proofs
 In LICS: 24th Symp. on Logic in Computer Science
, 2009
"... Abstract—We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicativeadditive linear logic are derived as fragments of LKU by increasing the sensitivity of specialize ..."
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Cited by 13 (7 self)
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Abstract—We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicativeadditive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cutelimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classicallinear hybrid logics. KeywordsProof theory; focused proof systems; linear logic I.
On the Linear Decoration of Intuitionistic Derivations
, 1993
"... We define an optimal proofbyproof embedding of intuitionistic sequent calculus into linear logic and analyse the (purely logical) linearity information thus obtained. 1 Introduction Uniform translations of intuitionistic into linear logic, with their plethoric use of exponentials, are bound to gi ..."
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Cited by 13 (1 self)
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We define an optimal proofbyproof embedding of intuitionistic sequent calculus into linear logic and analyse the (purely logical) linearity information thus obtained. 1 Introduction Uniform translations of intuitionistic into linear logic, with their plethoric use of exponentials, are bound to give only `universal linearity information' about proofs. This paper aims at displaying the structure of `specific linearity information ' hidden in a given derivation. How can we apply this to intuitionistic proofs? We have to build a translation into linear logic such that reductions of the intuitionistic proof can be simulated by reductions of its linear image. A necessary condition for this to hold, is that the `skeleton' of the original proof is preserved by the translation. We call translations with this property `decorations '. Specifically, we construct a proofbyproof embedding of IL into LL (formulated as sequent calculi) such that: 1/ the skeleton of the original proof is preserve...
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.
On the unity of duality
 Special issue on “Classical Logic and Computation
, 2008
"... Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the operational semantics may become visible becaus ..."
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Cited by 12 (2 self)
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Most type systems are agnostic regarding the evaluation strategy for the underlying languages, with the value restriction for ML which is absent in Haskell as a notable exception. As type systems become more precise, however, detailed properties of the operational semantics may become visible because properties captured by the types may be sound under one strategy but not the other. For example, intersection types distinguish between callbyname and callbyvalue functions, because the subtyping law (A → B) ∩ (A → C) ≤ A → (B ∩ C) is unsound for the latter in the presence of effects. In this paper we develop a prooftheoretic framework for analyzing the interaction of types with evaluation order, based on the notion of polarity. Polarity was discovered through linear logic, but we propose a fresh origin in Dummett’s program of justifying the logical laws through alternative verificationist or pragmatist “meaningtheories”, which include a bias towards either introduction or elimination rules. We revisit Dummett’s analysis using the tools of MartinLöf’s judgmental method, and then show how to extend it to a unified polarized logic, with Girard’s “shift ” connectives acting as intermediaries. This logic safely combines intuitionistic and dual intuitionistic reasoning principles, while simultaneously admitting a focusing interpretation for the classical sequent calculus. Then, by applying the CurryHoward isomorphism to polarized logic, we obtain a single programming language in which evaluation order is reflected at the level of types. Different logical notions correspond directly to natural programming constructs, such as patternmatching, explicit substitutions, values and callbyvalue continuations. We give examples demonstrating the expressiveness of the language and type system, and prove a basic but modular type safety result. We conclude with a brief discussion of extensions to the language with additional effects and types, and sketch the sort of explanation this can provide for operationallysensitive typing phenomena. 1
About Translations of Classical Logic into Polarized Linear Logic
 In Proceedings of the eighteenth annual IEEE symposium on Logic In Computer Science
, 2003
"... We show that the decomposition of Intuitionistic Logic into Linear Logic along the equation A ! B = !A ( B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly we build a categorical model of classical logic (a Control Category) from a categ ..."
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We show that the decomposition of Intuitionistic Logic into Linear Logic along the equation A ! B = !A ( B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly we build a categorical model of classical logic (a Control Category) from a categorical model of Linear Logic by a construction similar to the coKleisli category. Secondly we analyse two standard ContinuationPassing Style (CPS) translations, the Plotkin and the Krivine's translations, which are shown to correspond to two embeddings of LLP into LL.
An isomorphism between a fragment of sequent calculus and an extension of natural deduction
"... ..."
A Focused Approach to Combining Logics
, 2010
"... We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicativeadditive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structura ..."
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Cited by 8 (5 self)
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We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicativeadditive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut elimination holds in such fragments. From cut elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classicallinear hybrid logics.