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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...
Concurrent Games and Full Completeness
, 1998
"... A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is ..."
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Cited by 50 (17 self)
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A new concurrent form of game semantics is introduced. This overcomes the problems which had arisen with previous, sequential forms of game semantics in modelling Linear Logic. It also admits an elegant and robust formalization. A Full Completeness Theorem for MultiplicativeAdditive Linear Logic is proved for this semantics. 1 Introduction This paper contains two main contributions: ffl the introduction of a new form of game semantics, which we call concurrent games. ffl a proof of full completeness of this semantics for MultiplicativeAdditive Linear Logic. We explain the significance of each of these in turn. Concurrent games Traditional forms of game semantics which have appeared in logic and computer science have been sequential in format: a play of the game is formalized as a sequence of moves. The key feature of this sequential format is the existence of a global schedule (or polarization) : in each (finite) position, it is (exactly) one player's turn to move 1 . This seq...
Focusing and Polarization in Linear, Intuitionistic, and Classical Logics
, 2009
"... A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitioni ..."
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Cited by 43 (18 self)
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A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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Cited by 38 (11 self)
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
Strong Normalisation of CutElimination in Classical Logic
, 2000
"... In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos ..."
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Cited by 35 (4 self)
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In this paper we present a strongly normalising cutelimination procedure for classical logic. This procedure adapts Gentzen's standard cutreductions, but is less restrictive than previous strongly normalising cutelimination procedures. In comparison, for example, with works by Dragalin and Danos et al., our procedure requires no special annotations on formulae and allows cutrules to pass over other cutrules. In order to adapt the notion of symmetric reducibility candidates for proving the strong normalisation property, we introduce a novel term assignment for sequent proofs of classical logic and formalise cutreductions as term rewriting rules.
A Semantic analysis of control
, 1998
"... This thesis examines the use of denotational semantics to reason about control flow in sequential, basically functional languages. It extends recent work in game semantics, in which programs are interpreted as strategies for computation by interaction with an environment. Abramsky has suggested that ..."
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Cited by 32 (5 self)
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This thesis examines the use of denotational semantics to reason about control flow in sequential, basically functional languages. It extends recent work in game semantics, in which programs are interpreted as strategies for computation by interaction with an environment. Abramsky has suggested that an intensional hierarchy of computational features such as state, and their fully abstract models, can be captured as violations of the constraints on strategies in the basic functional model. Nonlocal control flow is shown to fit into this framework as the violation of strong and weak ‘bracketing ’ conditions, related to linear behaviour. The language µPCF (Parigot’s λµ with constants and recursion) is adopted as a simple basis for highertype, sequential computation with access to the flow of control. A simple operational semantics for both callbyname and callbyvalue evaluation is described. It is shown that dropping the bracketing condition on games models of PCF yields fully abstract models of µPCF.
Focusing and polarization in intuitionistic logic
 CSL 2007: Computer Science Logic, volume 4646 of LNCS
, 2007
"... dale.miller at inria.fr Abstract. A focused proof system provides a normal form to cutfree proofs that structures the application of invertible and noninvertible inference rules. The focused proof system of Andreoli for linear logic has been applied to both the proof search and the proof normaliza ..."
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Cited by 32 (14 self)
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dale.miller at inria.fr Abstract. A focused proof system provides a normal form to cutfree proofs that structures the application of invertible and noninvertible inference rules. The focused proof system of Andreoli for linear logic has been applied to both the proof search and the proof normalization approaches to computation. Various proof systems in literature exhibit characteristics of focusing to one degree or another. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems. 1
A semantic view of classical proofs  typetheoretic, categorical, and denotational characterizations (Extended Abstract)
 IN PROCEEDINGS OF LICS '96
, 1996
"... Classical logic is one of the best examples of a mathematical theory that is truly useful to computer science. Hardware and software engineers apply the theory routinely. Yet from a foundational standpoint, there are aspects of classical logic that are problematic. Unlike intuitionistic logic, class ..."
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Cited by 30 (2 self)
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Classical logic is one of the best examples of a mathematical theory that is truly useful to computer science. Hardware and software engineers apply the theory routinely. Yet from a foundational standpoint, there are aspects of classical logic that are problematic. Unlike intuitionistic logic, classical logic is often held to be nonconstructive, and so, is said to admit no proof semantics. To draw an analogy in the proofsas programs paradigm, it is as if we understand well the theory of manipulation between equivalent specifications (which we do), but have comparatively little foundational insight of the process of transforming one program to another that implements the same specification. This extended abstract outlines a semantic theory of classical proofs based on a variant of Parigot's λµcalculus [24], but presented here as a type theory. After reviewing the conceptual problems in the area and the potential benefits of such a theory, we sketch the key steps of our approach in ...
Linear Logic & Elementary Time
 Information and Computation
, 2001
"... Introduction Think of elementary linear logic as an idealized functional programming language with a severe typing mechanism. Definition by recursion is, of course, forbidden, but some sort of iteration still is possible and the purpose of this paper is to show that enough computing power remains s ..."
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Cited by 28 (0 self)
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Introduction Think of elementary linear logic as an idealized functional programming language with a severe typing mechanism. Definition by recursion is, of course, forbidden, but some sort of iteration still is possible and the purpose of this paper is to show that enough computing power remains so that elementary recursive functions can be implemented. Actually, the whole paper can be considered an exercise in programming elegantly with a rather desolate language. To zero in on an interesting class of functions, one usually tries to weaken in the given logic whatever corresponds to induction or iteration. Here we're following a di#erent strand, rather specific to the linear logic decomposition of the implication as !A#B, by fiddling with the rules handling `!'. The standard rules are enough to embed the full power of intuitionistic computations. So the game is to find a sensible way to make them harder to use than in full linear logic. There ar
Naming proofs in classical propositional logic
 IN PAWE̷L URZYCZYN, EDITOR, TYPED LAMBDA CALCULI AND APPLICATIONS, TLCA 2005, VOLUME 3461 OF LECTURE
"... We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentiali ..."
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Cited by 24 (8 self)
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We present a theory of proof denotations in classical propositional logic. The abstract definition is in terms of a semiring of weights, and two concrete instances are explored. With the Boolean semiring we get a theory of classical proof nets, with a geometric correctness criterion, a sequentialization theorem, and a strongly normalizing cutelimination procedure. This gives us a “Boolean ” category, which is not a poset. With the semiring of natural numbers, we obtain a sound semantics for classical logic, in which fewer proofs are identified. Though a “real” sequentialization theorem is missing, these proof nets have a grip on complexity issues. In both cases the cut elimination procedure is closely related to its equivalent in the calculus of structures.