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36
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...
Full Abstraction for Functional Languages with Control
 In Proceedings, Twelfth Annual IEEE Symposium on Logic in Computer Science
, 1997
"... This paper considers the consequences of relaxing the bracketing condition on `dialogue games', showing that this leads to a category of games which can be `factorized' into a wellbracketed substructure, and a set of classically typed morphisms. These are shown to be sound denotations for control o ..."
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Cited by 62 (5 self)
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This paper considers the consequences of relaxing the bracketing condition on `dialogue games', showing that this leads to a category of games which can be `factorized' into a wellbracketed substructure, and a set of classically typed morphisms. These are shown to be sound denotations for control operators, allowing the factorization to be used to extend the definability result for PCF to one for PCF with control operators at atomic types. Thus we define a fully abstract and effectively presentable model of a functional language with nonlocal control as part of a modular approach to modelling nonfunctional features using games. 1.
On the computational content of the axiom of choice
 The Journal of Symbolic Logic
, 1998
"... We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the rea ..."
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Cited by 34 (1 self)
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We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.
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 ...
Dependent choices, ‘quote’ and the clock
 Th. Comp. Sc
, 2003
"... When using the CurryHoward correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem: to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an inst ..."
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Cited by 28 (10 self)
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When using the CurryHoward correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem: to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an instruction of our programming language. This problem
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.
Travelling on Designs  Ludics Dynamics
, 2002
"... Proofs in Ludics are represented by designs. Designs (desseins) can be seen as an intermediate syntax between sequent calculus and proof nets, carrying advantages from both approaches, especially w.r.t. cutelimination. To study interaction between designs and develop a geometrical intuition, we ..."
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Cited by 9 (4 self)
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Proofs in Ludics are represented by designs. Designs (desseins) can be seen as an intermediate syntax between sequent calculus and proof nets, carrying advantages from both approaches, especially w.r.t. cutelimination. To study interaction between designs and develop a geometrical intuition, we introduce an abstract machine which presents normalization as a token travelling along a net of designs. This allows a concrete approach, from which to carry on the study of issues such as: (i) which part of a design can be recognized interactively; (ii) how to reconstruct a design from the traces of its interactions in different tests.
A Classical Linear λcalculus
, 1997
"... This paper proposes and studies a typed λcalculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natu ..."
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Cited by 8 (0 self)
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This paper proposes and studies a typed λcalculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formulation of classical linear logic. This formulation is compared in detail to the sequent calculus formulation. In an appendix I shall also demonstrate a somewhat hidden connexion with the paradigm of control operators for functional languages which gives a new computational interpretation of Parigot's techniques.
Programming interfaces and basic topology
 Annals of Pure and Applied Logic
, 2005
"... A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We ..."
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Cited by 7 (0 self)
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A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We present a category in which the objects —called interaction structures in the paper — serve as descriptions of services provided across such handshaken interfaces. The morphisms —called (general) simulations— model components that provide one such service, relying on another. The morphisms are relations between the underlying sets of the interaction structures. The proof that a relation is a simulation can serve (in principle) as an executable program, whose specification is that it provides the service described by its domain, given an implementation of the service described by its codomain.
A Limiting First Order Realizability Interpretation
"... Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics ..."
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Cited by 5 (0 self)
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Constructive Mathematics might be regarded as a fragment of classical mathematics in which any proof of an existence theorem is equipped with a computable function giving the solution of the theorem. Limit Computable Mathematics (LCM) considered in this note is a fragment of classical mathematics in which any proof of an existence theorem is equipped with a function computing the solution of the theorem in the limit.