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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...
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.
On universes in type theory
 191 – 204
, 1998
"... The notion of a universe of types was introduced into constructive type theory by MartinLöf (1975). According to the propositionsastypes principle inherent in ..."
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Cited by 32 (8 self)
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The notion of a universe of types was introduced into constructive type theory by MartinLöf (1975). According to the propositionsastypes principle inherent in
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 ...
Modified Bar Recursion and Classical Dependent Choice
 In Logic Colloquium 2001
"... We introduce a variant of Spector's bar recursion in nite types to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of 1 formulas in classical analysis. We also give a bar recursive denition of the fan functional and ..."
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Cited by 26 (16 self)
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We introduce a variant of Spector's bar recursion in nite types to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of 1 formulas in classical analysis. We also give a bar recursive denition of the fan functional and study the relationship of our variant of bar recursion with others. x1.
On the computational content of the Krasnoselski and Ishikawa fixed point theorems
, 2000
"... This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general ..."
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Cited by 11 (10 self)
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This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general socalled KrasnoselskiMann iterations. These iterations converge to fixed points of f under certain compactness conditions. But, as we show, already for uniformly convex spaces in general no bound on the rate of convergence can be computed uniformly in f . This is related to the nonuniqueness of fixed points. However, the iterations yield even without any compactness assumption and for arbitrary normed spaces approximate fixed points of arbitrary quality for bounded C (asymptotic regularity, Ishikawa 1976). We apply proof theoretic techniques (developed in previous papers of us) to noneffective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...
The structure of nuprl’s type theory
, 1997
"... on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html) ..."
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Cited by 9 (3 self)
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on the World Wide Web (\the Web") (www.cs.cornell.edu/Info/NuPrl/nuprl.html)
Uniform Heyting arithmetic
 Annals Pure Applied Logic
, 2005
"... Abstract. We present an extension of Heyting Arithmetic in finite types called Uniform Heyting Arithmetic (HA u) that allows for the extraction of optimized programs from constructive and classical proofs. The system HA u has two sorts of firstorder quantifiers: ordinary quantifiers governed by the ..."
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Cited by 9 (0 self)
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Abstract. We present an extension of Heyting Arithmetic in finite types called Uniform Heyting Arithmetic (HA u) that allows for the extraction of optimized programs from constructive and classical proofs. The system HA u has two sorts of firstorder quantifiers: ordinary quantifiers governed by the usual rules, and uniform quantifiers subject to stronger variable conditions expressing roughly that the quantified object is not computationally used in the proof. We combine a Kripkestyle Friedman/Dragalin translation which is inspired by work of Coquand and Hofmann and a variant of the refined Atranslation due to Buchholz, Schwichtenberg and the author to extract programs from a rather large class of classical firstorder proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program. §1. Introduction. According to the BrouwerHeytingKolmogorov interpretation of constructive logic a proof is a construction providing evidence for the proven formula [20]. Viewing this interpretation from a dataoriented perspective one arrives at the socalled proofsasprograms paradigm associating a constructive proof with a program ‘realizing ’ the proven formula. This paradigm has been
The Greatest Common Divisor: A Case Study for Program Extraction From Classical Proofs
 Types for Proofs and Programs. International Workshop TYPES '95
, 1996
"... > 1 ; : : : ; v 6 of them; these will be formulated as we need them. Theorem. 8a 1 ; a 2 (0 ! a 2 ! 9k 1 ; k 2 (abs(k 1 a 1 \Gamma k 2 a 2 )ja 1 abs(k 1 a 1 \Gamma k 2 a 2 )ja 2 0 ! abs(k 1 a 1 \Gamma k 2 a 2 ))): Proof. Let a 1 ; a 2 be given and assume 0 ! a 2 . The ideal (a 1 ; a 2 ) genera ..."
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Cited by 9 (2 self)
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> 1 ; : : : ; v 6 of them; these will be formulated as we need them. Theorem. 8a 1 ; a 2 (0 ! a 2 ! 9k 1 ; k 2 (abs(k 1 a 1 \Gamma k 2 a 2 )ja 1 abs(k 1 a 1 \Gamma k 2 a 2 )ja 2 0 ! abs(k 1 a 1 \Gamma k 2 a 2 ))): Proof. Let a 1 ; a 2 be given and assume 0 ! a 2 . The ideal (a 1 ; a 2 ) generated from a 1 ; a 2 has a least positive element<F27.