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26
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
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.
Operational domain theory and topology of a sequential language
 In Proceedings of the 20th Annual IEEE Symposium on Logic In Computer Science
, 2005
"... A number of authors have exported domaintheoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact ..."
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Cited by 11 (6 self)
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A number of authors have exported domaintheoretic techniques from denotational semantics to the operational study of contextual equivalence and order. We further develop this, and, moreover, we additionally export topological techniques. In particular, we work with an operational notion of compact set and show that total programs with values on certain types are uniformly continuous on compact sets of total elements. We apply this and other conclusions to prove the correctness of nontrivial programs that manipulate infinite data. What is interesting is that the development applies to sequential programming languages, in addition to languages with parallel features. 1
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
Computational interpretations of analysis via products of selection functions
 CIE 2010, INVITED TALK ON SPECIAL SESSION “PROOF THEORY AND COMPUTATION
, 2010
"... We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions. ..."
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Cited by 9 (8 self)
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We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions.
Realizability algebras: a program to well order R
, 2010
"... When transforming mathematical proofs into programs, the main problem is naturally due to the axioms: indeed, it has been a long time since we know how to transform a proof in pure (i.e. without axioms) intuitionistic logic, even at second order [2, 7, 4]. ..."
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Cited by 8 (4 self)
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When transforming mathematical proofs into programs, the main problem is naturally due to the axioms: indeed, it has been a long time since we know how to transform a proof in pure (i.e. without axioms) intuitionistic logic, even at second order [2, 7, 4].
The Peirce Translation and the Double Negation Shift
"... We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of classical minimal logic into minimal logic, which we refer to as the Peirce translation, and which we apply to interpret b ..."
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Cited by 5 (5 self)
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We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of classical minimal logic into minimal logic, which we refer to as the Peirce translation, and which we apply to interpret both a strengthening of the doublenegation shift and the axioms of countable and dependent choice, via infinite products of selection functions.
Continuous semantics for strong normalization
 In CiE’05, volume 3526 of LNCS
, 2005
"... Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of G"odel's system T, but also to various forms of bar recursion for ..."
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Cited by 4 (2 self)
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Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of G"odel's system T, but also to various forms of bar recursion for which strong normalization was hitherto unknown.
Realizability : a machine for analysis and set theory
, 2006
"... In this tutorial, we introduce the CurryHoward (proofprogram) correspondence which is usually restricted to intuitionistic logic. We explain how to extend this correspondence to the whole of mathematics and we build a simple suitable machine for this. ..."
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Cited by 4 (4 self)
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In this tutorial, we introduce the CurryHoward (proofprogram) correspondence which is usually restricted to intuitionistic logic. We explain how to extend this correspondence to the whole of mathematics and we build a simple suitable machine for this.