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22
A callbyname lambdacalculus machine
 Higher Order and Symbolic Computation
"... We present, in this paper, a particularly simple lazy machine which runs programs written in λcalculus. It was introduced by the present writer more than twenty years ago. It has been, since, used and implemented ..."
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We present, in this paper, a particularly simple lazy machine which runs programs written in λcalculus. It was introduced by the present writer more than twenty years ago. It has been, since, used and implemented
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 9 (1 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
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].
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.
Disjunctive Normal Forms and Local Exceptions
, 2003
"... All classical λterms typable with disjunctive normal forms are shown to share a common computational behavior: they implement a local exception handling mechanism whose exact workings depend on the tautology. Equivalent and more efficient control combinators are described through a specializ ..."
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All classical λterms typable with disjunctive normal forms are shown to share a common computational behavior: they implement a local exception handling mechanism whose exact workings depend on the tautology. Equivalent and more efficient control combinators are described through a specialized sequent calculus and shown to be correct.
Getting results from programs extracted from classical proofs.
, 2002
"... We present a new method to extract from a classical proof of ∀x(I[x] → ∃y(O[y] ∧ S[x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs ..."
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We present a new method to extract from a classical proof of ∀x(I[x] → ∃y(O[y] ∧ S[x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs and this is verified on a non trivial example: a proof of Dickson’s lemma. 1 Introduction. Since Griffin and Felleisen [7, 8], we know a relation between classical proofs and programs. However, it is not true that from a classical proof of the existence of an object you can compute this object. This would clearly be a contradiction with the existence of provably total but non computable functions (such as a function saying
A program for the Continuum Hypothesis
"... Classical realizability is a method to obtain programs from mathematical proofs in Analysis and set theory. In this talk, we limit ourselves to Analysis, a theory stated in second order logic. Since the discovery, by T. Griffin in 1990, that a control instruction was typed ..."
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Classical realizability is a method to obtain programs from mathematical proofs in Analysis and set theory. In this talk, we limit ourselves to Analysis, a theory stated in second order logic. Since the discovery, by T. Griffin in 1990, that a control instruction was typed
Realizability for programming languages.
"... We present a toy functional programming language inspired by our work on the PML language [22] together with a criterion ensuring safety and the fact that non termination can only occur via recursive programs. To prove this theorem, we use realizability techniques and a semantical notion of types. I ..."
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We present a toy functional programming language inspired by our work on the PML language [22] together with a criterion ensuring safety and the fact that non termination can only occur via recursive programs. To prove this theorem, we use realizability techniques and a semantical notion of types. Important features of PML like polymorphism, proofchecking, termination criterion for recursive function,... will be covered by forthcoming articles reusing the formalism introduced here. The paper contains the source of the algorithm (some boring parts like the parser are omitted) and the complete source are available from the author webpage.
Under consideration for publication in Math. Struct. in Comp. Science Krivine’s Classical Realizability from a Categorical Perspective
, 2010
"... introduced his notion of Classical Realizability for classical second order logic and ZermeloFraenkel set theory. Moreover, in more recent work (Krivine 2008) he has considered forcing constructions on top of it with the ultimate aim of providing a realizability interpretation for the axiom of choi ..."
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introduced his notion of Classical Realizability for classical second order logic and ZermeloFraenkel set theory. Moreover, in more recent work (Krivine 2008) he has considered forcing constructions on top of it with the ultimate aim of providing a realizability interpretation for the axiom of choice. The aim of this paper is to show how Krivine’s classical realizability can be understood as an instance of the categorical approach to realizability as started by Hyland in (Hyland 1982) and described in detail in (van Oosten 2008). Moreover, we will give an intuitive explanation of the iteration of realizability as can be found in (Krivine 2008). 1.