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Operational Aspects of Untyped Normalization by Evaluation
, 2003
"... A purely syntactic and untyped variant of Normalization by Evaluation for the λcalculus... ..."
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A purely syntactic and untyped variant of Normalization by Evaluation for the λcalculus...
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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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
Algebraic proofs of cut elimination
 Journal of Logic Algebraic Programming
"... Algebraic proofs of the cutelimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the doublenegation translation is also discussed: if ϕ ..."
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Algebraic proofs of the cutelimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the doublenegation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ) nf is provable in minimal logic, where θ nf denotes the negationnormal form of θ. The translation is used to show that cutelimination theorems for classical logic can be viewed as special cases of the cutelimination theorems for intuitionistic logic. 1
From Proof Normalization to Compiler Generation and TypeDirected ChangeofRepresentation
, 1997
"... The main part of this thesis is a synthesis of considerations from Type Theory, Mathematical Logic/Proof Theory, and (Denotational) Semantics to perform various automatic program transformations ranging from normalization over currying and coercioninsertion to compiler derivation. At the core of o ..."
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The main part of this thesis is a synthesis of considerations from Type Theory, Mathematical Logic/Proof Theory, and (Denotational) Semantics to perform various automatic program transformations ranging from normalization over currying and coercioninsertion to compiler derivation. At the core of our technique we have what has been described as "An Inverse of the Evaluation Functional for Typed calculus" [7]. It is essentially typedirected jexpansion followed by fireduction on certain terms. Quite independently of [7], jexpansion has been studied for its use in Partial Evaluation, where among other things it has been used to obtain a onepass CPStransformer [20]. It is some of the consequences of this coincidence [19] that are described in the following. Our approach will be purely syntactical and it is hoped that it marks a simplification on earlier treatments of the materiel. We have tried presenting the materiel based purely on the standard reduction properties for the simpl...
Realizability interpretation of proofs in constructive analysis
, 2006
"... We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language – a version of Gödel’s T – evaluation is ..."
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We prove constructively (in the style of Bishop) that every monotone continuous function with a uniform modulus of increase has a continuous inverse. The proof is formalized, and a realizing term extracted. It turns out that even in the logical term language – a version of Gödel’s T – evaluation is reasonably efficient.
Program development by proof transformation
"... We begin by reviewing the natural deduction rules for the!^8fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. ..."
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We begin by reviewing the natural deduction rules for the!^8fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. We also consider two alternative methods to deal with the strong or constructive existential quantifier 9\Lambda. Finally we discuss the wellknown notion of an extracted program of a derivation involving 9\Lambda, in order to set up a relation between the two alternatives. Section 2 deals with the computational content of classical proofs. As is wellknown a proof of a 89theorem with a quantifierfree kernel where 9 is viewed as defined by:8: can be used as a program. We describe a "direct method " to use such a proof as a program, and compare it with Harvey Friedman's Atranslation [3] followed by program extraction from the resulting constructive proof. It is shown that both algorithms coincide. In section 3 Goad's method of pruning of proof trees is introduced. It is shown how a proof can be simplified after addition of some further assumptions. In a first step some subproofs are replaced by different ones using the additional assumptions. In a second step parts of the proof tree are pruned, i.e. cut out. Note that the first step involves searching for new proofs using the new assumptions of formulas in the proof tree. Hence we also have to discuss proof search in minimal logic. Finally section 4 treats an example already considered by Goad in his thesis [5], the binpacking problem. The main difference to Goad's work is that he used a logic with the strong existential quantifier, whereas we work within the!8fragment. This example is particularly wellsuited to demonstrate that the pruning method can be applied to adapt programs to particular situations, and moreover that pruning can change the functions computed by programs. In this sense this method is essentially different from program development by program transformation. We would like to thank Michael Bopp and KarlHeinz Niggl for their help in preparing these notes.
Tait in one big step
 In MSFP 2006
, 2006
"... We present a Taitstyle proof to show that a simple functional normaliser for a combinatory version of System T terminates. The main interest in our construction is methodological, it is an alternative to the usual smallstep operational semantics on the one side and normalisation by evaluation on t ..."
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We present a Taitstyle proof to show that a simple functional normaliser for a combinatory version of System T terminates. The main interest in our construction is methodological, it is an alternative to the usual smallstep operational semantics on the one side and normalisation by evaluation on the other. Our work is motivated by our goal to verify implementations of Type Theory such as Epigram. Keywords: Normalisation,Strong Computability 1.
Extraction in Coq: An overview
 In CiE ’08, volume 5028 of Lecture
"... Abstract. The extraction mechanism of Coq allows one to transform Coq proofs and functions into functional programs. We illustrate the behavior of this tool by reviewing several variants of Coq de nitions for Euclidean division, as well as some more advanced examples. We then continue with a more ge ..."
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Abstract. The extraction mechanism of Coq allows one to transform Coq proofs and functions into functional programs. We illustrate the behavior of this tool by reviewing several variants of Coq de nitions for Euclidean division, as well as some more advanced examples. We then continue with a more general description of this tool: key features, main examples, strengths, limitations and perspectives. hal00338973, version 1 14 Nov 2008 1
An Arithmetic for NonSizeIncreasing PolynomialTime Computation
"... An arithmetical system is presented with the property that from every proof a realizing term can be extracted that is definable in a certain affine linear typed variant of Gödel's T and therefore defines a nonsizeincreasing polynomial time computable function. ..."
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An arithmetical system is presented with the property that from every proof a realizing term can be extracted that is definable in a certain affine linear typed variant of Gödel's T and therefore defines a nonsizeincreasing polynomial time computable function.