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Extracting constructive content from classical logic via controllike reductions
 In Bezem and Groote [12
, 1993
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Proof Transformations in HigherOrder Logic
, 1987
"... We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, ..."
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We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, nonanalytic proofs are represented by natural deductions. A nondeterministic translation algorithm between expansion proofs and Hdeductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cutelimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higherorder system, but is proven for the firstorder fragment. We extend the translations to a nonanalytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a nonextensional equality is definable in our system of type theory. Next we extend analytic and nonanalytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a
A Strong Normalization Result for Classical Logic
 Annals of Pure and Applied Logic
, 1995
"... In this paper we give a strong normalization proof for a set of reduction rules for classical logic. These reductions, more general then the ones usually considered in literature, are inspired to the reductions of Felleisen's lambda calculus with continuations. 1 Introduction Recently, in the ..."
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In this paper we give a strong normalization proof for a set of reduction rules for classical logic. These reductions, more general then the ones usually considered in literature, are inspired to the reductions of Felleisen's lambda calculus with continuations. 1 Introduction Recently, in the logic and theoretical computer science community, there has been an ever growing interest in the computational features of classical logic. The problem on which research is beginning to focus now is not the theoretical possibility of having constructive content present in classical proofs, established in old and well known results, but the practical applicability of such results. It was Kreisel in [12] who first pinpointed the presence of constructive content in classical proofs by proving the equality of the sets of \Sigma 0 1 sentences provable respectively in intuitionistic and classical logic. Friedman in [7] showed how to get the computational content of a classical proof of a \Sigma 0 1 ...
A Constructive Valuation Semantics for Classical Logic
"... Abstract The paper presents a constructive interpretation for proofs in classical logic of \Sigma 01 sentences and for a witness extraction procedure based on Prawitz's reduction rules. ..."
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Abstract The paper presents a constructive interpretation for proofs in classical logic of \Sigma 01 sentences and for a witness extraction procedure based on Prawitz's reduction rules.