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Strong Normalization in a NonDeterministic Typed LambdaCalculus
, 1994
"... In a previous paper [4], we introduced a nondeterministic λcalculus (λLK) whose type system corresponds exactly to Gentzen's cutfree LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for t ..."
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In a previous paper [4], we introduced a nondeterministic &lambda;calculus (&lambda;LK) whose type system corresponds exactly to Gentzen's cutfree LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, it is not possible to define an effective notion of reduction. In the present paper, we address this problem. We consider a weak version of the implicative fragment of &lambda;LK, and we define for it a relation of reduction that models, at the level of the terms, the appropriate prooftheoretic notion of proof reduction. This reduction relation satisøes several properties of interest, among others, the property of strong normalization. We prove this last result by using a reducibility argument à la Tait.
Niels Jakob Rehof Morten Heine Srensen
 THEORETICAL ASPECTS OF COMPUTER SOFTWARE
, 1994
"... . By restriction of Felleisen's control operator F we obtain an operator \Delta and a fully compatible, ChurchRosser control calculus \Delta enjoying a number of desirable properties. It is shown that \Delta contains a strongly normalizing typed subcalculus with a reduction corresponding ..."
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. By restriction of Felleisen's control operator F we obtain an operator \Delta and a fully compatible, ChurchRosser control calculus \Delta enjoying a number of desirable properties. It is shown that \Delta contains a strongly normalizing typed subcalculus with a reduction corresponding closely to systems of proof normalization for classical logic. The calculus is more than strong enough to express a callbyname catch=throw programming paradigm. 1 Background and motivation The first subsection describes previous work in the CurryHoward Isomorphism. The second subsection describes our contribution: a typed calculus with a number of desirable properties, not all shared by the systems mentioned in the first subsection. The CurryHoward Isomorphism and classical logic. The socalled CurryHoward Isomorphism states a correspondence between typed calculi and systems of formal logic. 2 At the heart of the isomorphism is the perception of proofs as functions, as formalized ...
Strong Normalization in a NonDeterministic Typed LambdaCalculus
"... Abstract. In a previous paper [4], we introduced a nondeterministic λcalculus (λLK) whose type system corresponds exactly to Gentzen’s cutfree LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, it is ..."
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Abstract. In a previous paper [4], we introduced a nondeterministic λcalculus (λLK) whose type system corresponds exactly to Gentzen’s cutfree LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, it is not possible to define an effective notion of reduction. In the present paper, we address this problem. We consider a weak version of the implicative fragment of λLK, and we define for it a relation of reduction that models, at the level of the terms, the appropriate prooftheoretic notion of proof reduction. This reduction relation satisfies several properties of interest, among others, the property of strong normalization. We prove this last result by using a reducibility argument a ̀ la Tait. 1