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Combinatory Reduction Systems: introduction and survey
 THEORETICAL COMPUTER SCIENCE
, 1993
"... Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simpl ..."
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Cited by 84 (9 self)
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Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and leftlinear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power, and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the wellknown confluence proof for λcalculus by Tait and MartinLof. There is a wellknown connection between the para...
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 this reason, i ..."
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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 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.
Denotations for classical proofs  Preliminary results
, 1992
"... This paper addresses the problem of extending the formulaeastypes principle to classical logic. More precisely, we introduce a typed lambdacalculus (LK ! ) whose inhabited types are exactly the implicative tautologies of classical logic and whose type assignment system is a classical sequent c ..."
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This paper addresses the problem of extending the formulaeastypes principle to classical logic. More precisely, we introduce a typed lambdacalculus (LK ! ) whose inhabited types are exactly the implicative tautologies of classical logic and whose type assignment system is a classical sequent calculus. Intuitively, the terms of LK ! correspond to constructs that are highly nondeterministic. This intuition is made much more precise by providing a simple model where the terms of LK ! are interpreted as nonempty sets of (interpretations of) untyped lambdaterms. We also consider the system (LK ! + cut) and investigate the relation existing between cut elimination and reduction. Finally, we show how to extend our system in order to take conjunction, disjunction and negation into account. 1 Introduction In: Logical Foundations of Computer Science Tver'92, A. Nerode, M. Taitslin (Eds.), Lecture Notes in Computer Science, Vol. 620, SpringerVerlag (1992), pp. 105116. In thi...