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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...
NonDeterministic Extensions of Untyped λcalculus
 INFO. AND COMP
, 1995
"... The main concern of this paper is the study of the interplay between functionality and non determinism. Indeed the first question we ask is whether the analysis of parallelism in terms of sequentiality and non determinism, which is usual in the algebraic treatment of concurrency, remains correct in ..."
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Cited by 6 (0 self)
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The main concern of this paper is the study of the interplay between functionality and non determinism. Indeed the first question we ask is whether the analysis of parallelism in terms of sequentiality and non determinism, which is usual in the algebraic treatment of concurrency, remains correct in presence of functional application and abstraction. We identify non determinism in the setting of λcalculus with the absence of the ChurchRosser property plus the inconsistency of the equational theory obtained by the symmetric closure of the reduction relation. We argue in favour of a distinction between non determinism and parallelism, due to the conjunctive nature of the former in contrast to the disjunctive character of the latter. This is the basis of our analysis of the operational and denotational semantics of non deterministiccalculus, which is the classical calculus plus a choice operator, and of our election of bounded indeterminacy as the semantical counterpart of conjunctive non determinism. This leads to operational semantics based on...
Origin Tracking for HigherOrder Term Rewriting Systems
 Proceedings of the International Workshop on HigherOrder Algebra, Logic and Term Rewriting HOA93
, 1993
"... Origin Tracking is a technique which, in the framework of firstorder term rewriting systems, establishes relations between each subterm t of a normal form and a set of subterms, the origins of t, in the initial term. Origin tracking is based on the notion of residuals. It has been used successfully ..."
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Cited by 3 (2 self)
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Origin Tracking is a technique which, in the framework of firstorder term rewriting systems, establishes relations between each subterm t of a normal form and a set of subterms, the origins of t, in the initial term. Origin tracking is based on the notion of residuals. It has been used successfully for the generation of error handlers and debuggers from algebraic specifications of programming languages. Recent experiments with the use of higherorder algebraic specifications for the definition of programming languages revealed a need to extend origin tracking for higherorder term rewriting systems. In this paper, we discuss how origin information can be maintained for fij reductions and expansions, during higherorder rewriting. We give a definition of higherorder origin tracking. The suitability of this definition is illustrated with a small, existing specification. AMS Subject Classification (1991): 68N20, 68Q55, 68Q65. CR Subject Classification (1991): D.2.5, D.2.6, D.3.4, F.3...
Structures for Lazy Semantics
 In Programming Concepts and Methods, PROCOMET'98. Chapman
, 1997
"... The paper explores different approaches for modeling the lazy calculus, which is a paradigmatic language for studying the operational behaviour of programming languages, like Haskell, using a callbyname and lazy evaluation mechanism. Two models for lazy calculus in the coherence spaces setting a ..."
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Cited by 1 (0 self)
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The paper explores different approaches for modeling the lazy calculus, which is a paradigmatic language for studying the operational behaviour of programming languages, like Haskell, using a callbyname and lazy evaluation mechanism. Two models for lazy calculus in the coherence spaces setting are built. They give a new insight in the behaviour of the language since their local structures are different from the one of all existing models in the literature. In order to compare different models, a class of models for lazy calculus is defined, namely, the lazy regular models class. All the models adequate for the lazy calculus studied in the literature belong to this class. Moreover, all the lazy regular model share important properties, like the approximation property, which is a key tool for studying their local structure. 1
Separating Developments in λCalculus
, 1996
"... We introduce a proof technique in calculus which can facilitate inductive reasoning on terms by separating certain fidevelopments from other fireductions. We present proofs based on this technique for several fundamental theorems in calculus such as the ChurchRosser theorem, the standardisatio ..."
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We introduce a proof technique in calculus which can facilitate inductive reasoning on terms by separating certain fidevelopments from other fireductions. We present proofs based on this technique for several fundamental theorems in calculus such as the ChurchRosser theorem, the standardisation theorem, the conservation theorem and the normalisation theorem. The appealing features of these proofs lie in their inductive styles and perspicuities. 1. Introduction Proofs based on structural inductions have certain desirable features. They usually enhance comprehensibility, yield more on the meaning of the proven theorems, and can be formalised relatively easily. Unfortunately, many theorems in calculus cannot be proven via structural induction on terms. This is mainly due to the fact that fireduction is not compositional, namely, a fireduction sequence from Mfx := Ng usually cannot be viewed as the composition of some fireduction sequences from M and N since new firedexes may...
Development Separation in LambdaCalculus
 WOLLIC 2005
, 2005
"... We present a proof technique in *calculus that can facilitate inductive reasoning on *terms by separating certain fidevelopments from other fireductions. We give proofs based on this technique for several fundamental theorems in *calculus such as the ChurchRosser theorem, the standardization ..."
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We present a proof technique in *calculus that can facilitate inductive reasoning on *terms by separating certain fidevelopments from other fireductions. We give proofs based on this technique for several fundamental theorems in *calculus such as the ChurchRosser theorem, the standardization theorem, the conservation theorem and the normalization theorem. The appealing features of these proofs lie in their inductive styles and perspicuities.