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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...
Correspondence between Operational and Denotational Semantics
 Handbook of Logic in Computer Science
, 1995
"... This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational ..."
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Cited by 23 (0 self)
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This course introduces the operational and denotational semantics of PCF and examines the relationship between the two. Topics: Syntax and operational semantics of PCF, Activity Lemma, undefinability of parallel or; Context Lemma (first principles proof) and proof by logical relations Denotational semantics of PCF induced by an interpretation; (standard) Scott model, adequacy, weak adequacy and its proof (by a computability predicate) Domain Theory up to SFP and Scott domains; non full abstraction of the standard model, definability of compact elements and full abstraction for PCFP (PCF + parallel or), properties of orderextensional (continuous) models of PCF, Milner's model and Mulmuley's construction (excluding proofs) Additional topics (time permitting): results on pure simplytyped lambda calculus, Friedman 's Completeness Theorem, minimal model, logical relations and definability, undecidability of lambda definability (excluding proof), dIdomains and stable functions Homepa...
A Conflict Between CallbyNeed Computation and Parallelism
, 1994
"... . In functional language implementation, there is a folklore belief that there is a conflict between implementing callbyneed semantics and parallel evaluation. In this note we illustrate this by proving that reduction algorithms of a certain general and commonly used form which give callbyneed ..."
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Cited by 15 (0 self)
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. In functional language implementation, there is a folklore belief that there is a conflict between implementing callbyneed semantics and parallel evaluation. In this note we illustrate this by proving that reduction algorithms of a certain general and commonly used form which give callbyneed semantics offer very little parallelism. The analysis of lazy patternmatching which leads to the above result also suggests an efficient sequential algorithm for the evaluation of a class functional programs satisfying certain constraints, an algorithm which respects the mathematical semantics of the program considered as a term rewrite system. 1 Introduction Huet and L'evy [Huet and L'evy, 1979, Huet and L'evy, 1991] have considered the problem of call by need computation of normal forms in orthogonal term rewrite systems. Call by need here means that no redex is ever reduced unless it must be reduced in order to compute the normal form. In general, such a redex cannot be effectiv...
Descendants and Origins in Term Rewriting
 Information and Computation
, 1999
"... In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first order term rewriting and calculus, their finitary as well as their infinitary variants. A recurrent theme is the Parallel Moves Lemma. Next to the ..."
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Cited by 8 (1 self)
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In this paper we treat various aspects of a notion that is central in term rewriting, namely that of descendants or residuals. We address both first order term rewriting and calculus, their finitary as well as their infinitary variants. A recurrent theme is the Parallel Moves Lemma. Next to the classical notion of descendant, we introduce an extended version, known as `origin tracking'. Origin tracking has many applications. Here it is employed to give new proofs of three classical theorems: the Genericity Lemma in calculus, the theorem of Huet and L'evy on needed reductions in first order term rewriting, and Berry's Sequentiality Theorem in (infinitary) calculus. Note: This article is based on a lecture given by Jan Willem Klop at RTA '98 held in Tsukuba, Japan. Contents 1 Introduction 2 2 Early views on descendants 3 3 Preliminaries 5 3.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Reduction . . . . . . . . . . . . . . . . . . . . . ....
Upper Bounds for Standardizations and an Application
 The Journal of Symbolic Logic
, 1996
"... We first present a new proof for the standardization theorem, a fundamental theorem in calculus. Since our proof is largely built upon structural induction on lambda terms, we can extract some bounds for the number of fireduction steps in the standard fireduction sequences obtained from transfor ..."
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Cited by 7 (1 self)
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We first present a new proof for the standardization theorem, a fundamental theorem in calculus. Since our proof is largely built upon structural induction on lambda terms, we can extract some bounds for the number of fireduction steps in the standard fireduction sequences obtained from transforming any given fireduction sequences. This result sharpens the standardization theorem and establishes a link between lazy and eager evaluation orders in the context of computational complexity. As an application, we establish a superexponential bound for the number of fireduction steps in fireduction sequences from any given simply typed terms. 1 Introduction The standardization theorem of Curry and Feys [CF58] is a very useful result, stating that if u reduces to v for terms u and v, then there is a standard fireduction from u to v. Using this theorem, we can readily prove the normalization theorem, i.e., a term has a normal form if and only if the leftmost fireduction sequence f...
Free Sequentiality in Orthogonal OrderSorted Rewriting Systems with Constructors
 In Proc. 11th Int. Conf. on Automated Deduction
, 1992
"... We introduce the notions of sequentiality and strong sequentiality in ordersorted rewriting systems, both closely related to the subsort order and to the form of declarations of the signature. We define free sequentiality for the class of orthogonal systems with constructors, a notion which does not ..."
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Cited by 1 (0 self)
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We introduce the notions of sequentiality and strong sequentiality in ordersorted rewriting systems, both closely related to the subsort order and to the form of declarations of the signature. We define free sequentiality for the class of orthogonal systems with constructors, a notion which does not impose conditions over the signature. We provide an effective decision procedure for free sequentiality that gives at the same time a simple construction of a nondeterministic pattern matching tree. These trees describe how the refinement of sorts and structures has to be done along the reduction sequence in such a way that wasteful computations are avoided.
A new proposal of Concurrent Process Calculus
"... In this paper, we present a new calculus to model concurrent systems, the Parallel LabelSelectivecalculus. This calculus integrates the (functional) expressiveness of thecalculus in a unified framework with some powerful features for expressing communication actions and supporting the independenc ..."
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In this paper, we present a new calculus to model concurrent systems, the Parallel LabelSelectivecalculus. This calculus integrates the (functional) expressiveness of thecalculus in a unified framework with some powerful features for expressing communication actions and supporting the independence of processes which can be a main source of improvement when performing parallel computations.
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...