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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...
Equational term graph rewriting
 FUNDAMENTA INFORMATICAE
, 1996
"... We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is wellknown in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bis ..."
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Cited by 71 (8 self)
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We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is wellknown in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the µrule, and translations are given between term graphs and µexpressions. Using these, a proof system is given for µexpressions that is complete for the semantics given by infinite tree unwinding. Next, orthogonal term graph rewrite ...
Higherorder narrowing
 PROC. NINTH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE
, 1994
"... We introduce several approaches for solving higherorder equational problems by higherorder narrowing and give first completeness results. The results apply to higherorder functionallogic programming languages and to higherorder unification modulo a higherorder equational theory. We lift the ge ..."
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Cited by 20 (8 self)
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We introduce several approaches for solving higherorder equational problems by higherorder narrowing and give first completeness results. The results apply to higherorder functionallogic programming languages and to higherorder unification modulo a higherorder equational theory. We lift the general notion of firstorder narrowing to socalled higherorder patterns and argue that the full higherorder case is problematic. Integrating narrowing into unification, called lazy narrowing, can avoid these problems and can be adapted to the full higherorder case. For the secondorder case, we develop a version where the needed secondorder unification remains decidable. Finally we discuss a method that combines both approaches by using narrowing on higherorder patterns with full higherorder constraints.
Comparing Combinatory Reduction Systems and HigherOrder Rewrite Systems
, 1993
"... In this paper two formats of higherorder rewriting are compared: Combinatory Reduction Systems introduced by Klop [Klo80] and Higherorder Rewrite Systems defined by Nipkow [Nipa]. Although it always has been obvious that both formats are closely related to each other, up to now the exact relations ..."
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Cited by 18 (3 self)
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In this paper two formats of higherorder rewriting are compared: Combinatory Reduction Systems introduced by Klop [Klo80] and Higherorder Rewrite Systems defined by Nipkow [Nipa]. Although it always has been obvious that both formats are closely related to each other, up to now the exact relationship between them has not been clear. This was an unsatisfying situation since it meant that proofs for much related frameworks were given twice. We present two translations, one from Combinatory Reduction Systems into HigherOrder Rewrite Systems and one vice versa, based on a detailed comparison of both formats. Since the translations are very `neat' in the sense that the rewrite relation is preserved and (almost) reflected, we can conclude that as far as rewrite theory is concerned, Combinatory Reduction Systems and HigherOrder Rewrite Systems are equivalent, the only difference being that Combinatory Reduction Systems employ a more `lazy' evaluation strategy. Moreover, due to this result...
Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Cited by 13 (6 self)
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
Contextsensitive Conditional Expression Reduction Systems
 In Proc. of the International Workshop on Graph Rewriting and Computation, SEGRAGRA'95
, 1995
"... We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems ..."
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Cited by 12 (4 self)
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We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed calculi possibly enriched with patternmatching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbertstyle proof systems, Gentzenstyle sequentcalculi, rewrite systems with rule priorities, and the ßcalculus into CERSs. This last encoding is an (important) example of real contextsensitive rewriting. 1 Introduction A term rewriting system is a pair consisting of an alphabet and a set of rewrite rules. The alphabet is used freely to gene...
Development Closed Critical Pairs
, 1996
"... . The class of orthogonal rewriting systems (rewriting systems where rewrite steps cannot depend on one another) is the main class of notnecessarilyterminating rewriting systems for which confluence is known to hold. Huet and Toyama have shown that for leftlinear firstorder term rewriting sys ..."
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Cited by 5 (1 self)
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. The class of orthogonal rewriting systems (rewriting systems where rewrite steps cannot depend on one another) is the main class of notnecessarilyterminating rewriting systems for which confluence is known to hold. Huet and Toyama have shown that for leftlinear firstorder term rewriting systems (TRSs) the orthogonality restriction can be relaxed somewhat by allowing critical pairs (arising from maximally general ways of dependence between steps), but requiring them to be parallel closed. We extend these results by replacing the parallel closed condition by a development closed condition. This also permits to generalise them to higherorder term rewriting, yielding a confluence criterion for Klop's combinatory reduction systems (CRSs), Khasidashvili's expression reduction systems (ERSs), and Nipkow's higherorder pattern rewriting systems (PRSs). 1 Introduction This paper is concerned with a method to prove confluence of rewriting systems. It's an extension of some co...
Normalization of Typable Terms by Superdevelopments
 Computer Science Logic'98, Springer LNCS 1584
, 1999
"... . We define a class of hyperbalanced lterms by imposing syntactic constraints on the construction of lterms, and show that such terms are strongly normalizing. Furthermore, we show that for any hyperbalanced term, the total number of superdevelopments needed to compute its normal form can be stati ..."
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Cited by 2 (1 self)
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. We define a class of hyperbalanced lterms by imposing syntactic constraints on the construction of lterms, and show that such terms are strongly normalizing. Furthermore, we show that for any hyperbalanced term, the total number of superdevelopments needed to compute its normal form can be statically determined at the beginning of reduction. To obtain the latter result, we develop an algorithm that, in a hyperbalanced term M, statically detects all inessential (or unneeded)subterms which can be replaced by fresh variables without effecting the normal form of M; that is, full garbage collection can be performed before starting the reduction. Finally, we show that, modulo a restricted hexpansion, all simply typable lterms are hyperbalanced, implying importance of the class of hyperbalanced terms. 1 Introduction The termination of breduction for typed terms is one of the most studied topics in l calculus. After classical proofs of Tait [21] and Girard [8], many interesting proo...
1 Contextsensitive Conditional Reduction Systems
"... We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems ( ..."
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Cited by 1 (0 self)
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We introduce Contextsensitive Conditional Expression Reduction Systems (CERS) by extending and generalizing the notion of conditional TRS to the higher order case. We justify our framework in two ways. First, we define orthogonality for CERSs and show that the usual results for orthogonal systems (finiteness of developments, confluence, permutation equivalence) carry over immediately. This can be used e.g. to infer confluence from the subject reduction property in several typed λcalculi possibly enriched with patternmatching definitions. Second, we express several proof and transition systems as CERSs. In particular, we give encodings of Hilbertstyle proof systems, Gentzenstyle sequentcalculi, rewrite systems with rule priorities, and the πcalculus into CERSs. This last encoding is an important example of real contextsensitive rewriting. ○c
1 2 HIGHERORDER MATCHING MODULO (SUPER)DEVELOPMENTS APPLICATIONS TO SECONDORDER MATCHING 3
, 2009
"... Abstract. To perform higherorder matching, we need to decide the βηequivalence on λterms. The first way to do it is to use simply typed λcalculus and this is the usual framework where higherorder matching is performed. Another approach consists in deciding a restricted equivalence. This restric ..."
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Abstract. To perform higherorder matching, we need to decide the βηequivalence on λterms. The first way to do it is to use simply typed λcalculus and this is the usual framework where higherorder matching is performed. Another approach consists in deciding a restricted equivalence. This restricted equivalence can be based on finite developments or more interestingly on finite superdevelopments. We consider higherorder matching modulo (super)developments over untyped λterms for which we propose terminating, sound and complete matching algorithms. This is in particular of interest since all secondorder βmatches are matches modulo superdevelopments. We further propose a restriction to secondorder matching that gives exactly all secondorder matches. We finally apply these results in the context of higherorder rewriting. Contents 1. Normalization in the lambdacalculus 3 2. Matching modulo beta (and eta) 9 3. Matching modulo superdevelopments (and eta) 11