Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, chapter 1, pages 1-117. Oxford University Press, Oxford, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the interest in higher order systems such as higher order logics and higherorder deduction systems [And86,Pau94,GLM97,And01,Pfe01] higher order (functional) programming languages [BMS80,Tur85,Pau91,Bar90,Bir98] higherorder logic programming languages [Mil91,HKMN95] higher order rewriting [Nip91,Klo92,DJ90] and higher order uni cation [Hue75,Dow01] It is well known that second order uni cation hence higher order uni cation is undecidable ( Gol81,Far91,LV00a] Higher order uni cation procedures were already described by in [Hue75,JP76] The undecidability results triggered research into ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, D.M. Gabbay, and T.S.E.Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 2-116. Oxford University Press, 1992.

....lazy rewriting strategy implements a fully data driven computation scheme and avoids useless branches of computation. Thus, there is no 5Conditions do not change this general scheme and are omitted from the presentation for the sake of simplicity. See for example [Dershowitz Plaisted 88] and [Klop 90] for a survey. 74 LIST [1: LIST 1 APPEND : LIST[ i: LIST] APPEND :LIST[ LISTJ , NIL , CONS[first: T T] Lrest: LIS APPENDO : APPEND1 : NIL fi] LIST rst: CONS rest :i] LIST NsIrst: Lrest: APPEND Figure 3: Rewrite rules for LIST and APPEND. need to have a secial ....

Jan Willera Klop. "Term rewriting systems". To appear in S. Abramsky, D. Gabbay and T. Maibaum. Handbook of Logic in Computer Science, Vol. 1, Oxford University Press.

....of normal forms in the interreduced TRS may be much more efficient than in the original one. Interreduction of a complete TRS can be performed by first normalizing all right hand sides, and then omitting all rules with a left hand side that is reducible by the remaining rules (cf. M et83] [Klo92]) a) R# : fl r# R j l r 2 Rg (right normalization) b) R del : fl r 2 R j l is (R n fl rg) irreducibleg (deletion of left reducible rules) If R is complete then R# and R# del are also complete, and their induced equational theories coincide: R# = R# del . These properties ....

....(1) 4) In particular, we show there that the important class of orthogonal TRSs enjoys all these preservation properties and, hence, allows for safe interreduction. 2 Preliminaries We assume familiarity with the basic no(ta)tions, terminology and theory of term rewriting (cf. e.g. DJ90] [Klo92]) but recall some no(ta)tions for the sake of readability. The set of terms over some given signature F and some (disjoint) countably infinite set V of variables is denoted by T (F ; V) We write T (F) for the set of all ground (i.e. variable free) terms over F . A term rewriting system (TRS) is ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, chapter 1, pages 2--117. Clarendon Press, Oxford, 1992.

....introduced in syntax productions to bind meta variables to sorts and introduce infix binary constructors. We do not delve further into the exciting details of the properties of rewriting systems in general and CRS in particular but refer the reader to the comprehensive literature on the subject [17, 18]. Instead we go straight to our basic example. 3.1.2. Example (2 level calculus) The 2 level calculus, denoted , is the single sorted CRS over the terms e : x j x:e j e 0 e 1 j x:e j e 0 e 1 where concatenation denotes application (the invisible infix application function symbol ....

Jan Willem Klop. Term rewriting systems. In Samson Abramsky, Dov M. Gabby, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, Vol. 2, chapter 1, pages 2--116. Oxford University Press, Oxford, 1992.

....a rewrite rule to a term is realised by initialising the machine with this term, and then successively executing the instructions of the compiled rule. 1 Introduction Combinatory Reduction Systems were introduced by Klop in 1980 [9] CRSs in their original form generalise applicative TRSs [10]. We shall concentrate here on functional CRSs, as defined by Kennaway in [8] they generalise ordinary TRSs. The techniques of this paper can easily be adapted to applicative CRSs. Functional CRSs extend TRSs in two respects. Firstly, they support a notion of variable binding. Substitution has ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. M. Gabbai, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 2, pages 1--116. Oxford University Press, 1992.

.... the interest in higher order systems such as higher order logics and higherorder deduction systems [And86,Pau94,GLM97,And01,Pfe01] higher order (functional) programming languages [BMS80,Tur85,Pau91,Bar90,Bir98] higherorder logic programming languages [Mil91,HKMN95] higher order rewriting [Nip91,Klo92,DJ90] and higher order uni cation [Hue75,Dow01] It is well known that second order uni cation hence higher order uni cation is undecidable ( Gol81,Far91,LV00a] Higher order uni cation procedures were already described by in [Hue75,JP76] The undecidability results triggered research into ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, D.M. Gabbay, and T.S.E.Maibaum, editors, Handbook pf Logic in Computer Science, volume 2, pages 2-116. Oxford University Press, 1992.

....transformation steps. They were studied the first time by Newman [80] and systematically applied in influential papers of Rosen [93] and Huet [52] In the following some basic notions and facts for abstract reduction systems are collected. Further concepts and results can be found, for example, in [9, 11, 16, 28, 52, 56, 66]. As the terminology in the literature is not completely uniform, alternative terms are given in parentheses. Definition 2.1 (Abstract reduction system) An abstract reduction system hA; i consists of a set A and a binary relation on A. For the rest of this section, let hA; i be an arbitrary ....

....bisimilarity. Collapsing sometimes speeds up the evaluation of term graphs considerably, which we show by an example. 4.1 Term rewriting We first recall some basic concepts of term rewriting systems. For a comprehensive introduction, the reader may consult the textbook [11] or one of the surveys [54, 10, 28, 66, 84, 57]. Let T Sigma;X be the set of all terms over Sigma and X. A substitution is a mapping oe : T Sigma;X T Sigma;X such that oe(c) c for every constant c, and oe(f(t 1 ; t n ) f(oe(t 1 ) oe(t n ) for every composite term f(t 1 ; t n ) A term rewrite rule is a ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 1--116. Oxford University Press, 1992.

....the letters u, v, w for variables; other small letters and numbers for ground symbols. The terms in such an L are applicative terms. The atomic formulae in L, all of the form s = t, are called equations between applicative terms, their (free) variables are to be seen as universal quantified. As [3] in page 16, we adopt the infix notation for (applicative) terms and the notational simplification by dropping parenthesis under the convention of association to the left: Instead of writing ap(s; t) we just write (st) we also write (t 0 ) for a term t 0 , and recursively, t 0 Delta Delta ....

....only if the equation s = t is derivable from instances of the equations in S with the schemata ref, trans, cons, dec, sep corresponding to the equality postulates. The set S is contradictory if and only if S j= s = t holds for a contradiction s = t (cf. definition 1) The literature, for example [3], page 41, is full of calculi similar to the above one. Most of the readers will find the above lemma trivial, some as an easy consequence of the results in [5] and few scrupulous ones would begin to calculate with terms. Well, all this is right. We remark: Lemma 1 remains valid if an equality ....

[Article contains additional citation context not shown here]

Jan Willem Klop, Term Rewriting Systems, In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors: Handbook of Logik in Computer Science, volume 2, chapter 1, pages 1--116. Oxford University Press, 1992.

....are applicable, then any one of them can be applied. If no rule is applicable, then the term cannot be rewritten any further. In practice, we often use abstract data types such as arrays and FIFO queues to make the descriptions more readable. More information about TRS s can be found elsewhere [1, 6]. A small but fascinating example of term rewriting is provided by the SK combinatory system, which has only two rules, and a simple grammar for generating terms. These two rules are sufficient to describe any computable function TERM j K [ S [ TERM.TERM K rule: K.x) y x S rule: ....

Jan Willem Klop. Term Rewriting System. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Computer Science, volume 2. Oxford University Press, 1992.

....proof would be interesting; one would then have confluence after critical pair completion without regard for termination. If these conditions are insufficient, the counterexamples will have to be (besides left linear) non rightlinear, non terminating, and non orthogonal (have critical pairs) See [Klo92]. Remark: Significant progress is reported in [OO97] A new criterion based on so called simultaneous critical pairs has been presented in [Oku98] ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, chapter 1, pages 1--117. Oxford University Press, Oxford, 1992.

....as in lambda lifting) See Barendsen and Smetsers [6] for examples. Say that subcommutes with if ; A rewrite relation is said to be subcommutative if it subcommutes with itself. It is a standard result that subcommutativity implies con uence (see eg [14]) Proposition 2.9. The rewrite relation generated by a simple functional GRS is subcommutative, hence con uent, up to . Proof. The restrictions placed on the form of rules allowed in a simple functional GRS ensure that the rules are non overlapping (non interfering in the terminology of [3] ....

Jan Willem Klop. Term rewriting systems. In Samson Abramsky, Dov M. Gabbay, and Tom Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 1-116. Oxford University Press, New York, 1992.

....to minimal ones. Then we prove that this transformation preserves termination and discuss that we can apply the OSO to any order sorted TRSs with non minimal signatures. Section 5 concludes this paper. 2. Order Sorted TRSs In this section, we give basic notions on order sorted TRSs, following [1][3] 5] An order sorted signature is a triple (S; 6) where S is a set of sorts, is a partial ordering over S and 6 is a family f6w;s jw 2 S 3 ; s 2 Sg of (not necessarily disjoint) set of operator symbols. Note that w is possibly an empty word . For any f 2 6w;s , a word w and a ....

....s, iii) f(t 1 ; t n ) 2 T 6 (V ) s if f : s 1 1 1 1 s n s 0 such that s 0 = s and t i 2 T6 (V ) s i for every 1 = i = n. T6 (V ) S s2S T6 (V ) s denotes the set of all terms over 6 and V . The set of all variables in a term t 2 T6 (V ) is written as V ar(t) Following [1], O(t) is the set of all occurrences of the term t. A symbol of t at 2 O(t) is described as t( tj is a subterm of t at . The result of the replacement of the subterm at in t by t 0 is denoted by t[ t 0 ] this term may be ill formed, even if t and t 0 are well formed. Let (S; ....

Jan Willem Klop. Term rewriting systems. In S.Abramsky, D.M. Gabbai,and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 2, pages 1-116. Oxford University Press, 1992.

....transformation steps. They were studied the first time by Newman [81] and systematically applied in influential papers of Rosen [95] and Huet [52] In the following some basic notions and facts for abstract reduction systems are collected. Further concepts and results can be found, for example, in [9,11,16,28,52,56,67]. As the terminology in the literature is not completely uniform, alternative terms are given in parentheses. Definition 1.2.1 (Abstract reduction system) An abstract reduction system hA; i consists of a set A and a binary relation on A. For the rest of this section, let hA; i be an ....

....Collapsing sometimes speeds up the evaluation of term graphs considerably, which we show by an example. 1.4.1 Term Rewriting We first recall some basic concepts of term rewriting systems. For a comprehensive introduction, the reader may consult the textbook [11] or one of the surveys [54,10,28,67,86,57]. 16 CHAPTER 1. TERM GRAPH REWRITING Let T Sigma;X be the set of all terms over Sigma and X. A substitution is a mapping oe : T Sigma;X T Sigma;X such that oe(c) c for every constant c, and oe(f(t 1 ; t n ) f(oe(t 1 ) oe(t n ) for every composite term f(t 1 ; ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 1--116. Oxford University Press, 1992.

....confluence and termination. Finally, we prove the modularity of confluence and termination in order sorted TRSs. The first author is currently with Nippon Telegraph and Telephone Corporation. 2 Order Sorted TRSs In this section, we give basic notions on order sorted TRSs, following [1][2] 5] An order sorted signature is a triple (S; 6) where S is a set of sorts, is a partial ordering over S and 6 is a family f6w;s jw 2 S 3 ; s 2 Sg of (not necessarily disjoint) set of operator symbols. Note that w is possibly an empty word . For any f 2 6 w;s , a word w and a pair (w; s) ....

....s 0 and s 0 s, iii) f(t 1 ; t n ) 2 T6 (V ) s if f : s 1 1 1 1 s n s 0 such that s 0 s and t i 2 T 6 (V ) s i for every 1 i n. T 6 (V ) S s2S T 6 (V ) s denotes the set of all terms over 6 and V . The set of all variables in a term t 2 T6 (V ) is written as V ar(t) Following [1], O(t) is the set of all occurrences of the term t. A symbol of t at 2 O(t) is described as t( tj is a subterm of t at . The result of the replacement of the subterm at in t by t 0 is denoted by t[ t 0 ] this term may be ill formed, even if t and t 0 are well formed. Let (S; ....

Jan Willem Klop. Term rewriting systems. In S.Abramsky, D.M. Gabbai,and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 2, pages 1-116. Oxford University Press, 1992.

....and variables, like f(x; 2; 3) Terms of applicative form consist of variables, constants and a binary symbol called application, like Ap(Ap(Ap(f; x) 2) 3) A famous example of applicative form of TRSs is Combinatory Logic(CL) which was developed by Shonfinkel and rediscovered by Curry. Klop [3] says that the TRS CL has universal computational power and that CL is used to implement functional programming languages. However, applicative forms of term rewriting are not usually treated in the theory of term rewriting. Kennaway et.al [1] insisted that one need to study the relationship ....

....5, we prove that compatibility and confluence of order sorted TRSs are preserved by currying. 2 Foundations We write N for the set of natural numbers. We describe ff for the reflexive and transitive closure of a relation ff (ff is a meta variable) and ff for the inverse relation, in [3]. Given a set A, A 3 is the set of finite words over A; is an empty word. Given an element a 2 A and a natural number n 2 N , a n is a word consisting of a of n pieces; a 0 is an empty word. v 1 w is concatenation of two words v and w. A 3 is partially ordered by the prefix ordering ....

Jan Willem Klop. Term rewriting systems. In S.Abramsky, D.M. Gabbai,and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 2, pages 1-116. Oxford University Press, 1992.

....be seen as a modification of Knuth Bendix algorithm, or as an algorithm for checking consistence of equations systems and finding term reduction systems as models for them: subsection 2.3 brings unification, completion and consistence checking together. Subsection 2. 2 puts Term Reduction Systems [9] in the context of predicate logic. The relevance of dealing with these equations systems and of 15 their solution lies in their immediate application to integrate logic and functional programming: we can substitute term unification in logic programming by cp completion, as [6] does with higher ....

....some occurrences of open subterms r i of r by s i , where each r i = s i is an instance of an equation of S. And that r n s or r s is derivable if we can do this recursively, n times in the first case. When we deal with derivation of these objects, we say that S acts as a term reduction system [9]. The expression r[r 1 ; r k ] denotes an open term t, each open term r i in this expression points to some occurrences of an open subterm of t formally equal to r i , the subterms pointed by different r i and r j do not overlap. After substituting the open subterms pointed by each r i by ....

Jan Willem Klop, Term Rewriting Systems, In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors: Handbook of Logik in Computer Science, volume 2, chapter 1, pages 1--116. Oxford University Press, 1992.

....of HRS, the so called orthogonal ones. An Orthogonal Higher Order Rewrite System (OHRS) is an HRS that is left linear and has no critical pairs. This means that there are no rules whose left hand sides overlap (see Def. 4. 1) Orthogonal term rewriting systems have been studied very closely [12] because they are similar to functional programs with pattern matching. The key property of orthogonal systems is their confluence, regardless of whether they terminate or not. We show that this holds for OHRS as well. 20 6.1 The Classical Proof In this section we generalize Aczel s [1] ....

Jan Willem Klop. Term rewriting systems. In Samson Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 2--116. Oxford University Press, 1992.

....Chomsky grammars, as the set of all terminal graphs G into which T transforms some distinguished start graph S, where terminality is usually defined by the absence of certain ( nonterminal ) labels in a graph. Graph transformation can be used to specify a function on graphs, like term rewriting [19] specifies functions on terms, by taking an arbitrary graph as input, and transforming it as long as possible. This function is partial if certain graphs can be transformed infinitely, and nondeterministic if a graph may be transformed in different ways. It is this last way of using graph ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 1--116. Oxford University Press, 1992.

....T c T j . The objective will be to simplify ( j ; j ) expressions using a decision procedure for T c (hence T c is assumed to be decidable) 3. 2 Contextual Reduction Systems We now introduce the notion of contextual reduction system by generalizing that of abstract reduction system given in [16]. This will be useful to specify CCR(X) in a way that is precise, concise, and incremental at the same time. Let L be a set of labels. For all 2 L, let C , E be sets of expressions and S(C ; E ) be the set of sequents of the form c : e e 0 for all c 2 C and e; e 0 2 E . A ....

Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, chapter 1, pages 1-117. Oxford University Press, Oxford, 1992.

....A term rewrite rule l r consists of two terms l and r over Sigma [ X such that l is not a variable and all variables in r occur also in l. A set R of term rewrite rules is a term rewriting system. We assume that the reader is familiar with basic concepts of term rewriting (see [DJ90, Klo92] for overviews) For the following we fix an arbitrary term rewiting system R. The term rewrite relation associated with R is denoted by , its transitive closure by , and its reflexive transitive closure by . Given a term t, we write Mt for the tree representing t. Moreover, Sigmat ....

....be found in [Plu93b] as part of the proof of the so called Critical Pair Lemma. 2 Call the relation ) subcommutative if whenever G 1 ( G ) G 2 , there is a term graph G 3 such that G 1 ) G 3 ( G 2 . It is well known that subcommutativity implies confluence (for arbitrary binary relations; see [Klo92] Corollary 5.3 If R is non overlapping, then ) is subcommutative. For the rest of this section we assume that R is an arbitrary non overlapping system. The following property of subcommutative relations will be needed in showing that the full substitution strategy is cofinal. Corollary 5.4 ....

[Article contains additional citation context not shown here]

Jan Willem Klop. Term rewriting systems. In S. Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 1--116. Oxford University Press, 1992.

....and r over and X such that l is not a variable and all variables in r occur also in l. A set R of term rewrite rules is a term rewriting system. We assume that the reader is familiar with basic concepts of term rewriting. For an introduction, see the textbook [BN98] or one of the surveys [DJ90, Klo92] For the following we x an arbitrary term rewiting system R. The term rewrite relation associated with R is denoted by , its transitive closure by , and its re exive transitive closure by . Given a term t, we write Mt for the tree representing t. Moreover, t denotes the term ....

....in [Plu93b] as part of the proof of the so called Critical Pair Lemma. 2 Call the relation ) subcommutative if whenever G 1 ( G ) G 2 , there is a term graph G 3 such that G 1 ) G 3 ( G 2 . It is well known that subcommutativity implies con uence (for arbitrary binary relations; see [Klo92] Corollary 5.3 If R is non overlapping, then ) is subcommutative. For the rest of this section we assume that R is an arbitrary non overlapping system. The following property of subcommutative relations will be needed in showing that the full substitution strategy is co nal. Corollary 5.4 ....

[Article contains additional citation context not shown here]

Jan Willem Klop. Term rewriting systems. In S. Abramsky, Dov M. Gabbay, and T.S.E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, pages 1-116. Oxford University Press, 1992.

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Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, chapter 1, pages 1-117. Oxford University Press, Oxford, 1992.

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Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, chapter 1, pages 1--117. Oxford University Press, Oxford, 1992.

No context found.

Jan Willem Klop. Term rewriting systems. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, volume 2, chapter 1, pages 1--117. Oxford University Press, Oxford, 1992.

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Jan Willem Klop. Term rewriting systems. In S. Abramsky,D.M.Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science,volume 2, chapter 1, pages 1#117. Oxford University Press, Oxford, 1992.

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