J. W. Klop. Term rewriting systems: A tutorial. Bulletin of the EATCS, 32:143--182, June 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 ; s n i 35 from which we conclude rpo hs 1 ; s n i by Proposition 44, part (4) Hence we are in case (2b) of the proposition. 2 This nishes the proof of Theorem 38. An alternative de nition of recursive path order in which all recursion has been eliminated was given in [41, 42]. However, that de nition is not equivalent to ours, and it does not directly imply an e ective procedure for deciding whether s rpo t for two given terms s; t. 4.3 Extensions of recursive path order Many variations and extensions of the basic version of recursive path order have been ....

Klop, J. W. Term rewriting systems: a tutorial. Bulletin of the EATCS 32 (1987), 143-183.

.... property of the relation associated with a term rewrite system is that of confluence, another one is local confluence (also known as the Church Rosser and weakly Church Rosser properties respectively) For an account of term rewrite systems in general and these properties in particular see [13]. Newman s lemma states that any relation that admits induction and is locally confluent is also confluent. First we formulate these two properties in the relational calculus. Spec R is confluent is equivalent to R ffi (R[ v (R[ ffi R : 84) Spec R is locally confluent is R ffi R[ v ....

J.W. Klop. Term rewriting systems: a tutorial. Bulletin of the EATCS, 32:143--182, 1987.

....states. A terminating rule set is confluent if for each initial database state db 0 the order of rule execution does not influence the final database state db 1 . That is, db 1 is uniquely determined by db 0 and the rule set. Confluency of rule sets is thus similar to confluency of rewrite systems [7] in that the execution order is immaterial. The difference, however, is that a rule sets behaviour is affected by the underlying database state, and a rule set is confluent if it is confluent on all database states. Whether a rule set is confluent or terminates depends, of course, on the rule ....

J.W. Klop. Term rewriting systems: A tutorial. In Bull. European Assoc. Theoretical Computer Science, volume 32, pages 143--183, 1987.

.... Fortunately, there is an ample literature on these subjects, and we are able to refer the reader to, e.g. 4] 13] for basics of algebraic specifications) 8] 6] for order sorted logic) 7] for hidden sorts) 10] for coinduction) 12] for rewriting logic) 5] for institutions) and [11], 1] for term rewriting systems) as primers. The logical aspects of CafeOBJ are explained in detail in [2] and [3] This manual is for the initiated, and we sometimes abandon the theoretical rigour for the sake of intuitiveness. For a very brief introduction, we just highlight a couple of ....

.... true xor a xor a and b, p:Prop true xor b 1 [9] b xor true xor a xor a and b xor b xor true xor true true xor a xor a and b xor false 1 [10] rule: eq p:Prop xor false = p:Prop p:Prop a and b xor true xor a 1 [10] true xor a xor a and b xor false a and b xor true xor a 1 [11] rule: eq p:Prop q:Prop = p:Prop xor q:Prop xor true p:Prop a and b xor a xor true, q:Prop a and b xor true xor a 1 [11] a and b xor a xor true a and b xor true xor a a and b xor a xor true xor a and b xor true xor a xor true 1 [12] rule: eq AC xor p:Prop xor p:Prop = ....

[Article contains additional citation context not shown here]

Klop, J.W., "Term Rewriting Systems: A Tutorial", EATCS Bulletin, No.32, EATCS, 1987, pp.143--182

....the static semantics is proved in Sect.3. 2 Church Rosser Property for CMS Before proving CR for CMS we introduce some standard de nition and properties on Term Rewriting Systems (TRSs) and Combinatory Reduction Systems (CRSs) which will be used later on. For more technical details we refer to (Klop, 1987; Klop et al. 1993) 2.1 Technical Preliminaries De nition 2.1 An Abstract Reduction System (ARS) is a pair A; 2I consisting of a set A and a sequence of binary relations on A, also called reduction or rewrite relations. If for a; b 2 A we have (a; b) 2 we write a b. The re ....

..... Fact 2.3 If and strongly commute, then they commute. De nition 2.4 A reduction is Church Rosser (or con uent) if is self commuting. Theorem 2.5 (Hindley) Let A; 2I be an ARS s.t. for all ; 2 I , and commutes. Then the union S 2I is CR. We refer to (Klop, 1987) and (Klop et al. 1993) for the notions of Term Rewriting System and of Combinatory Reduction System, respectively. The following de nitions and theorem on TRSs apply also to CRSs (by replacing the word variable with meta variable ) De nition 2.6 A term is linear if it contains no multiple ....

Klop, J.W. (1987). Term rewriting systems: a tutorial. Bull. of EATCS, 32, 143-182.

.... property of the relation associated with a term rewrite system is that of confluence, another one is local confluence (also known as the Church Rosser and weakly Church Rosser properties respectively) For an account of term rewrite systems in general and these properties in particular see [18]. 26 Newman s lemma states that any relation that admits induction and is locally confluent is also confluent. First we formulate these two properties in the relational calculus. Spec R is confluent is equivalent to R ffi (R[ R[ ffi R : 48) Spec R is locally confluent is R ffi R[ ....

J.W. Klop. Term rewriting systems: a tutorial. Bulletin of the EATCS, 32:143--182, 1987.

....employ CafeOBJ properly. Fortunately, there is an ample literature on these subjects, and we are able to refer the reader to, e.g. 3] 12] for basics of algebraic specifications) 8] 6] for order sorted logic) 7] for hidden sorts) 11] for rewriting logic) 5] for institutions) and [10], 1] for term rewriting systems) as primers. The logical aspects of CafeOBJ are explained in detail in [2] and [4] This manual is for the initiated, and we sometimes abandon the theoretical rigour for the sake of intuitiveness. For a very brief introduction, we just highlight a couple of ....

.... 0 [condition] not s(s(0) 0 not false [condition] not false true [7] gcd(s(s(s(s(0) s(s(0) gcd(s(s(s(s(0) s(s(0) s(s(0) 8] gcd(s(s(s(s(0) s(s(0) s(s(0) gcd(s(s(s(0) s(0) s(s(0) 9] gcd(s(s(s(0) s(0) s(s(0) gcd(s(s(0) 0,s(s(0) [10]: gcd(s(s(0) 0,s(s(0) gcd(s(s(0) s(s(0) condition] s(s(0) s(s(0) s(0) s(0) condition] s(0) s(0) 0 0 [condition] 0 0 false [condition] not s(0) s(0) not 0 0 [condition] not 0 0 not false [condition] not false true [17] ....

Klop, J.W., "Term Rewriting Systems: A Tutorial", Bulletin of the EATCS, Vol.32, EATCS, 1987, pp.143--182

....and ffl eval i (e) 1 ; n ; 0; 0] zeros) where is given by i = h i. 3.2 Rewrite rules To finish the above definition we need to define the system of rewrite rules on which the rewrite operators are based. Thereto the classical notion of a term rewrite rule [12] must be adapted to our setting. Let S be a schema and let = 1 ; n ] be a type. Let C f1; ng be the set of expression columns of , and for j 2 C let j be given by j = h j i. Definition 3.2 A rewrite rule over S with respect to is a rule of the form ff fi, ....

J.-W. Klop. Term rewriting systems: A tutorial. Bulletin of the EATCS, 32:143-- 183, 1987.

....Chapter Six of [34] So functions symbols are not allowed to occur in patterns; for example, a definition like Pair(In left(x) In right(x) x. is, given the two rules above, not allowed. A difficulty with these three rules together, is that they form Klop s famous Surjective Pairing example [27]; this function cannot be expressed in LC because when added to LC, the Church Rosser property no longer holds. This implies that, although both LC and TRS are Turing machine complete, there is no general syntactic solution for patterns in LC, so a full purpose translation (interpretation) of TRS ....

J.W. Klop. Term Rewriting Systems: a tutorial. EATCS Bulletin, 32:143--182, 1987.

....or to better understand and improve the actual computing practice. Various combinations of these two formalisms have been studied extensively in recent years, both in typed and untyped contexts. In the absence of types, the two systems do not interact in a very smooth manner. For instance, in [21] Klop showed that confluence, a highly desirable property in practice, is lost if a surjective pairing operation is added to the untyped LC. In [16] Dougherty provided some restrictions on terms, thus ensuring that properties that LC and TRS both possess can be preserved when these systems are ....

J.W. Klop. Term Rewriting Systems: a tutorial. EATCS Bulletin, 32:143--182, 1987.

....sequence n 0 ; n 1 ; such that (n i ; n i 1 ) 2 r for all i 0. Well founded relations are the essence of induction. In particular, they are crucial for establishing the absence of infinite loops in a program. Well founded relations are also used for proving termination of rewriting systems [4]. When it is difficult to establish well foundedness directly, we may try a divide and conquer approach: First we decompose the given relation, then we establish that each part is well founded, and finally, we attempt to deduce well foundedness of the whole. For example, we may wish to prove that ....

J. Klop. Term rewriting systems: a tutorial. Bulletin of the EATCS, 32(143-182), 1987.

....n 0 , n 1 , such that (n i , n i 1 ) # r for all i # 0. Well founded relations are the essence of induction. In particular, they are crucial for establishing the absence of infinite loops in a program. Well founded relations are also used for proving termination of rewriting systems [4]. When it is di#cult to establish well foundedness directly, we may try a divide andconquer approach: First we decompose the given relation, then we establish that each part is well founded, and finally, we attempt to deduce well foundedness of the whole. For example, we may wish to prove that the ....

J. Klop. Term rewriting systems: a tutorial. Bulletin of the EATCS, 32(143-182), 1987. Received 12 February

....the author was at LRI, Universit e de Paris Sud. The study of systems based on calculus and algebraic rewriting has been carried out both in untyped and typed contexts. If no type discipline is imposed on the languages, the interactions between these computational models raise several problems [Klo87, Dou91]. For typed languages things work out nicely. In [BTG90] and [Oka89] it is shown that the system obtained by combining a terminating first order many sorted term rewrite system with the second order typed calculus is again terminating with respect to fi reduction and the algebraic reductions ....

J. W. Klop. Term rewriting systems: a tutorial. EATCS Bulletin, 32:143--182, 1987.

....results were first obtained under the supervision of Sh. Pkhakadze and Kh. Rukhaia for some reductions in the Notation Theory of Sh. Pkhakadze [16] The results were generalized for OTRSs and reported in [8] after I became acquainted with the TRS theory by J. W. Klop s introductory paper [9], pointed to me out by G. Mints. I am grateful to them, as well as to H. Barendregt and J. J. L evy, for constant support. I thank H. Barendregt, P. L. Curien, H. Ganzinger, D. Kesner, J. W. Klop, G. Kucherov, and D. Rosner for organizing my talks at Nijmegen, Paris, Saarbrucken, Orsay, ....

Klop J.W. Term Rewriting Systems: a tutorial. Bulletin of the EATCS 32, 1987, p. 143-182.

....of strong normalisation of terms) property and why it is difficult to obtain. Finally we give an overview of the following sections. The reader is expected to be familiar with the untyped calculus (Barendregt 1984) and we make free use of abstract reduction system concepts (Huet 1980, Klop 1987), in particular the diagram stencils of Rosen (1973) Due to space limitations, most proofs are omitted; the full paper includes detailed proofs and will be available as a technical report. Infinity of the lambda calculus. Recall that the essential definition for reduction in calculus is ....

Klop, J. W. (1987). Term rewriting systems: a tutorial. Bulletin of the European Association for Theoretical Computer Science 32: 143--182.

.... extended to birewriting and the development of the article follows the same pattern as equational rewriting: the Church Rosser property is proved by means of a critical pair lemma, and we use a completion process to ensure the confluence of the critical pairs (Knuth and Bendix, 1970; Huet, 1980; Klop, 1987; Dershowitz and Jouannaud, 1990) However there are also some differences. Equational rewriting is in essence a theory of normal forms, while bi rewriting disregards this notion. Bi rewriting can also be seen as a generalization of equational rewriting: equations can be translated to pairs of ....

....bi rewrite systems by means of an example in section 9. In section 10 we present related work and in section 11 we conclude summarizing present and further work. 2 Inclusions and Bi rewrite Systems If nothing is said, we follow the notation and the standard definitions used in (Huet, 1980; Klop, 1987; Dershowitz and Jouannaud, 1990) We are concerned with first order terms T (F ; X ) over a nonempty signature F = S n2IN F n of function symbols, and a denumerable set X of variables. 1 The set of variables of a term t is denoted by FV(t) A position p is a sequence of positive integers. ....

Klop, J. W. (1987). Term rewriting systems: A tutorial. Bulletin of the EATCS, 32:143--183.

....project 432 (METEOR) x Supported in part by the U.S. National Science Foundation under Grant CCR 8802282. 2 N. Dershowitz, S. Kaplan, D. A. Plaisted has important applications in abstract data type specifications and functional programming languages. For surveys of the theory of rewriting, see [13, 16], or [8] our notations conform to the latter. A key property for rewrite systems is that every term rewrites to a unique normal form. This is usually decomposed into two requirements: normalization , which ensures that at least one normal form always exists; and confluence , which ensures that ....

....form t 1 in T 1 , then there exists a fair derivation t 0 R t 1 R Delta Delta Delta R t 1 with limit t 1 . 5 This property should not be confused with the (finite) unique normal form (i.e. at most one normal form) of property (finitely) confluent systems in the terminology of [16]. 10 N. Dershowitz, S. Kaplan, D. A. Plaisted Of course, unfair derivations can also need lead to normal forms. For example, either rule in f(x) f(f(x) f(x) f(f(f(x) 11) can be forever ignored. Proof. Suppose that t 0 R t 1 R Delta Delta Delta R t 1 , and t 1 is a normal ....

[Article contains additional citation context not shown here]

J. W. Klop, Term rewriting systems: A tutorial, Bulletin European Assoc. for Theoret. Comput. Sci. 32, pp. 143--183 (June 1987).

....like ML) in a unified framework. The study of systems based on calculi and algebraic rewriting has been carried out both in untyped and typed contexts. If no type discipline is imposed on the languages the interactions between these computational models raise several problems, as shown in [21] and [14] For typed languages (typed versions of calculus and typed term rewriting systems) things work out nicely. In [8] and [24] it is shown that the system obtained by combining a terminating first order many sorted term rewrite system with the second order typed calculus is again ....

J. W. Klop. Term rewriting systems: a tutorial. EATCS Bulletin, 32:143--182, June 1987.

....2 Let T be an optimally reducing term rewriting system, s be an irreducible term, and be an irreducible substitution. Then (s) can be reduced to its normal form in less than or equal to jsj steps. Proof sketch: It can be shown that doing reduction innermost (i.e. from the leaves upward see [Klo87]) would involve at the most jsj steps. 2 Lemma 3 Every confluent, optimally reducing term rewriting system has a decidable unification problem. Proof sketch: Let T be an optimally reducing canonical term rewriting system and s and t be terms to be unified modulo T . By Lemma 2, if is an ....

J. W. Klop. Term rewriting systems: A tutorial. Bulletin of the EATCS, 32:143--182, June 1987.

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J. W. Klop. Term rewriting systems: A tutorial. Bulletin of the EATCS, 32:143--182, June 1987.

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J. W. Klop, `Term rewriting systems: a tutorial', Note CS-N8701, Centre for Mathematics and Computer Science, CWI Amsterdam, The Netherlands, 1987.

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