N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1):69-- 116, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and variable values. It has to be shown that when a previous state is reached again, then the values of arguments and variables will have decreased in some Noetherian ordering. Such orderings for termination proofs have extensively been studied in the area of term rewriting systems, see e.g. [Der87]. Automatic termination provers for functional languages often stay in the tradition of term rewriting and try to generate sufficient ordering relations [Wal94, Gie95a, Gie95b] Such methods can only give termination proofs for applications to arguments that have a finite normal form, i.e. they ....

....means that the path between a recursive pair minimizes an expression. Here we can plug in modules which try to find one of the numerous different orderings for termination proofs proposed in the literature, e.g. polynomial orderings as proposed in [Ste92, Lan79] one of the orderings presented in [Der87] or some sort of generalized orderings based on multi sets as presented in [Mar87] A simple ordering which in many cases is sufficient enough is the ordering which is based on the number of constructors the normal form of an expression has. be a recursive pair such that its ordering ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1):69--116, 1987.

....R OBDD. It su#ces to prove termination: We apply TRS rules to a given BDD, until we reach a normal form after a finite number of steps, which is guaranteed by termination. The so derived BDD is the R OBDD. We prove termination by means of a powerful tool, the recursive path ordering (# rpo ) [13, 29]. This is a standard way to extend a (total) well founded order on a set of labels to a (total) well founded order on trees over these labels. To this end, we view guards as labels, ordered by Definition 8, and BDDs are viewed as binary trees, so ITE(g, T 1 , T 2 ) corresponds to the tree g(T 1 , ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1--2):69--115, 1987.

....Traditionally, termination has been investigated in the context of term rewriting systems. Research has focused on developing more and more powerful algorithms which accepted more and more programs as terminating. To this end, increasingly strong term orderings have been described (cf. Dershowitz [16], Steinbach [32] Giesl and Arts [7] The methods of term rewriting are transferable to functional and logic programming as follows: The given (untyped) program is translated into a term rewriting system which then is checked for termination. The advantage of this procedure is that the full ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-- 115, 1987.

....A number of methods have been developed for proving termination of CSR [FR99, GM99, GM01, Luc96, Zan97] they are transformations Theta from TRSs R and replacement maps that produce TRSs R Theta . Then, if we are able to prove termination of R Theta (using the standard methods, see [Der87] for a classic survey) termination of CSR under is ensured for R. Example 2 Consider R and as in Example 1. According to [Luc96] termination of CSR for TRS R under can be proved by proving termination of the following TRS R L : first(0,x) from(x) x) first(s(x) y) y) This ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-115, 1987.

....implemented our results in the automated termination prover AProVE and evaluated them on large collections of examples. 1 Introduction Termination is an essential property of term rewrite systems. Most traditional methods to prove termination of TRSs (automatically) use simplification orders [8, 26], where a term is greater than its proper subterms (subterm property) Examples for simplification orders include lexicographic or recursive path orders [7, 17] the Knuth Bendix order [18] and (most) polynomial orders [20] However, there are numerous important TRSs which are not simply ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--116, 1987.

....R OBDD. It su#ces to prove termination: We apply TRS rules to a given BDD, until we reach a normal form after a finite number of steps, which is guaranteed by termination. The so derived BDD is the R OBDD. We prove termination by means of a powerful tool, the recursive path ordering (# rpo ) [13, 29]. This is a standard way to extend a (total) well founded order on a set of labels to a (total) well founded order on trees over these labels. To this end, we view guards as labels, ordered by Definition 8, and BDDs are viewed as binary trees, so ITE(g,T 1 ,T 2 ) corresponds to the tree g(T 1 ,T 2 ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1--2):69--115, 1987.

....are successful for many special cases. Roughly these methods can be divided into two main types: syntactical methods and semantical methods. In a syntactical method terms are ordered by a careful analysis of the term structure. A well known representative of this type is the recursire path order ([4]) All of these orderings are simplification orderings, i.e. a term is always greater than its proper subterms. An overview and comparison of simplification orderings is given in [14] In a semantical method terms are interpreted in some well known well founded ordered set in such a way that ....

....examples are not claimed to be new but are included for completeness and for illustrating the setting of monotone algebras. A survey of the theory of term rewriting systems can be found in [5] Overviews of existing techniques for termination detection of term rewriting systems can be found in [4, 14]. In the literature termination is also called strong normalization. Term rewriting and termination First we give some standard terminology. Let be a set of operation symbols, each having a fixed arity 0, and let J be a set of variables. Let T( be the set of terms over and An term ....

DERSHOWITZ, N. Termination of rewriting. Journal of Symbolic Computation 3, 1 and 2 (1987), 69-116.

....the scheme can be combined with results from [9, 10] in order to ensure well foundedness of the orders. Many if not all of the orders known in the literature are instances of the schemes presented. As an example we show how four of the most representative path orders, namely recursive path order [3, 4], semantic path order [13] lexicographic path order [13] and Knuth Bendix order [16] fit in the scheme. The rest of the paper is organized as follows. In sec. 2 we introduce some needed notions about Complete Partial Orders and most of the terminology notions on terms that are used through out ....

....within our framework (see [9, 8] for the sake of simplicity and because we believe that dealing with the more complicated cases would not add anything to the understanding of the mechanism, we choose not to treat those cases here. 3.4. 1 rpo The following definition of rpo is due to Dershowitz [4]. Definition 3.30. rpo) Let Xi be a quasi order in the set F . The recursive path order denoted by rpo on the set T (F) is defined as follows: s = f(s 1 ; s k ) rpo g(t 1 ; t m ) t if one of the following conditions holds: 1. s i rpo t, for some i = 1; k; or 2. ....

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Dershowitz, N. Termination of rewriting. Journal of Symbolic Computation 3, 1 and 2 (1987), 69--116.

....are successful for many special cases. Roughly these methods can be divided into two main types: syntactical methods and semantical methods. In a syntactical method terms are ordered by a careful analysis of the term structure. A well known representative of this type is the recursire path order ([5]) All of these orderings are simplification orderings, i.e. a term is always greater than its proper subterms. An overview and comparison of simplification orderings is given in [22] In a semantical method terms are interpreted in some well known well founded ordered set in such a way that ....

....proof is simpler than the existing proofs and gives a stronger result: we prove that the system is even simply terminating. A survey of the theory of term rewriting systems can be found in [6] Overviews of existing techniques for termination detection of term rewriting systems can be found in [5, 22]. In the literature termination is also called strong normalization. 2 Term rewriting and termination First we give some standard terminology. Let be a set of operation symbols, each having a fixed arity 0, and let , be a set of variables. Let T( 2t ) be the set of terms over . and , An ....

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DERSHOWITZ, N. Termination of rewriting. Journal of Symbolic Computation 3, 1 and 2 (19s7), 69-n6.

....in particular termination follows from termination, as was already remarked in [7] For proving termination of TRSs many techniques have been developed. Standard techniques for automatic proving termination include recursive path order and Knuth Bendix order; for overviews we refer to [4, 8]. A more recent approach for an automatic technique is proposed in [1] Often stronger but less automatic techniques are polynomial interpretations ( 3] transformation order ( 2] and semantic labelling ( 10] 3 3 Context sensitive reduction orders and interpretations Let be a replacement ....

Dershowitz, N. Termination of rewriting. Journal of Symbolic Computation 3, 1 and 2 (1987), 69--116.

....to the definition of then (a 0 ; a 1 ; is an infinite descending chain on Tr(A) contradicting well foundedness of . For the only if part we will use the recursive path order , rpo , on trees, based on and with multiset status (for a definition of rpo see for example [2, 3]) given by (a; M) rpo (b; N) 9u 2 M : u rpo (b; N) or (a b) and (8u 2 N : a; M) rpo u) or (a = b) and (M rpo;mul N) Since rpo is well founded whenever is well founded (for a simple proof see [7] we only need to check that rpo . We will prove that for any ....

Dershowitz, N. Termination of rewriting. Journal of Symbolic Computation 3, 1 and 2 (1987), 69--116.

....t by clause (2b) In all cases we are done. 2 4.2 Justi cation of recursive path order In this subsection we prove Theorem 38. As far as we know the rst full treatment was given in [22] which is roughly followed here. Surprisingly, in the literature where recursive path order is introduced ([17, 18]) a characterization as in Theorem 38 is posed as being a recursive de nition, followed by a case analysis for verifying transitivity and irre exivity. However, there it is left unclear what is meant by a multiset lifting or lexicographic lifting of a relation which is not yet known to be ....

....AC are given in [19, 5, 57] 4.4 Knuth Bendix order The order we describe here combines the semantical and the syntactical approaches as we discussed them until now. It is a generalization of the original Knuth Bendix order as described in [43] The idea of such a generalization goes back to [18]. De nition 57 A weakly monotone algebra (A; A ; is a algebra (A; A ) provided with a partial order on A such that every algebra operation is weakly monotone in all arguments. More precisely, for every f 2 and all a 1 ; an , b 1 ; b n 2 A for which a i b i for some ....

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Dershowitz, N. Termination of rewriting. Journal of Symbolic Computation 3, 1 and 2 (1987), 69-116.

....dealt with the base case u = a, and the inductive cases are dealt with by (d) when u = f 1 (s; t) by (b) and (a) when u = f 0 (s; t) Since every term is reducible, by (a) every term is in SN . ut Most standard methods in rewriting fail to prove Lemma 1. In particular, the recursive path ordering [3] cannot deal with rules (7) or (8) when i = 0. In fact, this rewrite system is not simply terminating, and being included in a recursive path ordering implies simple termination. Recall that a rewrite system R is simply terminating if and only if R plus the simplification rules f i (s; t) s, f i ....

....Modify the J K translation so that: w ; A = f A (J 1 K ; J 2 K) where for each atom A there is a new binary commutative function symbol f A . Then the termination of the rewrite system on proofs of Section 3 in this case can be shown by using a multiset path ordering [3] with the precedence f A f B iff A B. This applies since there are only finitely many atoms in any given refutation, therefore is well founded. 4.2 Positive hyperresolution. Let (C) be the set of negative atoms A in C if any, otherwise (C) C. A cut between two parent clauses C; A ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--116, 1987.

....termination have been developed (e.g. path orderings [Pla78, Der82, DH95, Ste95b] Knuth Bendix orderings [KB70, Mar87] semantic interpretations [Lan79, BCL87, BL93, Ste94, Zan94, Gie95b] transformation orderings [BD86, BL90, Ste95a] semantic labelling [Zan95] etc. for surveys see e.g. [Der87, Ste95b]) In this paper we are concerned with the automation of termination proofs for constructor systems (CS for short) Due to the special form of these rewrite systems it is possible to use a different approach for CSs than is necessary for termination of general rewrite systems. Therefore, in this ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1, 2):69-115, 1987.

....of clauses, with an adequate ordering on terms and formulas, and an ordered strategy, there is no possible superposition into the conclusions, neither into premises, so the presentation is saturated, hence consistent. The ordering may be built with a lexicographic path ordering (see for instance [4]) from a precedence including ; succ N. To order formulas, we may ignore the membership and equality relations, map atoms and terms to multisets of sequences, and compare sequences with the multiset extension of the lexicographic extension of . For instance to compare the two atomic formulas ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1 & 2):69--116, 1987.

....orders, Program Analysis, Specialisation and Transformation, Logic Programming, Functional Logic Programming. 1 Introduction The problem of ensuring termination arises in many areas of computer science and a lot of work has been devoted to proving termination of term rewriting systems (e.g. [9 11, 52] and references therein) or of logic programs (e.g. 8, 55] and references therein) It is also an important issue within all areas of program analysis, specialisation and transformation: one usually strives for methods which are guaranteed to terminate. It can also be an issue in model checking ....

....contain V (i.e. v V v ) v V v ) the corresponding strict partial order OE V is a wfo. This property has been exploited in the context of static termination analysis to dynamically construct well founded orders from well quasi ones and led to the initial use of wqo s in the offline setting [9, 10]. The use of well quasi orders in an online setting has only emerged recently (it is mentioned, e.g. in [5] but also [59] and has never been compared to well founded approaches. There has been some comparison between wfo s and wqo s in the offline setting, e.g. in [52] it is argued that (for ....

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-- 116, 1987.

....recursive path orderings. Keywords: Termination, well foundedness, path orderings, Knuth Bendix orderings, calculus, higher order path orderings, graphs, automata. 1 Introduction The use of well founded orderings is a well established technique to show that term rewrite systems terminate [4]. On the other hand, the tradition in calculus circles, exemplified by the Tait Girard technique [10] is to show termination by structural induction on terms, backed by auxiliary well founded inductions. In fact, the recursive path ordering can be proved terminating by structural induction on ....

....auxiliary well founded inductions. In fact, the recursive path ordering can be proved terminating by structural induction on terms, as noticed in [14] Prompted by [12] we wrote a direct inductive proof of the termination of the recursive path ordering rpo based on a well founded precedence [4], which turned out to be surprisingly short. Our point is that this proof generalizes considerably, while remaining short and constructive, and still using only elementary principles of logic. Chasing generalizations and simplifications, we arrived at Theorem 1, which is the core of this paper. We ....

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--116, 1987.

....computation in the system can take place. In general it is undecidable whether a rewrite system terminates, but numerous techniques have been developed that with varying degrees of success may be used to prove termination of unsorted rewrite systems. These techniques include the path orderings [10,2,3], interpretation techniques [12] and not to forget the seminal Knuth and Bendix ordering [11] The dominating method for proving termination of order sorted systems has been to simply ignore sort information, and use the methods developed for unsorted rewriting [7, 5] The problem with this ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation,3:69-- 116, 1987.

....or( list(demodu7558d24 or(x,y) or(y,x) or(x,or(y,z) or(y,or(x,z) end of list. the term or(or(d,b) or(a,c) will be demodulated to or(a,or(b,or(c,d) in several steps) 8.2 LRPO 8.2. 1 Term Ordering (lrpo) The lexicograph c recursive path ordering (lrpo,orrpo with status) [5, 8, 10] is a method for comparing terms. The important theoretical property of lrpo is that it is a termination ordering.Thatis,letR be a set of demodulators in which in each demodulator, the left side is lrpo greater than the right side; then demodulation (applying the demodulators left to right) is ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--116, 1987.

....These works propose transformational methods: termination of CSR for a given TRS R and replacement map is demonstrated by proving termination of a transformed TRS which is obtained from R and . In this way, we can use with CSR the standard methods for proving termination of rewriting (see [Der87] for a survey) We prove that (the two) transformations of [GM99] are also correct for proving innermost termination of CSR. The transformation of [Luc96] is correct for left linear TRSs and replacement maps ensuring that the non variable subterms of the lhs s of the rules are replacing (see ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-115, 1987.

....is bounded then j[L]j denotes the maximum that j j takes on [L] Then we say that L is bounded by l if l j[L]j. A general goal G = L 1 ; L n is called bounded w.r.t. j j if every L i is bounded w.r.t. j j, for all i 2 [1; n] If G is bounded then j[G]j denotes the (finite) multiset (see [Der87]) consisting of the natural numbers j[L 1 ]j; j[L n ]j. The following results hold. Lemma 1.1 ( ApBe91] Let j j be a level mapping and L a bounded literal. Then, for every substitution , L is bounded and j[L ]j j[L]j. Lemma 1.2 ( ApBe91] Let P be acyclic w.r.t. j j. Then, for every ....

N. Dershowitz. Termination of Rewriting. Journal of Symbolic Computation, 3, pp. 69-116, 1987. 27

....of arbitrary terms are ensured to be finite. Imposing a well founded ordering on the terms of a sequence guarantees its finiteness. This application of orderings on terms is based on the tree theorem of Kruskal [Kru60] and later work on the termination of rewriting systems by Dershowitz [Der87] This is not the only method for ensuring strong normalisation, but one of the most powerful [Klo92] The basic idea behind the local control is that computations are arrested when the ordering of the terms is violated in the sequence. In this algorithm, the redexes used in a computation step ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1):69-- 116, 1987.

....l r 2 R and t is a subterm of r with defined root symbol then the rewrite rule l ] t ] is called a dependency pair of R. The set of all dependency pairs of R is denoted by DP(R) In examples we often write F for f ] For instance, consider the following well known one rule TRS R from [8]: f(f(x) f(e(f(x) 1) Here f is defined, e is a constructor, and DP(R) consists of the two dependency pairs F(f(x) F(e(f(x) F(f(x) F(x) An argument filtering [2] for a signature F is a mapping that associates with every n ary function symbol an argument position i 2 f1; ng ....

N. Dershowitz, Termination of Rewriting, Journal of Symbolic Computation 3, pp. 69--116, 1987.

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-115, 1987.

....issues, it is very important to authorize extended matching on these kinds of rules. In the rest of the paper, we always consider flat AC rewriting, so for easier reading, we abbreviate into . 2. 3 Term orderings For general notions about term orderings and termination, we refer to [15]. We briefly recall here what we need. With dependency pair criteria, both quasiorderings and strict orderings are used to compare terms. We follow here the notion of ordering pair [10, 20] although we still call it term ordering for simplicity. Definition 2.7 A term ordering is a pair ( of ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1):69--115, Feb. 1987.

....languages and computational systems. For instance, in term rewriting systems (TRS s) a proof of termination can be achieved if we are able to find a (monotone and stable) well founded ordering on terms (i.e. a reduction ordering) such that l r for every rule l r of the rewrite system [10, 40]. In practice, if we want to implement a tool for proving termination of a TRS R , we need to make this problem decidable. It is well known that termination of TRSs is an undecidable problem, even for TRSs containing only one rule [7] Hence, we can only provide effective approaches (which yield ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1):69--116, 1987.

.... : y) is obtained by removing the non replacing arguments of terms that integrate the rules of R (and by appropriately decreasing the arities of symbols) Termination of R L ensures termination of CSR under for R (see [Luc96] Here, L is terminating: use a recursive path ordering (rpo [Der87, Zan02] with precedence terms : recip; sqr; sqr dbl; s; and first [ Friedman and Wise also use replacement restrictions to provide alternative (more efficient) definitions to logical connectives and, or. In fact, they implement their short cut definitions of these boolean ....

....with left normality, inductive sequentiality, strong sequentiality, etc. on the shape of rules of the TRSs [Ant92, AM96, DM97, HL91, Ken89, O Do77, O Do85, SR93, Toy92] Formal techniques for proving termination are much more general since they usually apply to arbitrary TRSs [AG00, BFR00, Der87] On the other hand, in contrast to termination analysis, checking whether a TRS satisfies the syntactic requirements for applying a given normalizing strategy is usually easy (e.g. with (almost) orthogonality, left normality, inductive sequentiality, etc. In order to formalize the claim that ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-115, 1987.

....under strategy annotations. Termination of CSR has been studied in [GM99,Luc96,Zan97] In these works, termination of CSR for a given TRS is demonstrated by proving termination of a transformed TRS. In this way, with CSR we can use the standard methods for proving termination of rewriting (see [Der87] for a survey) We prove that the (two) transformations of [GM99] are correct for proving the innermost termination of CSR. The transformation of [Luc96] is correct in the cases that we characterize below. Zantema s transformation [Zan97] does not provide correct proofs of innermost termination of ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-115, 1987.

....languages and computational systems. For instance, in term rewriting systems (TRS s) a proof of termination can be achieved if we are able to find a (monotone and stable) well founded ordering on terms (i.e. a reduction ordering) such that l r for every rule l r of the rewrite system [10, 40]. In practice, if we want to implement a tool for proving termination of a TRS R , we need to make this problem decidable. It is well known that termination of TRSs is an undecidable problem, even for TRSs containing only one rule [7] Hence, we can only provide effective approaches (which yield ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1):69--116, 1987.

....which is a contradiction to the induction hypothesis. ut We now prove that the top rules are crucial for L(R; p) s termination behavior. Lemma 13 Let L (R; p) L(R; p) n f(1) 2)g. Then L (R; p) is terminating. Proof. Termination of L (R; p) can be proved by the recursive path order [4] using the precedence active check match proper start f ok found mark for all f 2 [ fX j x 2 V(p)g. ut Before relating L(R; p) and G , we study the connection of L(R; p) and R . Lemma 14 Let t; u 2 T ( Then we have active(t) mark(u) i t R u and top(active(t) ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-116, 1987.

....applications of term rewrite systems (TRSs) termination is an im portant property. A TRS is said to be terminating if it does not allow infinite reductions. Since termination is in general undecidable [HL78] several methods for proving this property have been developed; for surveys see e.g. Der87,Ste95b] Practically all known methods that are amenable to automation use simplifica tion orderings [Der79,Der87,Ste95b,MZ97] However, there exist numerous term rewrite systems for which termination cannot be proved by this kind of orderings. For that reason, Arts and Giesl ....

....if it does not allow infinite reductions. Since termination is in general undecidable [HL78] several methods for proving this property have been developed; for surveys see e.g. Der87,Ste95b] Practically all known methods that are amenable to automation use simplifica tion orderings [Der79,Der87,Ste95b,MZ97] However, there exist numerous term rewrite systems for which termination cannot be proved by this kind of orderings. For that reason, Arts and Giesl [AG97a,AG97b,AG98,AG00,GA01,GAO01] developed the so called dependency pair approach. Given a TRS, the dependency pair technique ....

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1-2):69- 116, 1987.

....implies termination of the original CSRS (i.e. all these transformations are sound ) Direct approaches to termination analysis of CSRSs and transformational approaches both have their advantages. Techniques for proving termination of ordinary term rewriting have been studied extensively (e.g. [21, 22, 7, 3, 30, 31, 1, 4]) and the main advantage of the transformational approach is that in this way, all termination techniques for ordinary TRSs including future developments can be used to infer termination of CSRSs. For instance, the methods of [5, 20] are unable to handle systems like Example 1. Of the ve ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-116, 1987.

....many applications of term rewrite systems (TRSs) termination is an important property. A TRS is said to be terminating if it does not allow in nite reductions. Since termination is in general undecidable [HL78] several methods for proving this property have been developed; for surveys see e.g. [Der87,Ste95b]. Practically all known methods that are amenable to automation use simpli cation orderings [Der79,Der87,Ste95b,MZ97] However, there exist numerous term rewrite systems for which termination cannot be proved by this kind of orderings. For that reason, Arts and Giesl ....

....terminating if it does not allow in nite reductions. Since termination is in general undecidable [HL78] several methods for proving this property have been developed; for surveys see e.g. Der87,Ste95b] Practically all known methods that are amenable to automation use simpli cation orderings [Der79,Der87,Ste95b,MZ97]. However, there exist numerous term rewrite systems for which termination cannot be proved by this kind of orderings. For that reason, Arts and Giesl [AG97a,AG97b,AG98,AG00,GA01,GAO01] developed the so called dependency pair approach. Given a TRS, the dependency pair technique automatically ....

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1-2):69{ 116, 1987.

....that of another, some kinds of transformation of TRSs have been proposed [FZ95] Zan95] Zan94] These transformations are especially useful to show the termination of non simply terminating TRSs, while well established methods based on precedence orderings, e.g. the recursive path ordering (c.f. [Der87]) are 1 effective only to show the termination of simply terminating TRSs. Here, a terminating TRS is said to be simply terminating if its termination can be proved by means of a simplification order, otherwise it is non simply terminating [MZ97] Among the proposed transformations, the ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--116, 1987.

.... proved to be terminating by using Arts and Giesl s dependency pairs technique, i.e. by checking that there is no infinite chain of dependency pairs associated to R 1 [1] It is not difficult to see that more conventional techniques such as Knuth Bendix, polynomial, or recursive path orderings [4] do not apply to this example. Given replacement maps and D , ODR permits reductions on positions that are not replacing (according to ) this is not captured by the transformation of Definition 2. In the following, we extend Giesl and Middeldorp s transformation to deal with ODR. According to ....

....We have provided two transformations that can be used to analyze termination of ODR as termination of rewriting. Our transformations are correct for left linear TRSs. Thus, it is possible to analyze the termination of ODR by using the standard methods for proving termination of rewriting [4]. Moreover, our second transformation is also complete thus showing that proving termination of ODR is equivalent to proving termination of rewriting (for left linear TRSs) We have shown that CSR and ODR can be used to implement and analyze computations with OBJ programs. CSR suffices for ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--115, 1987.

....i.e. TRS s.t. the set of rules that can be applied to a term is always finite. A term is strongly normalizable (SN) if it cannot reduce indefinitely. A rewrite relation is strongly normalizing or terminates if any term is SN. Termination is usually proven with the help of reduction orderings [5] or quasi orderings with dependency pairs. We briefly recall what we need. A term ordering, also known as ordering pair [10] is a pair ( of relations over T (F ; X) such that: is a quasi ordering, i.e. reflexive and transitive; is a strict ordering, i.e. irreflexive and transitive and ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1):69--115, Feb. 1987.

....procedure [27] attempts to transform an equational speci cation into a complete rewrite system. The termination of the procedure cannot be guaranteed and its execution may require human intervention. The diculty stems from the undecidability of whether or not a rewrite system is canonical [9, 22]. For this reason, we do not attempt to convert an equational speci cation in the corresponding complete rewrite system. Rather, we ask speci ers to structure their speci cations as rewrite systems with the above characteristics. The task is eased considerably by two strategies used in designing ....

....In fact, every term has a normal form which obviously contains only constructor symbols because any term containing a de ned operation is reducible. Underspeci cation and overspeci cation are easily checked syntactic properties. However, the termination of a rewrite system is undecidable [9]. In the next section, we discuss syntactic properties sucient to ensure termination and show how to obtain them through our design strategies. 3.3 Design Strategies for Axioms Con uence and sucient completeness are undecidable, although essential, properties of a speci cation. Lack of con uence ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-116, 1987.

....S complete. Completeness of rewriting computations with respect to different semantics have already been studied in the literature. For instance, nf complete TRSs are usually known as weakly normalizing TRSs [BN98] Terminating TRSs are weakly normalizing; hence, proofs of termination of TRSs (see [Der87] for a survey on this topic) can be used for ensuring nf completeness of TRSs. The eval completeness is related to the standard notion of completely defined (CD) TRS s (e.g. see [Han94,HH80] A function symbol is completely defined if it does not occur in any ground term in normal form. A TRS R ....

N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-115, 1987.

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-115, 1987.

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3(1):69-- 116, 1987.

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--116, 1987.

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N. Dershowitz. Termination of Rewriting. Journal of Symbolic Computation, 3(1&2):69--115, 1987.

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N. Dershowitz. Termination of Rewriting. Journal of Symbolic Computation, 3(1&2):69--115, 1987.

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Dershowitz, N., Termination of rewriting, Journal of Symbolic Computation 3 (1987), pp. 69--116.

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N. Dershowitz and Y-J. Lee. Logical Debugging. Journal of Symbolic Computation, 15(5/6):745-- 773, May 1993.

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--116, 1987.

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69--116, 1987.

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 8:69--116, 1987.

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N. Dershowitz. Termination of rewriting. Journal of Symbolic Computation, 3:69-116, 1987.

No context found.

N. Dershowitz. Termination of Rewriting. Journal of Symbolic Computation, 3(1,2):69-116, 1987.

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