D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ac-rewrite systems. In Foundations of Software Technology and Theoretical Computer Science (FSTTCS), volume 472 of Lecture Notes in Computer Science, pages 133--148, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....methods with built in associativity and commutativity (AC, for short) properties for some of the operators are well known to be crucial in theorem proving and programming. Therefore a lot of work has been done on the development of suitable AC compatible reduction or simplification orderings, like [6, 3, 8, 2, 11, 12, 1, 7, 13, 9]. An essential additional property of the ordering that is needed in order to preserve the completeness of most rewrite based theorem proving techniques (modulo AC) is AC totality, i.e. the totality on (AC different) ground terms. Since the initial attempts, it has always been an aim to obtain ....

....better approach seems to be to directly apply an RPO like scheme, treating as the only special case the AC equal top case, that is, when both terms to be compared are headed by the same AC symbol. In this direction the first AC compatible simplification ordering with an RPO scheme was defined in [11] and the first one AC total on ground terms in [9] Other simpler proposals for AC orderings with RPO scheme were given in [14] and in [10] Partially supported by the spanish CICYT project HEMOSS ref. TIC98 0949 C02 01 However, all these AC orderings need to interpret terms (apart from ....

D. Kapur, G. Sivakumar and H. Zhang. A new method for proving termination of AC-rewrite systems. In Conf. Found. of Soft. Technology and Theor. Comput. Science, LNCS 472, pp. 134--148, New Delhi, 1990. Springer-Verlag.

....methods with built in associativity and commutativity (AC) properties for some of the operators are well known to be crucial in theorem proving and programming. Therefore a lot of work has been done on the development of suitable AC compatible reduction or simplification orderings, like [DHJP83,BP85,GL86,BCL87,KSZ90,NR91,Bac92,DP93,RN95,KS97]. An essential additional property of the ordering that is needed in order to preserve the completeness of most rewrite based theorem proving techniques (modulo AC) is AC totality, i.e. the totality on (AC different) ground terms. Since the initial attempts, it has always been an aim to obtain ....

....is, when both terms to be compared are headed by the same AC symbol. In this direction the first AC compatible simplification ordering with an RPO Partially supported by the ESPRIT working group CCL II, ref. WG # 22457 and the CICYT project HEMOSS ref. TIC98 0949 C02 01 scheme was defined in [KSZ90] and the first one AC total on ground terms in [KS97] Other simpler proposals for AC orderings with RPO scheme were given in [Rub97] and in [KS98] However, all these AC orderings need to interpret terms (apart from flattening) in some way, which makes their behaviour less intuitive, unlike it ....

D. Kapur, G. Sivakumar and H. Zhang. A new method for proving termination of AC-rewrite systems. In Conf. Found. of Soft. Technology and Theor. Comput. Science, LNCS 472, pp. 134--148, New Delhi, 1990. Springer-Verlag.

....R is precedence terminating then R=E is terminating. Proof. By de nition there is a well founded order = on F such that root(l) f for every rule l r 2 R and every function symbol f occurring in r. Any AC compatible recursive path order induced by = that is de ned on terms with variables (e.g. [13, 19]) orients the rules of R from left to right. The complicated case in which two terms with equal root symbols in G have to be compared never arises due to the assumption on = We conclude that R=E is terminating. ut 3 Semantic Labelling for Equational Rewriting In this section we present our ....

D. Kapur, G. Sivakumar, and H. Zhang, A New Method for Proving Termination of AC-Rewrite Systems, Proc. 10th FSTTCS, LNCS 472, pp. 133-148, 1990.

....term, which is fb; ag. Paramodulation based theorem proving 53 be quite difficult to find AC compatible reduction orderings, especially when ACtotality is also required. In fact, the first attempts were not total in general (see e.g. Bachmair and Plaisted 1985, Ben Cherifa and Lescanne 1987, Kapur, Sivakumar and Zhang 1990]) The first AC compatible AC total reduction ordering was exhibited in [Narendran and Rusinowitch 1991] while the first such ordering based on RPO appeared in [Rubio and Nieuwenhuis 1995] and further improvements on AC orderings with RPO scheme are developed in [Kapur and Sivakumar 1997, Rubio ....

Kapur D., Sivakumar G. and Zhang H. [1990], A new method for proving termination of acrewrite systems, in `Conf. Found. of Software Technology and Theoretical Computer Science', LNCS 472, Springer-Verlag, New Delhi, India, pp. 134--148.

....of conditional rewrite rules and conditional equivalences like (S6) and (S7) see section 5.4) Last but not least, we have to (F) provide a proper quasi order for the orientation of conditional equations of Pim t as rewrite rules and equivalences. We solve problem (F) by using the ACRPO ordering [13] in section 8. The proof techniques presented in the paper were implemented on the basis of the automated induction procedure [20] In the appendix, we present the proof trace for (L9s) generated by our proof procedure. The proof techniques developed in this paper are rather technical. So we ....

....AC on operator ffi s , so that permits the strict orientation of the axioms of Pim t other than (L3s) and (S7) First we tried the AC extensions of RPOs [17, 22] but the conditions imposed on the underlying precedence did not hold for Pim t . Finally, the ACRPO ordering proposed in [13] turned out to be a suitable one. The only restriction imposed on the precedence in ACRPO is that equivalent operators must have the same status. We present the relevant notions below. 8.1 Basic Notions We consider syntactic quasi ordering on terms which depend on precedences and statuses of ....

[Article contains additional citation context not shown here]

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ACrewrite systems. In 10th Conf. on Foundations of Software Technology and Theoretical Computer Science, volume 472 of LNCS, pages 133--148, 1990.

....Special AC termination orderings are also needed to show termination of AC rewriting systems. Many commonly used orderings, such as recursive path ordering and lexicographic path ordering, are no longer well founded when flattened terms are used. Several AC termination ordering have been devised [DHJP83, BP85, KSZ90]. We study the problem of AC equational reasoning with a different approach. We propose a procedure called consistent unfailing completion in which only consistent rules and equations are used for critical pair generation and rewriting. A consistent unfailing completion procedure can be regarded ....

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ac-rewrite systems. In Proc. of Tenth Conference on Foundations of Software Technology and Theoretical Computer Science, pages 133--148, December 1990. Springer Verlag LNCS 472.

....termination of AC rewriting systems. Many commonly used orderings, such as recursive path ordering and lexicographic path ordering, are no longer well founded when flattened terms are used. Several AC termination ordering have been devised [ Dershowitz et al. 1983; Bachmair and Plaisted, 1985; Kapur et al. 1990 ] We study the problem of AC equational reasoning with a different approach. We propose a procedure called consistent unfailing completion in which only consistent rules and equations are used for critical pair generation and rewriting. A consistent unfailing completion procedure can be ....

....1981 ] However, the input contained many additional clauses that facilitated the proof. Stickel was the first to prove the problem with a natural set of input equations [ Stickel, 1984 ] The proof took over 14 hours. Zhang and Kapur could find a proof in a few minutes using RRL [ Zhang and Kapur, 1990 ] Both [ Stickel, 1984 ] and [ Zhang and Kapur, 1990 ] used approaches based on AC unifications. The use of the cancellation law for additional group was important for them to get the proof efficiently. 6 Discussion In term rewriting systems with AC functions, a term can have an exponential ....

[Article contains additional citation context not shown here]

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ac-rewrite systems. In Proc. of Tenth Conference on Foundations of Software Technology and Theoretical Computer Science, pages 133--148, December 1990. Springer Verlag LNCS 472.

....of the polynomials are again integer polynomials. Surprisingly, it turned out to be rather hard to construct AC compatible reduction orderings by appropriately modifying standard orderings such as recursive path orderings [7] The main idea underlying most proposals in this direction (e.g. [5, 3, 11, 6]) is to apply certain transformations such as flattening to the terms before comparing them with one of the standard path orderings. A major drawback of these approaches is that they impose rather strong restrictions on the precedence orderings on function symbols that may be used. One consequence ....

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of AC-rewrite systems. In Proceedings of the Tenth International Conference of Foundations of Software Technology and Theoretical Computer Science, volume 472 of Springer LNCS, pages 133--148, Berlin, 1990.

....of the polynomials are again integer polynomials. Surprisingly, it turned out to be rather hard to construct AC compatible reduction orderings by appropriately modifying standard orderings such as recursive path orderings [7] The main idea underlying most proposals in this direction (e.g. [5, 3, 12, 6]) is to apply certain transformations such as flattening to the terms before comparing them with one of the standard path orderings. A major drawback of these approaches is that they impose rather strong restrictions on the precedence orderings on function symbols that may be used. One consequence ....

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of AC-rewrite systems. In Proceedings of the Tenth International Conference of Foundations of Software Technology and Theoretical Computer Science, volume 472 of Springer LNCS, pages 133--148, Berlin, 1990.

....interpreting function symbols as multivariate polynomials; see [Dershowitz, 1987] for a survey of the area. In this thesis, we develop a precedence based binary relation for proving termination of extended rewriting, modulo associativity and commutativity. Our ordering was inspired by the one in [Kapur et al. 1990]. Similar research has been reported in [Bachmair and Plaisted, 1985; Bachmair, 1992] and recently in [Delor and Puel, 1993; Rubio and Nieuwenhuis, 1993] 1.2 Outline of the Thesis In Chapter 2 we introduce most of the notations that we are going to use, and briefly provide some historical ....

.... the fact that they do not establish termination in the AC case (see the counterexamples in [Dershowitz et al. 1983] Extensions of path orderings that do handle associative and commutative functions properly ( Bachmair and Plaisted, 1985] for example) have been proposed, most recently in [Kapur et al. 1990]. However, the ordering of [Kapur et al. 1990] is difficult to implement, because it requires many nondeterministic operations (like pseudocopying ; see Section 6.2) In this chapter, we show that if a rewrite system can be proved terminating using the recursive path ordering (RPO) then it is ....

[Article contains additional citation context not shown here]

D. Kapur, G. Sivakumar and H. Zhang. A new method for proving termination of AC-rewrite systems. In Proceedings of the Tenth International Conference of Foundations of Software Technology and Theoretical Computer Science, 1990. Volume 472, pages 133--148, of Lecture Notes in Computer Science, Springer Verlag.

....methods with built in associativity and commutativity(AC) properties for some of the operators are well known to be crucial in theorem proving and programming. Therefore a lot of work has been done on the development of suitable AC compatible reduction or simplification orderings, like [DHJP83, BP85, GL86, BCL87, KSZ90, NR91, Bac92, DP93, RN95, KS97]. An essential additional property of the ordering that is needed in order to preserve the completeness of most rewrite based theorem proving techniques (modulo AC) is AC totality, i.e. the totality on (AC different) ground terms. Since the initial attempts, it has always been an aim to obtain ....

....better approach seems to be to directly apply an RPO like scheme, treating as the only special case the AC equal top case, that is, when both terms to be compared are headed by the same AC symbol. In this direction the first AC compatible simplification ordering with an RPO scheme was defined in [KSZ90] and the first one AC total on ground terms in Partially supported by the ESPRIT working group CCL II, ref. WG # 22457. KS97] Other simpler proposals for AC orderings with RPO scheme were given in [Rub97] and in [KS98] However, all these AC orderings need to interpret terms (apart from ....

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ac-rewrite systems. In Conf. Found. of Software Technology and Theoretical Computer Science, LNCS 472, pages 134--148, New Delhi, India, December 1990. Springer-Verlag.

....of TRS s do not carry over to rewriting modulo equational theories so that the theory developed to study termination of TRS s needs to be adapted to the equational case. Along these lines, a lot of work has been done on the development of suitable AC compatible orderings, as for example [4, 13, 14, 18, 19, 24, 23], exploring the possibilities of adapting the recursive path ordering (RPO) 5, 12] to the AC case. It is well known that the RPO technique can not always be applied for proving termination of TRS s and the same remark applies to AC extensions of it with respect to AC termination. Let us see that ....

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ac-rewrite systems. In Proc. of the 10th Conf. on Foundations of Software Technology and Theoretical Computer Science, volume 472 of LNCS, pages 133 -- 148. Springer, 1990.

....Especially interesting because of its many practical applications is the case when the equational theory is associativity and commutativity (AC) for some of the operators. Therefore a lot of work has been done on the development of suitable AC compatible reduction or simplification orderings, like [DHJP83, BP85, GL86, BCL87, KSZ90, Bac92]. An essential additional property of the ordering that is needed in order to preserve the completeness of most rewrite based theorem proving techniques for first order clauses (modulo AC) is its totality on (AC different) ground terms. Such a total ordering was finally found by Narendran and ....

....of the ordering to the orientation of AC rewrite systems. As shown in example 8.1, in many cases the orientations obtained by our ordering are different from the usual ones. An AC compatible path ordering that does orient the distributivity axiom in the appropriate way is the one given in [KSZ90] (this is also the case using polynomial interpretations [BCL87] But this ordering is not total on ground terms because of its way of comparing terms with the same AC top symbol, since sometimes the subterms can be used only once in a comparison. For instance, if a F b F c then the terms f(a; ....

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ac-rewrite systems. In Conf. Found. of Software Technology and Theoretical Computer Science, LNCS 472, pages 134--148, New Delhi, India, December 1990. Springer-Verlag.

....Klop, 1980; Nipkow, 1991; Takahashi, 1993 ] are quite close, which is encouraging, as it may hint at a canonical framework for higher order rewriting. AC termination Recent work on proving termination of associative commutative rewriting (the most prevalent extension of term rewriting) includes [ Kapur et al. 1990; Rubio and Nieuwenhuis, 1993; Delor and Puel, 1993 ] It would be nice to somehow combine these results in an ordering that could orient distributivity the right way and be total when the precedence is. The ordering in [ Kapur et al. 1990 ] was incorporated in the Rrl system, but most of this ....

....most prevalent extension of term rewriting) includes [ Kapur et al. 1990; Rubio and Nieuwenhuis, 1993; Delor and Puel, 1993 ] It would be nice to somehow combine these results in an ordering that could orient distributivity the right way and be total when the precedence is. The ordering in [ Kapur et al. 1990 ] was incorporated in the Rrl system, but most of this work has yet to filter down into widespread implemented tests that can be used within those rewrite based theorem provers which support associativity and commutativity. Hierarchical systems From the point of view of software engineering, it ....

Deepak Kapur, G. Sivakumar, and Hantao Zhang. A new method for proving termination of AC-rewrite systems. In Proceedings of the Tenth International Conference of Foundations of Software Technology and Theoretical Computer Science, volume 472 of Lecture Notes in Computer Science, pages 133--148, Berlin, 1990. SpringerVerlag.

....methods with built in associativity and commutativity (AC) properties for some of the operators are well known to be crucial in theorem proving and programming. Therefore a lot of work has been done on the development of suitable AC compatible reduction or simplification orderings, like [DHJP83, BP85, GL86, BCL87, KSZ90, NR91, Bac92, DP93, RN95, KS97]. An essential additional property of the ordering that is needed in order to preserve the completeness of most rewrite based theorem proving techniques (modulo AC) is AC totality, i.e. the totality on (AC different) ground terms. Since the initial attempts, it has always been an aim to obtain ....

....t i for all i. f = g 2 AC and : AC equal top case where args AC is the extension (e.g. multiset or lexicographic) of AC we use to compare recursively the arguments of non AC function symbols. Hence the definition of two such AC compatible simplification orderings with RPO scheme in [KSZ90] and [KS97] has been an important step forwards. However, in our opinion further improvements are still needed in order to obtain AC orderings that are as useful as RPO is for the standard case. The ordering [KSZ90] is not AC total and it has a rather complicated AC equal top case, while RPO has a ....

[Article contains additional citation context not shown here]

D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ac-rewrite systems. In Conf. Found. of Software Technology and Theoretical Computer Science, LNCS 472, pages 134--148, New Delhi, India, December 1990. Springer-Verlag.

....a study of termination orderings has been an important research area of rewriting techniques. In RRL, rules can be made from equations manually or using an algorithm for lexicographic recursive path ordering (lrpo) 8] and the associative commutative lexicographic recursive path ordering (aclrpo) [34]. Both lrpo and aclrpo extend a precedence relation on function symbols to terms. Whenever an equation cannot be oriented into a terminating rewrite rule by the algorithm using an existing precedence relation, RRL presents the user with various options: ffl postpone the equation for consideration ....

Kapur, D., Sivakumar, G., and Zhang, H., A new method for proving termination of ACrewrite systems. In: Proc. Tenth Conference on Foundations of Software Technology and Theoretical Computer Science, Bangalore, India, 1990.

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D. Kapur, G. Sivakumar, and H. Zhang. A new method for proving termination of ac-rewrite systems. In Foundations of Software Technology and Theoretical Computer Science (FSTTCS), volume 472 of Lecture Notes in Computer Science, pages 133--148, 1990.

No context found.

D. Kapur, G. Sivakumar, and H. Zhang. A New Method for Proving Termination of AC-Rewrite Systems. To be presented at the Conference on the Foundations of Software Technology and Theoretical Computer Science, New Delhi, India, December 1990.

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