Thomas Arts and Jurgen Giesl. Modularity of termination using dependency pairs. In Tobias Nipkow, editor, Proc. 9th Int. Conf. on Rewriting Techniques and Applications (RTA'98), LNCS 1379, pages 226--240, Tsukuba, Japan, 1998. Springer-Verlag.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a suitable ordering, via RPS in particular. This question has been already studied by Arts [1, 4] who proposed to try RPS of a certain form. Thirdly, there is the question of termination of modular or hierarchical rewrite systems, which have been studied by Arts in the case of innermost strategy [3], and seems to be very promising since the hierarchy of rules corresponds to some hierarchy between parts of the dependency graphs. It could be interesting also to study Normalized rewriting [24] which is a special case of hierarchical rewriting. ....

T. Arts and J. Giesl. Modularity of termination using dependency pairs. In Nipkow [27], pages 226--240. 35

....tool will lead us to Sec. 6 and thus to new criteria for modular termination (Thm 1 and Thm 2) Those results provide an incremental proof method illustrated with the complete example of Sec. 7. We shall then discuss in Sec. 8 how this work compares to others, especially to results of Arts Giesl [2] and Dershowitz [6] We will eventually conclude and give ideas on how we intend to apply and extend our framework. 2 Preliminaries We recall usual notions about rewriting [7] and give our notations. A signature F is a finite set of symbols with arities. Let X be a countable set of variables; ....

....1 ) and [F 2 j R 2 ] be such that [F 1 j R 1 ] F 2 j R 2 ] 1. If R 1 is C E strongly normalizing and 2. If there is no infinite dependency chain of [F 2 j R 2 ] over R 1 [ R 2 , Then R 1 [ R 2 is strongly normalizing. 4 We do not consider here the restricted case of innermost termination [2] where usable rules modify the set of dependency pairs. By contradiction: We will assume that there is an infinite dependency chain of R 1 [ R 2 , then we will conclude either on an infinite dependency chain of [F 2 j R 2 ] over R 1 [ R 2 , or on non C E termination of R 1 which contradicts ....

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T. Arts and J. Giesl. Modularity of termination using dependency pairs. In Nipkow [13], pages 226--240.

....another TRS such that the termination of the latter implies that of the former and the latter can be proven terminating more easily. For instance, transformation orderings [BL90,Ste95a] semantic labelling [Zan95] and freezing [Xi98] belong to this category. Also the dependency pair approach [AG97,AG98] can be loosely classified into this category since it transforms a TRS into a set of dependency pairs. There are also various results on modular termination, which basically give the sufficient conditions on two terminating TRSs that imply the termination of their union. The importance of ....

....R 0 1 is the conservative erasure of R 1 . In this way, we have reduced the innermost termination of R to that of R 1 . If R 1 is empty, then we have proven that R is innermost terminating. Clearly, there is no need for splitting R before applying Theorem 1. The following TRS R wt is taken from [AG98] Note that m;n are variables, is the infix operator for cons , for nil and [n] for cons(n; nil) The function weight computes a weighted sum of natural numbers: weight(n 0 : n 1 : Delta Delta Delta : n k : nil) n 0 Sigma k i=1 i n i . 1) sum(s(m) x; n : y) sum(m : ....

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Thomas Arts and Jurgen Giesl. Modularity of termination using dependency pairs. In Tobias Nipkow, editor, Proceedings of the 9th Conference on Rewriting Techniques and Applications, pages 226--240. Springer-Verlag LNCS 1379, 1998.

....[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 automatically gen erates a set of constraints and the existence of a well founded (quasi )ordering satisfying these constraints is sufficient for termination. The advantage is that ....

....pairs corresponds to a cycle in the dependency graph. Hence, the dependency pairs that are not on a cycle in the dependency graph are insignificant for the termination proof. One can prove termination of a TRS in a modular way, by proving absence of infinite chains separately for every cycle [AG98,GAO01] Theorem 2 (Modular termination criterion) A TRS 7 is terminating if and only if for each cycle 7 in the dependency graph there exists no infinite 7 chain of dependency pairs from 7 . This theorem can be refined by narrowing certain dependency pairs [AGO0] Definition 4 (Narrowing) ....

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T. Arts and J. Giesl. Modularity of termination using dependency pairs. In Proceedings of the 9th International Conference on Rewriting Techniques and Applications, RTA-98, volume 1379 of Lecture Notes in Computer Science, pages 226-240, Tsukuba, Japan, 1998. Springer Verlag, Berlin.

....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 generates a set of constraints and the existence of a well founded (quasi )ordering satisfying these constraints is sucient for termination. The advantage is that standard (automatic) ....

....pairs corresponds to a cycle in the dependency graph. Hence, the dependency pairs that are not on a cycle in the dependency graph are insigni cant for the termination proof. One can prove termination of a TRS in a modular way, by proving absence of in nite chains separately for every cycle [AG98,GAO01]. Theorem 2 (Modular termination criterion) A TRS R is terminating if and only if for each cycle P in the dependency graph there exists no in nite R chain of dependency pairs from P. This theorem can be re ned by narrowing certain dependency pairs [AG00] De nition 4 (Narrowing) Let R be a ....

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T. Arts and J. Giesl. Modularity of termination using dependency pairs. In Proceedings of the 9th International Conference on Rewriting Techniques and Applications, RTA-98, volume 1379 of Lecture Notes in Computer Science, pages 226-240, Tsukuba, Japan, 1998. Springer Verlag, Berlin.

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T. Arts and J. Giesl, Modularity of Termination Using Dependency Pairs, in Proc. RTA '98, LNCS 1379, 226-240, 1998.

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T. Arts and J. Giesl, Modularity of Termination Using Dependency Pairs, in Proc. RTA '98, LNCS 1379, 226-240, 1998.

....5. 2 Preliminaries An introduction to term rewrite systems (TRSs) can be found in [4] for example. We first introduce the dependency pair technique. Our presentation combines features of [2, 13, 16] Apart from the presentation, all results stated below are due to Arts and Giesl. We refer to [2, 3] for motivations and proofs. Let R be a (finite) TRS over a signature F . As usual, all root symbols of left hand sides of rewrite rules are called defined, whereas all other function symbols are constructors. Let F ] denote the union of F and ff ] j f is a defined symbol of Rg where f ] has ....

....] for at least one dependency pair l ] t ] 2 C. In the above example, the dependency graph only contains an arrow from F(f(x) F(x) to itself and thus fF(f(x) F(x)g is the only cluster. Hence, with the refinement of Theorem 2 the inequality F(f(x) F(e) is no longer necessary. See [3] for further examples which illustrate the advantages of regarding clusters separately. Note that while in general the dependency graph cannot be computed automatically (since it is undecidable whether t ] 1 oe R l ] 2 oe holds for some oe) one can nevertheless approximate this graph ....

[Article contains additional citation context not shown here]

T. Arts and J. Giesl, Modularity of Termination Using Dependency Pairs, Proc. 9th RTA, Tsukuba, Japan, LNCS 1379, pp. 226--240, 1998.

....5. 2 Preliminaries An introduction to term rewrite systems (TRSs) can be found in [4] for example. We rst introduce the dependency pair technique. Our presentation combines features of [2, 13, 16] Apart from the presentation, all results stated below are due to Arts and Giesl. We refer to [2, 3] for motivations and proofs. Let R 2 be a ( nite) TRS over a signature F . As usual, all root symbols of left hand sides of rewrite rules are called de ned, whereas all other function symbols are constructors. Let F ] denote the union of F and ff ] j f is a de ned symbol of Rg where f ] has ....

....] for at least one dependency pair l ] t ] 2 C. In the above example, the dependency graph only contains an arrow from F(f(x) F(x) to itself and thus fF(f(x) F(x)g is the only cluster. Hence, with the re nement of Theorem 2 the inequality F(f(x) F(e) is no longer necessary. See [3] for further examples which illustrate the advantages of regarding clusters separately. Note that while in general the dependency graph cannot be computed automatically (since it is undecidable whether t ] 1 R l ] 2 holds for some ) one can nevertheless approximate this graph ....

[Article contains additional citation context not shown here]

T. Arts and J. Giesl, Modularity of Termination Using Dependency Pairs, Proc. 9th RTA, Tsukuba, Japan, LNCS 1379, pp. 226-240, 1998.

....been adapted for termination proofs in other areas. In particular, a method for termination proofs of term rewriting systems based on the comparison of arguments (instead of rules) and on our estimation technique can be found in (Arts and Giesl, 1996; Arts and Giesl, 1997a; Arts and Giesl, 1997b; Arts and Giesl, 1998). datei.tex; 13 02 1998; 10:59; p.22 TERMINATION ANALYSIS FOR FUNCTIONAL PROGRAMS 23 4. TERMINATION ANALYSIS FOR PARTIAL FUNCTIONS A functional program P (without mutual recursion) can be written as a sequence h f 1 ; f k i of algorithms, such that no algorithm f i is called by an ....

Arts, T. and J. Giesl: 1998, `Modularity of Termination Using Dependency Pairs'. In: Proc. RTA '98. Tsukuba, Japan. LNCS.

....checking of this criterion enables us to prove innermost termination automatically, even if the TRS is not terminating. Additionally, for several classes of TRSs innermost termination already suffices for termination [35, 36] Moreover, numerous modularity results exist for innermost termination [4, 5, 6, 35, 44, 45], which do not hold for termination. Therefore, for those classes of TRSs termination can be proved by splitting the TRS and proving innermost termination of the subsystems separately. The advantage of this approach is that there are several interesting TRSs where a direct termination proof is not ....

....is therefore well suited to be used for more general rewriting problems, too. For example, the framework of dependency pairs can easily be extended for termination modulo associativity and commutativity [48] Moreover, several well known and new modularity results can be derived in this framework [4, 6]. 5 Examples This collection of examples demonstrates the power of the described method. The majority of them occurred as challenge problems in the literature, whereas the other examples are added to point out specific failures of existing techniques. Sect. 5.1 contains a collection of TRSs where ....

T. Arts and J. Giesl, Modularity of termination using dependency pairs, Technical Report IBN 97/45, TH Darmstadt, Germany, 1997.

....that can be satis ed by a simpli cation ordering and herewith can prove termination of the TRS. This paper describes a tool that implements the dependency pair approach with its most recent additions and re nements, such as modularity results that can e ectively be used on larger TRSs [AG98] and operations on the dependency pairs such as narrowing, rewriting and instantiation of pairs. These re nements all increase the power of the method and turned out to be useful for larger examples from a veri cation case study [GA00] The tool is described from the user s point of view via the ....

T. Arts and J. Giesl. Modularity of termination using dependency pairs. In Proc. of RTA-98, LNCS 1379, pages 226-240, Tsukuba, Japan, March/April 1998. Springer Verlag, Berlin.

....veri cation described above. Moreover, due to the complexity and the safety requirements arising with practical applications in industry, a high degree of automation is desirable for the termination proofs required. These reasons motivate why we chose to apply the dependency pair technique [2,3,5,8] (i.e. the currently most powerful termination proof method that is amenable to automation) However, it turned out that (without further extensions) even the dependency pair technique could not perform the required termination proofs automatically. In Sect. 3 we show that termination problems ....

....cation orderings for automated termination and innermost termination proofs where they were not applicable before. In this section we brie y recapitulate the basic concepts of this approach and we present the theorems that we need for the rest of the paper. For further details and explanations see [3,5,8]. In contrast to the standard approaches for termination proofs, which compare left and right hand sides of rules, we only examine those subterms that are responsible for starting new reductions. For that purpose we concentrate on the subterms in the right hand sides of rules that have a de ned ....

[Article contains additional citation context not shown here]

Arts, T., Giesl, J.: Modularity of termination using dependency pairs. In: Proc. RTA-98, Tsukuba, Japan. LNCS, Vol. 1379, pp. 226-240. Springer 1998

....checking of this criterion enables us to prove innermost termination automatically, even if the TRS is not terminating. Additionally, for several classes of TRSs innermost termination already suffices for termination [27,28] Moreover, numerous modularity results exist for innermost termination [5,7,27,35,36], which do not hold for termination. Therefore, for those classes of TRSs termination can be proved by splitting the TRS and proving innermost termination of the sub29 systems separately. The advantage of this approach is that there are several interesting TRSs where a direct termination proof is ....

....is therefore well suited to be used for more general rewriting problems, too. For example, the framework of dependency pairs can easily be extended for termination modulo associativity and commutativity [39] Moreover, several well known and new modularity results can be derived in this framework [5,7,26]. Acknowledgement We would like to thank Hans Zantema, Aart Middeldorp, Thomas Kolbe, and Bernhard Gramlich for constructive criticism and many helpful comments. This work was partially supported by the Deutsche Forschungsgemeinschaft under grants no. Wa 652 7 1,2 as part of the focus program ....

T. Arts and J. Giesl. Modularity of termination using dependency pairs. In T. Nipkow, editor, Proceedings of the 9th International Conference on Rewriting Techniques and Applications, RTA-98, volume 1379 of Lecture Notes in Computer Science, pages 226--240, Tsukuba, Japan, March/April 1998. Springer Verlag, Berlin.

....very labour intensive [AD99] We describe one of the properties which had to be verified in Sect. 2 and show that it can be represented as a non trivial termination problem of a CTRS. But standard techniques (see e.g. Der87,Ste95,DH95] and even recent advances like the dependency pair technique [AG97a,AG97b,AG98,AG99] could not perform the required termination proof automatically. In Sect. 3 we show that termination problems of CTRSs can be reduced to termination problems of unconditional TRSs. After recapitulating the basic notions of dependency pairs in Sect. 4, we present two important extensions, viz. ....

....orderings for automated termination and innermost termination proofs where they were not applicable before. In this section we briefly recapitulate the basic concepts of this approach and we present the theorems that we need for the rest of the paper. For further details and explanations see [AG97b,AG98,AG99]. In contrast to the standard approaches for termination proofs, which compare left and right hand sides of rules, we only examine those subterms that are responsible for starting new reductions. For that purpose we concentrate on the subterms in the right hand sides of rules that have a defined ....

T. Arts & J. Giesl, Modularity of termination using dependency pairs. In Proc. RTA-98, LNCS 1232, pp. 226--240, Tsukuba, Japan, 1998.

....in our example are f(6)g and f(5) 6)g. In the remainder of the paper, we always restrict ourselves to finite TRSs (and to finite signatures) Then any infinite chain corresponds to a cycle, i.e. it suffices to prove that there is no infinite chain of dependency pairs from any cycle, cf. Arts and Giesl, 1998 ] For an automation of this criterion, we generate a set of inequalities such that the existence of a well founded quasi ordering satisfying these inequalities is sufficient for the absence of infinite chains. As usual, a quasi ordering is a reflexive and transitive relation. The ....

.... . For the sake of brevity, we write instead of ss , i.e. in this paper always denotes the stable strict relation corresponding to . Analogously, we will call a quasi ordering well founded if the corresponding stable strict relation is well founded. The following theorem is from [ Arts and Giesl, 1998 ] where instead of the strict relation corresponding to the quasi ordering we now use the stablestrict relation. Note that the present formulation of Thm. 6 with stablestrict relations is more powerful than the formulation with strict relations. To use the strict relation s of a ....

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T. Arts and J. Giesl. Modularity of Termination Using Dependency Pairs. In Proc. RTA '98, pages 226--240. LNCS 1379, 1998.

....to use this framework in a modular way. Similarly, in Sect. 3 we present a modular approach for innermost termination proofs using dependency pairs. As shown in Sect. 4, these results imply new modularity criteria (which can also be used independently from the dependency pair technique) See [AG97c] for a collection of examples to demonstrate the power of these results. In Sect. 5 we give a comparison with related work and we conclude in Sect. 6. 2 Modular Termination with Dependency Pairs In [AG97a] we introduced the dependency pair technique to prove termination automatically. In this ....

....In this paper we introduced a refinement of the dependency pair approach in order to perform termination and innermost termination proofs in a modular way. This refinement allows automated termination and innermost termination proofs for many TRSs where such proofs were not possible before, cf. [AG97c]. We showed that our new modularity results extend previous results for modularity of innermost termination. Due to the framework of dependency pairs, we also obtain easy proofs for existing modularity theorems. ....

T. Arts & J. Giesl, Modularity of termination using dependency pairs. Tech. Rep. IBN 97/45, TU Darmstadt, 1997. http://www.inferenzsysteme. informatik.tu-darmstadt.de/~reports/notes/ibn-97-45.ps

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Thomas Arts and Jurgen Giesl. Modularity of termination using dependency pairs. In Tobias Nipkow, editor, Proc. 9th Int. Conf. on Rewriting Techniques and Applications (RTA'98), LNCS 1379, pages 226--240, Tsukuba, Japan, 1998. Springer-Verlag.

No context found.

Thomas Arts and Jurgen Giesl. Modularity of termination using dependency pairs. In Tobias Nipkow, editor, Proc. 9th Int. Conf. on Rewriting Techniques and Applications (RTA'98), LNCS 1379, pages 226--240, Tsukuba, Japan, 1998. Springer-Verlag.

No context found.

Thomas Arts and Jrgen Giesl. Modularity of termination using dependency pairs. In Tobias Nipkow, editor, 9th International Conference on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 226240, Tsukuba, Japan, April 1998. SpringerVerlag.

No context found.

Arts, T. and J. Giesl: 1998, `Modularity of Termination Using Dependency Pairs'. In: T. Nipkow (ed.): 9th International Conference on Rewriting Techniques and Applications, Vol. 1379 of Lecture Notes in Computer Science. Tsukuba, Japan, pp. 226--240.

No context found.

Arts, T. and J. Giesl: 1998, `Modularity of Termination Using Dependency Pairs'. In: T. Nipkow (ed.): 9th International Conference on Rewriting Techniques and Applications, Vol. 1379 of Lecture Notes in Computer Science. Tsukuba, Japan, pp. 226--240.

No context found.

T. Arts and J. Giesl. Modularity of termination using dependency pairs. In T. Nopkow, editor, 9th International Conference on Rewriting Techniques and Applications (RTA), LNCS 1379, pages 226--240, Tsukuba, Japan, 1998. Springer-Verlag.

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