A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175:127-158, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 automatically ....

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175:127-158, 1997.

....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 ....

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175:127-158, 1997.

....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 ....

....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 simply terminating rewrite systems) approaches based upon quasi orders are less interesting than ones based upon a partial orders. In this paper we will show that the situation is somewhat reversed in an online setting. Furthermore, in the online setting, transitivity of ....

[Article contains additional citation context not shown here]

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175(1):127--158, 1997.

....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 semantic labelling by Zantema [Zan95] is powerful enough to resolve the non simple termination of a TRS into the simple termination of another TRS [MOZ96] Transformations by the semantic labelling, however, can not be proceeded automatically (we have to ....

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175(1):127--158, 1997.

....to find a wellfounded ordering such that for all rules of the TRS the left hand sides are greater than the corresponding right hand sides. In most practical applications the synthesized orderings are total on ground terms [25] and therefore virtually all orderings used are simplification orderings [17, 18, 50, 55]. However, numerous TRSs are not simply terminating, i.e. not compatible with a simplification ordering. Hence, standard techniques like the recursive path ordering, polynomial interpretations, and the Knuth Bendix ordering fail in proving termination of these TRSs. In Sect. 2 we introduce a new ....

A. Middeldorp and H. Zantema, Simple termination of rewrite systems, Theoret. Comput. Sci. 175 (1997) 127--158.

....find a well founded ordering such that for all rules of the TRS the left hand sides are greater than the corresponding right hand sides. In most practical applications the synthesized orderings are total on ground terms [20] and therefore virtually all orderings used are simplification orderings [15,16,41,45]. However, numerous TRSs are not simply terminating, i.e. not compatible with a simplification ordering. Hence, standard techniques like the recursive path ordering, polynomial interpretations, and the Knuth Bendix ordering fail in proving termination of these TRSs. In Sect. 2 we introduce a new ....

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175:127--158, 1997.

....formal proof methods in functional programming or orderings of rewriting systems. For instance we mention polynomial interpretations [3, 9, 21] recursive path orderings [14] and Knuth Bendix orderings [7, 12] The latter methods are characterized by orderings called simplification orderings [5, 18] and deal with the termination of functions called simply terminating functions. Some functions that are non simply terminating can be proven to terminate with methods based on structural inductive proofs because they focus on recursive functions which can be viewed as sorted constructor systems ....

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science 175, 127-158, 1997.

....is quasisimplifying according to Proposition 16. In order to rectify [ALS94, Lemma 3.1] it is sufficient to replace reduction order with simplification order . This yields the third sufficient condition for quasi simplifyingness. 1 This is true because we consider finite signatures only; see [MZ97] for details on infinite signatures. Proposition 17. Let (F ; R) be a deterministic 3 CTRS. If its backward substituted system (F [ fcg; R) is simplifying, then (F ; R) is quasi simplifying. Proof. If (F [ fcg; R) is simplifying, then there is a simplification order such that l r and l s i ....

....a certain class of CTRSs if, for all CTRSs (F 1 ; R 1 ) and (F 2 ; R 2 ) belonging to that class and having property P , their union (F 1 [ F 2 ; R 1 [ R 2 ) also belongs to that class and has the property P . It is well known that simple termination is modular for constructor sharing TRSs; see [KO92,MZ97]. It readily follows from Lemma 14 that simplifyingness is also modular for finite constructor sharing 1 CTRSs. Therefore, if quasisimplifyingness of two constructor sharing 3 CTRSs R 1 and R 2 can be shown by Proposition 17, then R 1 [ R 2 is also quasi simplifying. This is because the ....

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175(1):127--158, 1997.

....of the modularity results are often not applicable in practice. For example, collapsing and duplicating rules occur naturally in most TRSs. In contrast to this, since most methods for automated termination proofs work with so called simplification orderings [ Dershowitz, 1987; Steinbach, 1995; Middeldorp and Zantema, 1997 ] Kurihara and Ohuchi s [ 1992 ] result for constructor sharing systems is thus of practical relevance. They showed that the combination of finite simply terminating TRSs (systems whose termination can be verified by a simplification ordering) is again simply terminating. Their result was ....

....(systems whose termination can be verified by a simplification ordering) is again simply terminating. Their result was extended to composable systems [ Ohlebusch, 1995 ] and to certain hierarchical combinations [ Krishna Rao, 1994 ] Moreover, all these results also hold for infinite TRSs; see [ Middeldorp and Zantema, 1997 ] However, there are numerous relevant TRSs where simplification orderings fail in proving termination. For that purpose, a new technique for automated termination proofs, viz. the so called dependency pair approach, was developed by Arts and Giesl [ 1997a; 1997b; 1997c; 1998 ] Given a TRS, ....

[Article contains additional citation context not shown here]

A. Middeldorp and H. Zantema. Simple Termination of Rewrite Systems. Theoretical Computer Science, 175:127--158, 1997.

....of R does not imply termination of R q . On the other hand, it is easy to see that termination of R q is a sufficient criterion for quasi reductivity of R (in this case R is quasi reductive w.r.t. the reduction order Rq ) 1 This is true because we consider finite signatures only; see [MZ97] for details on infinite signatures. Proposition 18. If U(R) is simply terminating, then R is quasi simplifying. Proof. By Lemma 13, U(R) Emb(F 0 ) is terminating. We show Rq[Emb(F) U(R) Emb(F 0 ) The proposition then follows from Proposition 16. If s Emb(F) t, then s Emb(F ....

A. Middeldorp and H. Zantema. Simple Termination of Rewrite Systems. Theoretical Computer Science 175(1), pages 127--158, 1997.

....Analysis, Specialisation and Transformation, Logic Programming, Functional Logic Programming, Metaprogramming. 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. [14, 15, 16, 60] and references therein) or of logic programs (e.g. 5, 12, 64] and references therein) It is also an important issue within all areas of automatic program analysis, synthesis, specialisation and transformation: Part of the work was done while the author was Post doctoral Fellow of the Fund ....

....in the offline setting [14, 15] The use of well quasi orders in an online setting has only emerged recently (it is mentioned, e.g. in [6] but also [69] 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 [60] it is argued that (for simply terminating rewrite systems) approaches based upon quasi orders are less interesting than ones based upon partial orders. In this paper we will show that the situation is somewhat reversed in an online setting. Furthermore, in the online setting, transitivity of a ....

[Article contains additional citation context not shown here]

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175(1):127--158, 1997.

....subsystems with disjoint signatures. Therefore, partitions into subsystems which may at least have constructors This work was partially supported by the Deutsche Forschungsgemeinschaft under grants no. Wa 652 7 1,2 as part of the focus program Deduktion . in common have also been considered [KO92, MT93, Gra95, MZ97]. Nevertheless, in practice these results often cannot be applied for automated termination proofs, either. For example, many systems are hierarchical combinations of TRSs that do not only share constructors, but where one subsystem contains defined symbols of the other subsystem. Termination is ....

A. Middeldorp & H. Zantema, Simple termination of rewrite systems. TCS, 175:127--158, 1997.

.... that both parts are complete and have disjoint sets of defined symbols [MT93] This result can also be generalized to overlay systems [Gra95] Simple termination is modular for TRSs with shared constructors and disjoint defined symbols [KO92] and this result can be extended to composable TRSs [MZ97]. Nevertheless, in practice these results often cannot be applied for automated termination proofs. For example, many systems are hierarchical combinations of TRSs that have not only constructors in common, but where one subsystem contains defined symbols of the other subsystem. Termination is ....

A. Middeldorp & H. Zantema, Simple termination of rewrite systems. Theoretical Computer Science, 175:127--158, 1997.

....20 A TRS is called simply terminating if it admits a compatible simple well founded monotone algebra. 18 The above propositions state that for nite the well foundedness condition can be removed without changing this de nition. For in nite this is not true as we saw in Example 10. In [52, 53] an alternative more complicated de nition of simple termination is given, which coincides with the de nition given here for nite signatures, and also implying well foundedness in case of in nite signatures. The following proposition gives a characterization of simple termination not ....

....cation order is a reduction order; by Proposition 3 then R [ Emb( is terminating, concluding the proof. 2 As we saw in Example 10 this proposition does not extend directly to in nite signatures. A rst investigation of this kind of problems for in nite signatures has been given in [54] In [52, 53] the notion of simple termination is revisited in such a way that all these notions coincide, and Proposition 19 can be generalized to in nite signatures. For nite signatures modularity of simple termination has been proved in [45] 3.2 Total termination If 1 2 for two well founded orders ....

[Article contains additional citation context not shown here]

Middeldorp, A., and Zantema, H. Simple termination of rewrite systems. Theoretical Computer Science 175 (1997), 127-158.

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A. Middeldorp, H. Zantema, Simple Termination of Rewrite Systems, Theoretical Computer Science 175, pp. 127-158, 1997.

....signatures, but which is also well founded over infinite signatures and covers orders like the recursive path order. We investigate the properties of the ensuing class of simply terminating systems. This paper is a completely revised and extended version of [32] A short abstract appeared in [33]. The first author is partially supported by the Grant in Aid for Scientific Research (C) 06680300 and the Grant in Aid for Encouragement of Young Scientists 06780229 of the Ministry of Education, Science and Culture of Japan. 1 Introduction One of the main problems in the theory of term ....

A. Middeldorp and H. Zantema, Simple Termination of Rewrite Systems, Bulletin of the Section of Logic 24 (1995) 31--36.

.... R C[toe] for some term t, context C, and substitution oe. A TRS R is called cyclic if it admits a reduction t R t for some term t. A TRS R is called self embedding if it admits a reduction t R u Emb t for some terms t, u. Recent investigations of these notions include [4, 5, 17, 20, 21, 24]. Validity of most of the implications in the termination hierarchy is direct from the definitions; only TT ) ST requires some well known argument, see e.g. 21] and NSE ) SN requires Kruskal s theorem. None of the implications are equivalences: for all implications X ) Y in the termination ....

....hierarchy a TRS exists satisfying Y but not X. For infinite TRSs over infinite signatures the termination hierarchy is more complicated: if the notion of embedding is not changed then NSE ) SN does not hold any more, if the notions of embedding and simple termination are adjusted as motivated in [17], then the implication TT ) ST no longer holds ( 17] In this paper however we consider only finite TRSs over finite signatures. All TRSs needed for the termination hierarchy are modifications of two basic TRSs parameterized by an arbitrary PCP instance P . For any string ff = a 1 a 2 : an 2 ....

[Article contains additional citation context not shown here]

A. Middeldorp and H. Zantema, Simple Termination of Rewrite Systems, Theoretical Computer Science 175 (1997) 127--158.

....t, some context C and some substitution oe. A TRS R is called cyclic if it admits a reduction t R t for some term t. A TRS R over a signature F is called self embedding if it admits a reduction t R u Emb(F) t for some terms t, u. Recent investigations of these notions include [5, 7, 8, 14, 19]. For the proofs we use Post s Correspondence Problem (PCP) which can be described as follows: given a finite alphabet Gamma and a finite set P ae Gamma Theta Gamma , is there some natural number n 0 and (ff i ; fi i ) 2 P for i = 1; n such that ff 1 ff 2 Delta Delta ....

A. Middeldorp and H. Zantema. Simple termination of rewrite systems. Theoretical Computer Science, 175, 1997. To appear.

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