C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proc. 17th CADE, LNAI 1831, pages 346--364, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... 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 relations over T var (F; X) such that: is a quasi ordering, that is reflexive and transitive; is a strict ordering, that is irreflexive and transitive; that is for all ....

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In D. McAllester, editor, 17th International Conference on Automated Deduction, volume 1831 of Lecture Notes in Computer Science, Pittsburgh, PA, USA, June 2000. Springer-Verlag.

....Thm. 11 implies R#EmbD # S #Emb C , i.e. simple termination of R# S. ## The restriction to simple termination excludes many relevant TRSs. Thm. 12 illustrates that the size change principle cannot compete with new techniques (e.g. dependency pairs [1] or monotonic semantic path ordering [4]) where simplification orderings may be applied to non simply terminating TRSs as well. However, these new techniques require methods to generate underlying base orderings. Hence, there is still an urgent need for powerful simplification orderings. Now we clarify the connection between size change ....

....plus(0, y) y is trivial to prove with polynomial orderings or KBO, but the TRS is not size change terminating w.r.t. any reduction pair. 5 Comparison and Combination with Dependency Pairs Now we compare the size change principle with dependency pairs. In contrast to other recent techniques [4, 6], dependency pairs and size change graphs are both built from recursive calls which suggests to combine these approaches to benefit from their respective advantages. We recapitulate the concepts of dependency pairs; see [1, 7, 8] for refinements and motivations. Let f # D be a set of ....

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proc. 17th CADE, LNAI 1831, pages 346--364, 2000.

....together 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 ....

C. Borralleras, M.C.F. Ferreira, and A. Rubio. Complete Monotonic Semantic Path Ordering. In D.A. McAllester, editor, Proc. of 17th International Conference on Automated Deduction, CADE'00, LNCS 1831:346-364, Springer-Verlag, Berlin, 2000.

....on our experiments can be found in the appendix. Thus, the contributions of this paper are also very helpful for other tools based on dependency pairs (e.g. 1] CiME [6] TTT [14, 15] TALP [24] Moreover, we conjecture that they can also be used in other recent approaches for termination [5, 10] which have several aspects in common with dependency pairs. 2 Dependency Pairs We briefly present the dependency pair approach of Arts and Giesl and refer to [2, 11, 12] for refinements and motivations. We assume familiarity with term rewriting (see, e.g. 4] For a TRS over a signature ....

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proc. 17th CADE, LNAI 1831, pages 346--364, 2000.

....size change termination is not a ected when adding F emb . ut The restriction to simple termination excludes many practically relevant TRSs. Thm. 13 illustrates that the size change principle cannot compete with 10 new techniques like dependency pairs [1] or the monotonic semantic path ordering [4] which were developed to handle non simply terminating TRSs as well. However, these new techniques require methods to generate underlying base orderings. Hence, there is still an urgent need for powerful simpli cation orderings. In the remainder of this section, we clarify the connection between ....

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proc. 17th CADE, LNAI 1831, pages 346-364, 2000.

....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 ....

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proceedings of the 17th International Conference on Automated Deduction, volume 1831 of Lecture Notes in Arti cial Intelligence, pages 346-364, 2000.

....z x (y)1 z x (x y)0 z These extended rules are needed to ensure monotonicity with respect to the ACsymbols. But adding them to the system usually makes the termination proof harder. An alternative to the dependency pair method is the use of the monotonic semantic path ordering (MSPO; [BFR00]) which is a monotonic version of the semantic path ordering (SPO; KL80] The SPO generalizes path orderings like Dershowit s recursive path ordering (RPO; Der82] by replacing the use of a precedence by the use of any measure, defined by an underlaying quasi ordering, involving the whole term ....

....argument of s than as argument of t. Hence, after linearization all occurrences of variables in L s;t that are arguments in t have been renamed. under substitutions. 4 The Monotonic AC Semantic Path Ordering Now we will obtain a monotonic version by using the same technique as for the MSPO in [BFR00] but considering AC reduction triplets of the form h I ; Q ; q i. Definition 11. A quasi ordering Q is monotonic wrt. I if s I t implies f( s : Q f( t : for every f 2 F . Note that if f 2 FAC then we may need to apply some flattening. Then h I ; Q ; q i is an ....

[Article contains additional citation context not shown here]

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. Proc. of 17th Int. Conf. on Automated Deduction, LNAI 1831:346--364, 2000.

....z x (y)1 z x (x y)0 z These extended rules are needed to ensure monotonicity with respect to the ACsymbols. But adding them to the system usually makes the termination proof harder. An alternative to the dependency pair method is the use of the monotonic semantic path ordering (MSPO; [BFR00]) which is a monotonic version of the semantic path ordering (SPO; KL80] The SPO generalizes path orderings like Dershowit s recursive path ordering (RPO; Der82] by replacing the use of a precedence by the use of any measure, defined by an underlaying quasi ordering, involving the whole term ....

....of variables in L s;t that are arguments in t have been renamed. 10 Theorem 2. acspo is a well founded AC compatible ordering which is stable under substitutions. 4 The Monotonic AC Semantic Path Ordering Now we will obtain a monotonic version by using the same technique as for the MSPO in [BFR00] but considering AC reduction triplets of the form h I ; Q ; q i. Definition 10. A quasi ordering Q is monotonic wrt. I if s I t implies f( s : Q f( t : for every f 2 F . Note that if f 2 FAC then we may need to apply some flattening. Then, h I ; Q ; q i is an ....

[Article contains additional citation context not shown here]

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. Proc. of 17th Int. Conf. on Automated Deduction, LNAI 1831:346--364, 2000.

....However, RPO has some weaknesses. For instance, since it is a simplification ordering, it can only handle simply terminating systems. Several techniques have been developed for overcoming these weaknesses of RPO. A very recent such technique is the monotonic semantic path ordering (MSPO) [BFR00], a simple and easily automatizable ordering which generalizes other more ad hoc methods. Another recent extension of RPO is its higher order version HORPO [JR99] HORPO is an ordering on terms of a typed lambda calculus generated by a signature of higher order function symbols. Althgough many ....

....in practice, since, although it is well founded, it is not, in general, monotonic. Hence, in order to ensure termination, apart from checking that the rules of the rewrite system are included in the ordering, in addition the monotonicity for contexts of the rewrite rules has to be proved. In [BFR00], a monotonic version of SPO, called MSPO, has been presented. MSPO overcomes the weaknesses of RPO, it is automatable and it is shown to generalize other existing transformation methods. Due to the fact that RPO and SPO share the same path ordering nature , our aim is to obtain for SPO and MSPO ....

[Article contains additional citation context not shown here]

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. Proc. of the 17th International Conference on Automated Deduction, LNAI 1831, pp. 346--364, Pittsburgh, USA, 2000. Springer-Verlag.

....However, RPO has some weaknesses. For instance, since it is a simplification ordering, it can only handle simply terminating systems. Several techniques have been developed for overcoming these weaknesses of RPO. A very recent such technique is the monotonic semantic path ordering (MSPO) [BFR00], a simple and easily automatizable ordering which generalizes other more ad hoc methods. Another recent extension of RPO is its higher order version HORPO [JR99] HORPO is an ordering on terms of a typed lambda calculus generated by a signature of higher order function symbols. Althgough many ....

....in practice, since, although it is well founded, it is not, in general, monotonic. Hence, in order to ensure termination, apart from checking that the rules of the rewrite system are included in the ordering, in addition the monotonicity for contexts of the rewrite rules has to be proved. In [BFR00], a monotonic version of SPO, called MSPO, has been presented. MSPO overcomes the weaknesses of RPO, it is automatable and it is shown to generalize other existing transformation methods. Due to the fact that RPO and SPO share the same path ordering nature , our aim is to obtain for SPO and MSPO ....

[Article contains additional citation context not shown here]

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. Proc. of the 17th International Conference on Automated Deduction, LNAI 1831, pp. 346--364, Pittsburgh, USA, 2000. Springer-Verlag.

....of a typed lambda calculus generated by a signature of polymorphic higher order function symbols [JR99] Many examples can be shown to be terminating using HORPO. However, the HORPO inherits the same weaknesses the recursive path ordering (RPO) has in the first order case. In a recent work [BFR00], an automatable general termination proof method based on a monotonic version of the semantic path ordering (SPO) has been presented. This ordering based method overcomes the weaknesses of RPO and generalizes other existing methods, developed as well to avoid these weaknesses. In this paper ....

....since, although it is well founded, it is not, in general, monotonic. Hence, in order to ensure termination, apart from checking that the rules of the rewrite system are included in the ordering, in addition the monotonicity for contexts of the rewrite rules has to be proved. 2 In a recent work [BFR00], a monotonic version of SPO, called MSPO, has been presented. MSPO overcomes the weaknesses of RPO, is automatable and it is shown to generalize other existing transformation methods. Due to the fact that RPO and SPO share the same path ordering nature , our aim is to obtain for SPO and MSPO ....

[Article contains additional citation context not shown here]

C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In David McAllester, ed., Proc. of the 17th Int. Conf. on Automated Deduction (CADE-17), LNAI 1831, pp. 346--364, Springer-Verlag, 2000.

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C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proc. 17th CADE, LNAI 1831, pages 346--364, 2000.

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C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proc. 17th CADE, pages 346--364, 2000. LNAI 1831.

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C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proc. 17th CADE, LNAI 1831, pages 346--364, 2000.

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C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In Proc. 17th CADE, LNAI 1831, pages 346--364, 2000.

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C. Borralleras, M. Ferreira, and A. Rubio. Complete monotonic semantic path orderings. In D. McAllester, editor, Proc. 17th CADE, LNAI 1831, pages 346--364, Pittsburgh, PA, USA, 2000. http://www-i2.informatik.rwth-aachen.de/AProVE. Our contributions are integrated in AProVE 1.1-beta, which does not yet contain all options of AProVE 1.0. 15

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Borralleras, C., M. Ferreira, and A. Rubio: 2000, `Complete Monotonic Semantic Path Orderings'. In: D. McAllester (ed.): 17th International Conference on Automated Deduction, Vol. 1831 of Lecture Notes in Computer Science. Pittsburgh, PA, USA.

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Borralleras, C., M. Ferreira, and A. Rubio: 2000, `Complete Monotonic Semantic Path Orderings'. In: D. McAllester (ed.): Proceedings of the 17th Conference on Automated Deduction, Vol. 1831 of Lecture Notes in Artificial Intelligence. Pittsburgh, USA, pp. 346--364.

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