Order-sorted Homeomorphic Embedding modulo Combinations of Associativity and/or Commutativity Axioms

[EN] The Homeomorphic Embedding relation has been amply used for defining termination criteria of symbolic methods for program analysis, transformation, and verification. However, homeomorphic embedding has never been investigated in the context of order-sorted rewrite theories that support symbolic execution methods modulo equational axioms. This paper generalizes the symbolic homeomorphic embedding relation to order-sorted rewrite theories that may contain various combinations of associativity and/or commutativity axioms for different binary operators. We systematically measure the performance of different, increasingly efficient formulations of the homeomorphic embedding relation modulo axioms that we implement in Maude. Our experimental results show that the most efficient version indeed pays off in practice.M. Alpuente and S. Escobar have been partially supported by the EU (FEDER) and the Spanish MCIU under grant RTI2018-094403-B-C32, by the Spanish Generalitat Valenciana under grant PROMETEO/2019/098, and by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 952215 (TAILOR). J. Meseguer has been supported by NRL under contract number N00173-17-1-G002. A. Cuenca-Ortega has been supported by the SENESCYT, Ecuador (scholarship program 2013).Alpuente Frasnedo, M.; Cuenca-Ortega, A.; Escobar Román, S.; Meseguer, J. (2020). Order-sorted Homeomorphic Embedding modulo Combinations of Associativity and/or Commutativity Axioms. Fundamenta Informaticae. 177(3-4):297-329. https://doi.org/10.3233/FI-2020-1991S2973291773-

Simplification orders in term rewriting

Thema der Arbeit ist die Anwendung von Methoden der Beweistheorie auf Termersetzungssysteme, deren Termination mittels einer Simplifikationsordnung beweisbar ist. Es werden optimale Schranken für Herleitungslängen im allgemeinen Fall und im Fall der Termination mittels einer Knuth-Bendix-Ordnung (KBO) angegeben. Zudem werden die Ordnungstypen von KBOs vollständig klassifiziert und die unter KBO berechenbaren Funktionen vorgestellt. Einen weiteren Schwerpunkt bildet die Untersuchung der Löngen von Reduktionsketten, die bei einfach terminierenden Termersetzungssysteme auftreten und bestimmten Wachstumsbedingungen genügen

Etude de la stratégie de réécriture de termes k-bornée

We introduce k-bounded term rewriting for linear systems (bo(k), for k integer). This strategy is associated with the class of k-bounded systems LBO(k). We show that the systems in the class LBO (union of the LBO(k) for all k), inverse-preserve recognizability. We show that the problems of termination and inverse-termination for the bo(k) strategy are decidable and use this result to show the decidability of these two problems for subclasses of LBO: the classes of linear systems strongly k-bounded: LFBO(k). The class LFBO (union of the LFBO(k)) includes strictly many known classes: the inverse left-basic systems, the linear growing systems, the linear inverse Finite-Path-Overlapping systems. Membership to LFBO(k) is decidable but this is not hte case for LBO(0). For words, we show that the bo(k) strategy preserves algebricity. We extend k-bounded rewriting to left-linear systems. As in the linear case, we associate a class of systems to the strategy: the class of left-linear kbounded systems BO(k) which extends LBO(k). We show that the systems in BO(k) inversepreserve recognizability. As in the linear case, we define the class of strongly k-bounded systems FBO(k), which extends LFBO(k). Membership to FBO(k) is proved decidable. The FBO class contains stricly the class of left-linear growing systems.Nous introduisons la stratégie de réécriture de termes k-bornée (bo(k), pour k entier) pour les systèmes linéaires. Cette stratégie est associée à une classe de systèmes dits k-bornés LBO(k). Nous démontrons que les systèmes de la classe LBO (union des LBO(k) pour tous les k), inversent-préservent la reconnaissabilité. Nous montrons que les différents problèmes de terminaison et d'inverse-terminaison pour la stratégie bo(k) sont décidables et utilisons ce résultat pour démontrer la décidabilité de ces problèmes pour des sous-classes de LBO: les classes de systèmes linéaires fortement k-bornés: LFBO(k). La classe LFBO (union des LFBO(k)) inclut strictement de nombreuses classes de systèmes connues: les systèmes inverses basiques à gauche, linéaires growing, et linéaires inverses Finite-Path-Overlapping. Le problème de l'appartenance à LFBO(k) est décidable alors qu'il ne l'est pas pour LBO(0). Pour les mots, nous prouvons que la stratégie bo(k) préserve l'algébricité. Nous étendons la notion de réécriture k-bornée aux systèmes de réécriture de termes linéaires à gauche. Comme dans le cas linéaire, nous associons à cette stratégie la classe des systèmes linéaires à gauche k-bornés BO(k) qui étend la classe LBO(k). Nous démontrons que les systèmes de cette classe inverse-préservent la reconnaissabilité.Comme dans le cas linéaire, nous définissons ensuite la classe des systèmes fortement kbornés FBO(k), qui étend la classe LFBO(k). Nous montrons que le problème de l'appartenance à FBO(k) est décidable. La classe FBO contient strictement la classe des systèmes growing linéaires à gauche

Simple Termination is Difficult

A terminating term rewriting system is called simply terminating if its termination can be shown by means of a simplification ordering, an ordering with the property that a term is always bigger than its proper subterms. Almost all methods for proving termination yield, when applicable, simple termination. We show that simple termination is an undecidable property, even for one-rule systems. This contradicts a result by Jouannaud and Kirchner. The proof is based on the ingenious construction of Dauchet who showed the undecidability of termination for one-rule systems. Our results may be summarized as follows: being simply terminating, (non-)selfembedding, and (non-)looping are undecidable properties of orthogonal, variable preserving, one-rule constructor systems. 1. Introduction It is well-known that termination is an undecidable property of term rewriting systems. This result was obtained by Huet and Lankford [9] in 1978. They showed that every Turing machine can be coded as a strin..