A Combination Framework for Complexity

In this paper we present a combination framework for polynomial complexity analysis of term rewrite systems. The framework covers both derivational and runtime complexity analysis. We present generalisations of powerful complexity techniques, notably a generalisation of complexity pairs and (weak) dependency pairs. Finally, we also present a novel technique, called dependency graph decomposition, that in the dependency pair setting greatly increases modularity. We employ the framework in the automated complexity tool TCT. TCT implements a majority of the techniques found in the literature, witnessing that our framework is general enough to capture a very brought setting

A Dependency Pair Framework for AvC-Termination

The development of powerful techniques for proving termination of rewriting modulo a set of equational axioms is essential when dealing with rewriting logic-based programming languages like CafeOBJ, Maude, ELAN, OBJ, etc. One of the most important techniques for proving termination over a wide range of variants of rewriting (strategies) is the dependency pair approach. Several works have tried to adapt it to rewriting modulo associative and commutative (AC) equational theories, and even to more general theories. However, as we discuss in this paper, no appropriate notion of minimality (and minimal chain of dependency pairs) which is well-suited to develop a dependency pair framework has been proposed to date. In this paper we carefully analyze the structure of in nite rewrite sequences for rewrite theories whose equational part is any combination of associativity and/or commutativity axioms, which we call AvC-rewrite theories. Our analysis leads to a more accurate and optimized notion of dependency pairs through the new notion of stably minimal term. We then develop a suitable dependency pair framework for proving termination of AvC-rewrite theories.Alarcón Jiménez, B.; Gutiérrez Gil, R.; Lucas, S.; Meseguer, J. (2011). A Dependency Pair Framework for AvC-Termination. http://hdl.handle.net/10251/1079

Termination of Rewriting with and Automated Synthesis of Forbidden Patterns

We introduce a modified version of the well-known dependency pair framework that is suitable for the termination analysis of rewriting under forbidden pattern restrictions. By attaching contexts to dependency pairs that represent the calling contexts of the corresponding recursive function calls, it is possible to incorporate the forbidden pattern restrictions in the (adapted) notion of dependency pair chains, thus yielding a sound and complete approach to termination analysis. Building upon this contextual dependency pair framework we introduce a dependency pair processor that simplifies problems by analyzing the contextual information of the dependency pairs. Moreover, we show how this processor can be used to synthesize forbidden patterns suitable for a given term rewriting system on-the-fly during the termination analysis.Comment: In Proceedings IWS 2010, arXiv:1012.533

Reducing relative termination to dependency pair problems

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-21401-6_11Relative termination, a generalized notion of termination, has been used in a number of different contexts like proving the confluence of rewrite systems or analyzing the termination of narrowing. In this paper, we introduce a new technique to prove relative termination by reducing it to dependency pair problems. To the best of our knowledge, this is the first significant contribution to Problem #106 of the RTA List of Open Problems. The practical significance of our method is illustrated by means of an experimental evaluation.Germán Vidal is partially supported by the EU (FEDER) and the Spanish Ministerio de Economía y Competitividad under grant TIN2013-44742-C4-R and by the Generalitat Valenciana under grant PROMETEOII201/013. Akihisa Yamadais supported by the Austrian Science Fund (FWF): Y757Iborra, J.; Nishida, N.; Vidal Oriola, GF.; Yamada, A. (2015). Reducing relative termination to dependency pair problems. En Automated Deduction - CADE-25. Springer. 163-178. https://doi.org/10.1007/978-3-319-21401-6_11S163178Alarcón, B., Lucas, S., Meseguer, J.: A dependency pair framework for A $$\vee$$ C-termination. In: Ölveczky, P.C. (ed.) WRLA 2010. LNCS, vol. 6381, pp. 35–51. Springer, Heidelberg (2010)Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000)Arts, T., Giesl, J.: A collection of examples for termination of term rewriting using dependency pairs. Technical report AIB-2001-09, RWTH Aachen (2001)Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Dershowitz, N.: Termination of rewriting. J. Symb. Comput. 3(1&2), 69–115 (1987)Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reasoning 40(2–3), 195–220 (2008)Geser, A.: Relative termination. Dissertation, Fakultät für Mathematik und Informatik, Universität Passau, Germany (1990)Giesl, J., Kapur, D.: Dependency pairs for equational rewriting. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 93–107. Springer, Heidelberg (2001)Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: automatic termination proofs in the dependency pair framework. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 281–286. Springer, Heidelberg (2006)Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reasoning 37(3), 155–203 (2006)Hirokawa, N., Middeldorp, A.: Polynomial interpretations with negative coefficients. In: Buchberger, B., Campbell, J. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 185–198. Springer, Heidelberg (2004)Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 249–268. Springer, Heidelberg (2004)Hirokawa, N., Middeldorp, A.: Decreasing diagrams and relative termination. J. Autom. Reasoning 47(4), 481–501 (2011)Hullot, J.M.: Canonical forms and unification. CADE-5. LNCS, vol. 87, pp. 318–334. Springer, Heidelberg (1980)Iborra, J., Nishida, N., Vidal, G.: Goal-directed and relative dependency pairs for proving the termination of narrowing. In: De Schreye, D. (ed.) LOPSTR 2009. LNCS, vol. 6037, pp. 52–66. Springer, Heidelberg (2010)Kamin, S., Lévy, J.J.: Two generalizations of the recursive path ordering (1980, unpublished note)Klop, J.W.: Term rewriting systems: a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. 32, 143–183 (1987)Koprowski, A., Zantema, H.: Proving liveness with fairness using rewriting. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 232–247. Springer, Heidelberg (2005)Koprowski, A.: TPA: termination proved automatically. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 257–266. Springer, Heidelberg (2006)Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean termination tool 2. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 295–304. Springer, Heidelberg (2009)Lankford, D.: Canonical algebraic simplification in computational logic. Technical report ATP-25, University of Texas (1975)Liu, J., Dershowitz, N., Jouannaud, J.-P.: Confluence by critical pair analysis. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 287–302. Springer, Heidelberg (2014)Nishida, N., Sakai, M., Sakabe, T.: Narrowing-based simulation of term rewriting systems with extra variables. ENTCS 86(3), 52–69 (2003)Nishida, N., Vidal, G.: Termination of narrowing via termination of rewriting. Appl. Algebra Eng. Commun. Comput. 21(3), 177–225 (2010)Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer-Verlag, London (2002)Thiemann, R., Allais, G., Nagele, J.: On the formalization of termination techniques based on multiset orderings. In: RTA 2012. LIPIcs, vol. 15, pp. 339–354. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)Vidal, G.: Termination of narrowing in left-linear constructor systems. In: Garrigue, J., Hermenegildo, M.V. (eds.) FLOPS 2008. LNCS, vol. 4989, pp. 113–129. Springer, Heidelberg (2008)Yamada, A., Kusakari, K., Sakabe, T.: Nagoya termination tool. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 466–475. Springer, Heidelberg (2014)Yamada, A., Kusakari, K., Sakabe, T.: A unified ordering for termination proving. Sci. Comput. Program. (2014). doi: 10.1016/j.scico.2014.07.009Zantema, H.: Termination of term rewriting by semantic labelling. Fundamenta Informaticae 24(1/2), 89–105 (1995)Zantema, H.: Termination. In: Bezem, M., Klop, J.W., de Vrijer, R. (eds.) Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55, pp. 181–259. Cambridge University Press, Cambridge (2003

The 2D Dependency Pair Framework for conditional rewrite systems. Part I: Definition and basic processors

[EN] Different termination properties of conditional term rewriting systems have been recently described emphasizing the bidimensional nature of the termination behavior of conditional rewriting. The absence of infinite sequences of rewriting steps (termination in the usual sense), provides the horizontal dimension. The absence of infinitely many attempts to launch the subsidiary processes that are required to check the rule's condition and perform a single rewriting step has been called V-termination and provides the vertical dimension. We have characterized these properties by means of appropriate notions of dependency pairs and dependency chains. In this paper we introduce a 2D Dependency Pair Framework for automatically proving and disproving all these termination properties. Our implementation of the framework as part of the termination tool MU-TERM and the benchmarks obtained so far suggest that the 2D Dependency Pair Framework is currently the most powerful technique for proving operational termination of conditional term rewriting systems. (C) 2018 Elsevier Inc. All rights reserved.Partially supported by the EU (FEDER), Spanish MINECO project TIN2015-69175-C4-1-R, GV project PROMETEOII/2015/013, and NSF grant CNS 13-19109. Raul Gutierrez is also supported by Juan de la Cierva Fellowship JCI-2012-13528.Lucas Alba, S.; Meseguer, J.; Gutiérrez Gil, R. (2018). The 2D Dependency Pair Framework for conditional rewrite systems. Part I: Definition and basic processors. Journal of Computer and System Sciences. 96:74-106. https://doi.org/10.1016/j.jcss.2018.04.002S741069

The dependency pair framework: Combining techniques for automated termination proofs

Abstract. The dependency pair approach is one of the most powerful techniques for automated termination proofs of term rewrite systems. Up to now, it was regarded as one of several possible methods to prove termination. In this paper, we show that dependency pairs can instead be used as a general concept to integrate arbitrary techniques for termination analysis. In this way, the benefits of different techniques can be combined and their modularity and power are increased significantly. We refer to this new concept as the “dependency pair framework ” to distinguish it from the old “dependency pair approach”. Moreover, this framework facilitates the development of new methods for termination analysis. To demonstrate this, we present several new techniques within the dependency pair framework which simplify termination problems considerably. We implemented the dependency pair framework in our termination prover AProVE and evaluated it on large collections of examples.

Argument filterings and usable rules in higher-order rewrite systems

The static dependency pair method is a method for proving the termination of higher-order rewrite systems a la Nipkow. It combines the dependency pair method introduced for first-order rewrite systems with the notion of strong computability introduced for typed lambda-calculi. Argument filterings and usable rules are two important methods of the dependency pair framework used by current state-of-the-art first-order automated termination provers. In this paper, we extend the class of higher-order systems on which the static dependency pair method can be applied. Then, we extend argument filterings and usable rules to higher-order rewriting, hence providing the basis for a powerful automated termination prover for higher-order rewrite systems