Polygraphs for termination of left-linear term rewriting systems

We present a methodology for proving termination of left-linear term rewriting systems (TRSs) by using Albert Burroni's polygraphs, a kind of rewriting systems on algebraic circuits. We translate the considered TRS into a polygraph of minimal size whose termination is proven with a polygraphic interpretation, then we get back the property on the TRS. We recall Yves Lafont's general translation of TRSs into polygraphs and known links between their termination properties. We give several conditions on the original TRS, including being a first-order functional program, that ensure that we can reduce the size of the polygraphic translation. We also prove sufficient conditions on the polygraphic interpretations of a minimal translation to imply termination of the original TRS. Examples are given to compare this method with usual polynomial interpretations.Comment: 15 page

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Synthesis of sup-interpretations: a survey

In this paper, we survey the complexity of distinct methods that allow the programmer to synthesize a sup-interpretation, a function providing an upper- bound on the size of the output values computed by a program. It consists in a static space analysis tool without consideration of the time consumption. Although clearly related, sup-interpretation is independent from termination since it only provides an upper bound on the terminating computations. First, we study some undecidable properties of sup-interpretations from a theoretical point of view. Next, we fix term rewriting systems as our computational model and we show that a sup-interpretation can be obtained through the use of a well-known termination technique, the polynomial interpretations. The drawback is that such a method only applies to total functions (strongly normalizing programs). To overcome this problem we also study sup-interpretations through the notion of quasi-interpretation. Quasi-interpretations also suffer from a drawback that lies in the subterm property. This property drastically restricts the shape of the considered functions. Again we overcome this problem by introducing a new notion of interpretations mainly based on the dependency pairs method. We study the decidability and complexity of the sup-interpretation synthesis problem for all these three tools over sets of polynomials. Finally, we take benefit of some previous works on termination and runtime complexity to infer sup-interpretations.Comment: (2012

Decreasing Diagrams for Confluence and Commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

Termination orders for 3-dimensional rewriting

This paper studies 3-polygraphs as a framework for rewriting on two-dimensional words. A translation of term rewriting systems into 3-polygraphs with explicit resource management is given, and the respective computational properties of each system are studied. Finally, a convergent 3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In order to prove these results, it is explained how to craft a class of termination orders for 3-polygraphs.Comment: 30 pages, 35 figure

The 2D Dependency Pair Framework for Conditional Rewrite Systems¿Part II: Advanced Processors and Implementation Techniques

[EN] Proving termination of programs in `real-life¿ rewriting-based languages like CafeOBJ, Haskell, Maude, etc., is an important subject of research. To advance this goal, faithfully cap- turing the impact in the termination behavior of the main language features (e.g., conditions in program rules) is essential. In Part I of this work, we have introduced a 2D Dependency Pair Framework for automatically proving termination properties of Conditional Term Rewriting Systems. Our framework relies on the notion of processor as the main practical device to deal with proofs of termination properties of conditional rewrite systems. Processors are used to decompose and simplify the proofs in a divide and conquer approach. With the basic proof framework defined in Part I, here we introduce new processors to further improve the abil- ity of the 2D Dependency Pair Framework to deal with proofs of termination properties of conditional rewrite systems. We also discuss relevant implementation techniques to use such processors in practice.Partially supported by the EU (FEDER) and projects RTI2018-094403-B-C32, PROMETEO/2019/098, SP20180225. Jose Meseguer was supported by grants NSF CNS 13-19109 and NRL N00173-17-1-G002. Salvador Lucas' research was partly developed during a sabbatical year at the UIUC.Lucas Alba, S.; Meseguer, J.; Gutiérrez Gil, R. (2020). The 2D Dependency Pair Framework for Conditional Rewrite Systems¿Part II: Advanced Processors and Implementation Techniques. Journal of Automated Reasoning. 64(8):1611-1662. https://doi.org/10.1007/s10817-020-09542-3S16111662648Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000)Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with MU-TERM. In: Proceedings of AMAST’10, LNCS, vol. 6486, pp. 201–208 (2011)Baader, F., Nipkow, T.: Term Rewriting and all That. Cambridge University Press, Cambridge (1998)Barwise, J.: An introduction to first-order logic. In: Barwise, J. (ed.) Handbook of Mathematical Logic. North-Holland, Amsterdam (1977)Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.: All About Maude—A High-Performance Logical Framework. LNCS 4350, Springer, New York (2007)Contejean, E., Marché, C., Tomás, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. J. Autom. Reason. 34(4), 325–363 (2006)Dershowitz, N.: A note on simplification orderings. Inf. Process. Lett. 9(5), 212–215 (1979)Durán, F., Lucas, S., Meseguer, J.: MTT: the Maude termination tool (system description). In: Proceedings of IJCAR’08, LNAI, vol. 5195, pp. 313–319 (2008)Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reason. 40(2–3), 195–220 (2008)Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: Automatic Termination proofs in the dependency pair framework. In: Proceeding of IJCAR’06, LNAI, vol. 4130, pp. 281–286 (2006)Giesl, J., Thiemann, R., Schneider-Kamp, P.: The dependency pair framework: combining techniques for automated termination proofs. In: Proceedings of LPAR’04, LNAI, vol. 3452, pp. 301–331 (2004)Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reason. 37(3), 155–203 (2006)Goguen, J., Meseguer, J.: Models and equality for logical programming. In: Proceedings of TAPSOFT’87, LNCS, vol. 250, pp. 1–22 (1987)Gutiérrez, R., Lucas, S.: Automatic generation of logical models with AGES. In: Proceedings of CADE 2019, LNCS, vol. 11716, pp. 287–299 (2019). Tool page: http://zenon.dsic.upv.es/ages/Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: Proceedings of RTA’04, LNCS, vol. 3091, pp. 249–268 (2004)Hodges, W.: Elementary predicate logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 1, pp. 1–131. Reidel Publishing Company, Dordrecht (1983)Lankford, D.S.: On proving term rewriting systems are noetherian. Technical Report, Louisiana Technological University, Ruston, LA (1979)Lucas, S.: Using Well-founded relations for proving operational termination. J. Autom. Reason. to appear (2020). https://doi.org/10.1007/s10817-019-09514-2Lucas, S., Gutiérrez, R.: Automatic synthesis of logical models for order-sorted first-order theories. J. Autom. Reason. 60(4), 465–501 (2018)Lucas, S., Gutiérrez, R.: Use of logical models for proving infeasibility in term rewriting. Inf. Process. Lett. 136, 90–95 (2018)Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inf. Process. Lett. 95, 446–453 (2005)Lucas, S., Meseguer, J.: Models for logics and conditional constraints in automated proofs of termination. In: Proceedings of AISC’14, LNAI, vol. 8884, pp. 9–20 (2014)Lucas, S., Meseguer, J.: 2D Dependency pairs for proving operational termination of CTRSs. In: Escobar, S., (ed) Proceedings of the 10th International Workshop on Rewriting Logic and its Applications, WRLA’14, LNCS, vol. 8663, pp. 195–212 (2014)Lucas, S., Meseguer, J.: Dependency pairs for proving termination properties of conditional term rewriting systems. J. Log. Algebr. Methods Program. 86, 236–268 (2017)Lucas, S., Meseguer, J.: Normal forms and normal theories in conditional rewriting. J. Log. Algebr. Methods Program. 85(1), 67–97 (2016)Lucas, S., Meseguer, J., Gutiérrez, R.: Extending the 2D DP framework for conditional term rewriting systems. In: Selected Papers from LOPSTR’14, LNCS, vol. 8981, pp. 113–130 (2015)Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems. Part I: Definition and basic processors. J. Comput. Syst. Sci. 96, 74–106 (2018)McCune, W.: Prover9 & Mace4. http://www.cs.unm.edu/~mccune/prover9/ (2005–2010)Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, New York (2002)Schernhammer, F., Gramlich, B.: Characterizing and proving operational termination of deterministic conditional term rewriting systems. J. Log. Algebr. Program. 79, 659–688 (2010)Sternagel, T., Middeldorp, A.: Conditional confluence (system description). In: Proceedings of RTA-TLCA’14, LNCS, vol. f8560, pp. 456–465 (2014)Sternagel, T., Middeldorp, A.: Infeasible conditional critical pairs. In: Proceedings of IWC’15, pp. 13–18 (2014)Thiemann, R.: The DP Framework for Proving Termination of Term Rewriting. PhD Thesis, RWTH Aachen, Technical Report AIB-2007-17 (2007)Thiemann, R., Giesl, J., Schneider-Kamp, P.: Improved modular termination proofs using dependency pairs. In: Proceedings of IJCAR’04, LNAI, vol. 3097, pp. 75–90 (2004)Wang, H.: Logic of many-sorted theories. J. Symb. Log. 17(2), 105–116 (1952

Termination of rewrite relations on λ\lambda-terms based on Girard's notion of reducibility

In this paper, we show how to extend the notion of reducibility introduced by Girard for proving the termination of $\beta$-reduction in the polymorphic $\lambda$-calculus, to prove the termination of various kinds of rewrite relations on $\lambda$-terms, including rewriting modulo some equational theory and rewriting with matching modulo $\beta$$\eta$, by using the notion of computability closure. This provides a powerful termination criterion for various higher-order rewriting frameworks, including Klop's Combinatory Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems

Matrix Interpretations on Polyhedral Domains

We refine matrix interpretations for proving termination and complexity bounds of term rewrite systems we restricting them to domains that satisfy a system of linear inequalities. Admissibility of such a restriction is shown by certificates whose validity can be expressed as a constraint program. This refinement is orthogonal to other features of matrix interpretations (complexity bounds, dependency pairs), but can be used to improve complexity bounds, and we discuss its relation with the usable rules criterion. We present an implementation and experiments

Compression of Rewriting Systems for Termination Analysis

We adapt the TreeRePair tree compression algorithm and use it as an intermediate step in proving termination of term rewriting systems. We introduce a cost function that approximates the size of constraint systems that specify compatibility of matrix interpretations. We show how to integrate the compression algorithm with the Dependency Pairs transformation. Experiments show that compression reduces running times of constraint solvers, and thus improves the power of automated termination provers