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

Certifying Higher-Order Polynomial Interpretations

Higher-order rewriting is a framework in which one can write higher-order programs and study their properties. One such property is termination: the situation that for all inputs, the program eventually halts its execution and produces an output. Several tools have been developed to check whether higher-order rewriting systems are terminating. However, developing such tools is difficult and can be error-prone. In this paper, we present a way of certifying termination proofs of higher-order term rewriting systems. We formalize a specific method that is used to prove termination, namely the polynomial interpretation method. In addition, we give a program that processes proof traces containing a high-level description of a termination proof into a formal Coq proof script that can be checked by Coq. We demonstrate the usability of this approach by certifying higher-order polynomial interpretation proofs produced by Wanda, a termination analysis tool for higher-order rewriting

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

MU-TERM: Verify Termination Properties Automatically (System Description)

[EN] We report on the new version of mu-term, a tool for proving termination properties of variants of rewrite systems, including conditional, context-sensitive, equational, and order-sorted rewrite systems. We follow a unified logic-based approach to describe rewriting computations. The automatic generation of logical models for suitable first-order theories and formulas provide a common basis to implement the proofs.Supported by EU (FEDER), and projects RTI2018-094403-B-C32,PROMETEO/ 2019/098, and SP20180225. Also by INCIBE program "Ayudas para la excelencia de los equipos de investigación avanzada en ciberseguridad" (Raul Gutiérrez).Gutiérrez Gil, R.; Lucas Alba, S. (2020). MU-TERM: Verify Termination Properties Automatically (System Description). Springer Nature. 436-447. https://doi.org/10.1007/978-3-030-51054-1_28S436447Alarcón, B., et al.: Improving context-sensitive dependency pairs. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 636–651. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-89439-1_44Alarcón, B., Gutiérrez, R., Lucas, S.: Context-sensitive dependency pairs. Inf. Comput. 208(8), 922–968 (2010). https://doi.org/10.1016/j.ic.2010.03.003Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with mu-term. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 201–208. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-17796-5_12Alarcón, 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). https://doi.org/10.1007/978-3-642-16310-4_4Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000). https://doi.org/10.1016/S0304-3975(99)00207-8Clavel, M., et al.: All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71999-1Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reasoning 40(2–3), 195–220 (2008). https://doi.org/10.1007/s10817-007-9087-9Giesl, J., Arts, T.: Verification of erlang processes by dependency pairs. Appl. Algebra Eng. Commun. Comput. 12(1/2), 39–72 (2001). https://doi.org/10.1007/s002000100063Giesl, J., Thiemann, R., Schneider-Kamp, P.: Proving and disproving termination of higher-order functions. In: Gramlich, B. (ed.) FroCoS 2005. LNCS (LNAI), vol. 3717, pp. 216–231. Springer, Heidelberg (2005). https://doi.org/10.1007/11559306_12Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reasoning 37(3), 155–203 (2006). https://doi.org/10.1007/s10817-006-9057-7Goguen, J.A., Meseguer, J.: Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci. 105(2), 217–273 (1992). https://doi.org/10.1016/0304-3975(92)90302-VGutiérrez, R., Lucas, S.: Function calls at frozen positions in termination of context-sensitive rewriting. In: Martí-Oliet, N., Ölveczky, P.C., Talcott, C. (eds.) Logic, Rewriting, and Concurrency. LNCS, vol. 9200, pp. 311–330. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-23165-5_15Gutiérrez, R., Lucas, S.: Proving termination in the context-sensitive dependency pair framework. In: Ölveczky, P.C. (ed.) WRLA 2010. LNCS, vol. 6381, pp. 18–34. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16310-4_3Gutiérrez, R., Lucas, S.: Automatic generation of logical models with AGES. In: Fontaine, P. (ed.) CADE 2019. LNCS (LNAI), vol. 11716, pp. 287–299. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29436-6_17Gutiérrez, R., Lucas, S.: Automatically proving and disproving feasibility conditions. In: Peltier, N., Sofronie-Stokkermans, V. (eds.) IJCAR 2020. LNAI, vol. 12167, pp. 416–435. Springer, Heidelberg (2020)Lucas, S.: Context-sensitive computations in functional and functional logic programs. J. Funct. Log. Program. 1998(1), 1–61 (1998). http://danae.uni-muenster.de/lehre/kuchen/JFLP/articles/1998/A98-01/A98-01.htmlLucas, S.: Context-sensitive rewriting strategies. Inf. Comput. 178(1), 294–343 (2002). https://doi.org/10.1006/inco.2002.3176Lucas, S.: Proving semantic properties as first-order satisfiability. Artif. Intell. 277 (2019). https://doi.org/10.1016/j.artint.2019.103174Lucas, S., Gutiérrez, R.: Automatic synthesis of logical models for order-sorted first-order theories. J. Autom. Reasoning 60(4), 465–501 (2017). https://doi.org/10.1007/s10817-017-9419-3Lucas, S., Gutiérrez, R.: Use of logical models for proving infeasibility in term rewriting. Inf. Process. Lett. 136, 90–95 (2018). https://doi.org/10.1016/j.ipl.2018.04.002Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inf. Process. Lett. 95(4), 446–453 (2005). https://doi.org/10.1016/j.ipl.2005.05.002Lucas, S., Meseguer, J.: Order-sorted dependency pairs. In: Antoy, S., Albert, E. (eds.) Proceedings of the 10th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming, 15–17 July 2008, Valencia, Spain, pp. 108–119. ACM (2008). https://doi.org/10.1145/1389449.1389463Lucas, S., Meseguer, J.: Dependency pairs for proving termination properties of conditional term rewriting systems. J. Log. Algebraic Methods Program. 86(1), 236–268 (2017). https://doi.org/10.1016/j.jlamp.2016.03.003Lucas, 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). https://doi.org/10.1016/j.jcss.2018.04.002Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems—part II: advanced processors and implementation techniques. J. Autom. Reasoning (2020). https://doi.org/10.1007/s10817-020-09542-3McCune, W.: Prover9 & Mace4. Technical report (2005–2010). http://www.cs.unm.edu/~mccune/prover9/Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer (2002). https://doi.org/10.1007/978-1-4757-3661-8 . http://www.springer.com/computer/swe/book/978-0-387-95250-5Ölveczky, P.C., Lysne, O.: Order-sorted termination: the unsorted way. In: Hanus, M., Rodríguez-Artalejo, M. (eds.) ALP 1996. LNCS, vol. 1139, pp. 92–106. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61735-3_6Zantema, H.: Termination of term rewriting: interpretation and type elimination. J. Symb. Comput. 17(1), 23–50 (1994). https://doi.org/10.1006/jsco.1994.1003Zantema, H.: Termination of context-sensitive rewriting. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 172–186. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-62950-5_6

Termination of graph rewriting systems through language theory

International audienceThe termination issue that we tackle is rooted in Natural Language Processing where computations are performed by graph rewriting systems (GRS) that may contain a large number of rules, often in the order of thousands. This asks for algorithmic procedures to verify the termination of such systems. The notion of graph rewriting that we consider does not make any assumption on the structure of graphs (they are not "term graphs", "port graphs" nor "drags"). This lack of algebraic structure led us to proposing two orders on graphs inspired from language theory: the matrix multiset-path order and the rational embedding order. We show that both are stable by context, which we then use to obtain the main contribution of the paper: under a suitable notion of "interpretation", a GRS is terminating if and only if it is compatible with an interpretation

On graph rewriting systems termination through language theory

The termination issue that we tackle is rooted in Natural Language Processing where computations are performed by graph rewriting systems (GRS) that may contain a large number of rules, often in the order of thousands. Thus algorithms become mandatory to verify the termination of such systems. The notion of graph rewriting that we consider does not make any assumption on the structure of graphs (they are not "term graphs", "port graphs" nor "drags"). This lack of algebraic structure led us to proposing two orders on graphs inspired from language theory: the matrix multiset-path order and the rational embedding order. We show that both are stable by context, which we then use to obtain the main contribution of the paper: under a suitable notion of "interpretation", a GRS is terminating if and only if it is compatible with an interpretation

On termination of Graph Rewriting Systems through language theory

The termination issue we tackle is rooted in natural language processing where graph rewriting systems (GRS) may contain a large number of rules, often in the order of thousands. Decidable concepts thus become mandatory to verify the termination of such systems. The notion of graph rewriting consider does not make any assumption on the structure of graphs (they are not “term graphs”, “port graphs” nor drags). The lack of algebraic structure in our setting led us to proposing two orders on graphs inspired from language theory: the matrix multiset-path order and the rational embedding order. We show that both are stable by context, which we then use to obtain the main contribution of the paper: under a suitable notion of “interpretation”, a GRS is terminating if and only if it is compatible with an interpretation

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

CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

Termination is an important property of programs; notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting systems, where many methods and tools have been developed over the years to address this problem. Ensuring reliability of those tools is therefore an important issue. In this paper we present a library formalizing important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools