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

Modular Termination Proofs of Recursive Java Bytecode Programs by Term Rewriting

In earlier work we presented an approach to prove termination of non-recursive Java Bytecode (JBC) programs automatically. Here, JBC programs are first transformed to finite termination graphs which represent all possible runs of the program. Afterwards, the termination graphs are translated to term rewrite systems (TRSs) such that termination of the resulting TRSs implies termination of the original JBC programs. So in this way, existing techniques and tools from term rewriting can be used to prove termination of JBC automatically. In this paper, we improve this approach substantially in two ways: (1) We extend it in order to also analyze recursive JBC programs. To this end, one has to represent call stacks of arbitrary size. (2) To handle JBC programs with several methods, we modularize our approach in order to re-use termination graphs and TRSs for the separate methods and to prove termination of the resulting TRS in a modular way. We implemented our approach in the tool AProVE. Our experiments show that the new contributions increase the power of termination analysis for JBC significantly

Relative Termination via Dependency Pairs

[EN] A term rewrite system is terminating when no infinite reduction sequences are possible. Relative termination generalizes termination by permitting infinite reductions as long as some distinguished rules are not applied infinitely many times. Relative termination is thus a fundamental notion that has been used in a number of different contexts, like analyzing the confluence of rewrite systems or the termination of narrowing. In this work, we introduce a novel 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. We first present a general approach that is then instantiated to provide a concrete technique for proving relative termination. The practical significance of our method is illustrated by means of an experimental evaluation.Open access funding provided by Austrian Science Fund (FWF). We would like to thank Nao Hirokawa, Keiichirou Kusakari, and the anonymous reviewers for their helpful comments and suggestions in early stages of this work.Iborra, J.; Nishida, N.; Vidal Oriola, G.; Yamada, A. (2017). Relative Termination via Dependency Pairs. Journal of Automated Reasoning. 58(3):391-411. https://doi.org/10.1007/s10817-016-9373-5391411583Alarcón, B., Lucas, S., Meseguer, J.: A dependency pair framework for A $$\vee$$ ∨ C-termination. In: WRLA 2010, LNCS, vol. 6381, pp. 36–52. Springer (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)Bachmair, L., Dershowitz, N.: Critical pair criteria for completion. J. Symb. Comput. 6, 1–18 (1988)Bonacina, M., Hsiang, J.: On fairness of completion-based theorem proving strategies. In: RTA 1991, LNCS, vol. 488, pp. 348–360. Springer (1991)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. Reason. 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: RTA 2001, LNCS, vol. 2051, pp. 93–107. Springer (2001)Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: automatic termination proofs in the dependency pair framework. In: IJCAR 2006, LNCS, vol. 4130, pp. 281–286. Springer (2006)Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency Pairs. J. Autom. Reason. 37(3), 155–203 (2006)Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: RTA 2004, LNCS, vol. 3091, pp. 249–268. Springer (2004)Hirokawa, N., Middeldorp, A.: Polynomial interpretations with negative coefficients. In: AISC 2004, LNAI, vol. 3249, pp. 185–198. Springer (2004)Hirokawa, N., Middeldorp, A.: Tyrolean termination tool: techniques and features. Inf. Comput. 205(4), 474–511 (2007)Hirokawa, N., Middeldorp, A.: Decreasing diagrams and relative termination. J. Autom. Reason. 47(4), 481–501 (2011)Hullot, J.M.: Canonical forms and unification. In: CADE 1980, LNCS, vol. 87, pp. 318–334. Springer (1980)Iborra, J., Nishida, N., Vidal, G.: Goal-directed and relative dependency pairs for proving the termination of narrowing. In: LOPSTR 2009, LNCS, vol. 6037, pp. 52–66. Springer (2010)Iborra, J., Nishida, N., Vidal, G., Yamada, A.: Reducing relative termination to dependency pair problems. In: CADE-25, LNAI, vol. 9195, pp. 163–178. Springer (2015)Kamin, S., Lévy, J.J.: Two generalizations of the recursive path ordering (1980). Unpublished noteKlop, J.W.: Term rewriting systems: a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. 32, 143–183 (1987)Koprowski, A.: TPA: termination proved automatically. In: RTA 2006, LNCS, vol. 4098, pp. 257–266. Springer (2006)Koprowski, A., Zantema, H.: Proving liveness with fairness using rewriting. In: FroCoS 2005, LNCS, vol. 3717, pp. 232–247. Springer (2005)Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean termination tool 2. In: RTA 2009, LNCS, vol. 5595, pp. 295–304. Springer (2009)Kusakari, K., Toyama, Y.: On proving AC-termination by AC-dependency pairs. IEICE Trans. Inf. Syst. E84–D(5), 439–447 (2001)Lankford, D.: Canonical algebraic simplification in computational logic. Technical report ATP-25, University of Texas (1975)Marché, C., Urbain, X.: Modular and incremental proofs of AC-termination. J. Symb. Comput. 38(1), 873–897 (2004)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, London (2002)Slagle, J.: Automated theorem-proving for theories with simplifiers commutativity and associativity. J. ACM 21(4), 622–642 (1974)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: FLOPS 2008, LNCS, vol. 4989, pp. 113–129. Springer (2008)Yamada, A., Kusakari, K., Sakabe, T.: Nagoya termination tool. In: RTA-TLCA 2014, LNCS, pp. 466–475. Springer (2014)Yamada, A., Kusakari, K., Sakabe, T.: A unified ordering for termination proving. Sci. Comput. Program. 111, 110–134 (2015)Zantema, H.: Termination of term rewriting by semantic labelling. Fundam. Inf. 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, chap. 6, vol. 55, pp. 181–259. Cambridge University Press, Cambridge (2003

A static higher-order dependency pair framework

We revisit the static dependency pair method for proving termination of higher-order term rewriting and extend it in a number of ways: (1) We introduce a new rewrite formalism designed for general applicability in termination proving of higher-order rewriting, Algebraic Functional Systems with Meta-variables. (2) We provide a syntactically checkable soundness criterion to make the method applicable to a large class of rewrite systems. (3) We propose a modular dependency pair framework for this higher-order setting. (4) We introduce a fine-grained notion of formative and computable chains to render the framework more powerful. (5) We formulate several existing and new termination proving techniques in the form of processors within our framework. The framework has been implemented in the (fully automatic) higher-order termination tool WANDA

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

Non-simplifying Graph Rewriting Termination

So far, a very large amount of work in Natural Language Processing (NLP) rely on trees as the core mathematical structure to represent linguistic informations (e.g. in Chomsky's work). However, some linguistic phenomena do not cope properly with trees. In a former paper, we showed the benefit of encoding linguistic structures by graphs and of using graph rewriting rules to compute on those structures. Justified by some linguistic considerations, graph rewriting is characterized by two features: first, there is no node creation along computations and second, there are non-local edge modifications. Under these hypotheses, we show that uniform termination is undecidable and that non-uniform termination is decidable. We describe two termination techniques based on weights and we give complexity bound on the derivation length for these rewriting system.Comment: In Proceedings TERMGRAPH 2013, arXiv:1302.599

Modularity of Convergence and Strong Convergence in Infinitary Rewriting

Properties of Term Rewriting Systems are called modular iff they are preserved under (and reflected by) disjoint union, i.e. when combining two Term Rewriting Systems with disjoint signatures. Convergence is the property of Infinitary Term Rewriting Systems that all reduction sequences converge to a limit. Strong Convergence requires in addition that redex positions in a reduction sequence move arbitrarily deep. In this paper it is shown that both Convergence and Strong Convergence are modular properties of non-collapsing Infinitary Term Rewriting Systems, provided (for convergence) that the term metrics are granular. This generalises known modularity results beyond metric \infty