Proving Confluence in the Confluence Framework with CONFident

This article describes the *Confluence Framework*, a novel framework for proving and disproving confluence using a divide-and-conquer modular strategy, and its implementation in CONFident. Using this approach, we are able to automatically prove and disprove confluence of *Generalized Term Rewriting Systems*, where (i) only selected arguments of function symbols can be rewritten and (ii) a rather general class of conditional rules can be used. This includes, as particular cases, several variants of rewrite systems such as (context-sensitive) *term rewriting systems*, *string rewriting systems*, and (context-sensitive) *conditional term rewriting systems*. The divide-and-conquer modular strategy allows us to combine in a proof tree different techniques for proving confluence, including modular decompositions, checking joinability of (conditional) critical and variable pairs, transformations, etc., and auxiliary tasks required by them, e.g., joinability of terms, joinability of conditional pairs, etc

Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories

[EN] In program analysis, the synthesis of models of logical theories representing the program semantics is often useful to prove program properties. We use order-sorted first- order logic as an appropriate framework to describe the semantics and properties of programs as given theories. Then we investigate the automatic synthesis of models for such theories. We use convex polytopic domains as a flexible approach to associate different domains to different sorts. We introduce a framework for the piecewise definition of functions and predicates. We develop its use with linear expressions (in a wide sense, including linear transformations represented as matrices) and inequalities to specify functions and predicates. In this way, algorithms and tools from linear algebra and arithmetic constraint solving (e.g., SMT) can be used as a backend for an efficient implementation.Partially supported by the EU (FEDER), projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/ 013. R. Gutiérrez also supported by Juan de la Cierva Fellowship JCI-2012-13528.Lucas Alba, S.; Gutiérrez Gil, R. (2018). Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories. Journal of Automated Reasoning. 60(4):465-501. https://doi.org/10.1007/s10817-017-9419-3S465501604Alarcón, 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)Alarcón, B., Lucas, S., Navarro-Marset, R.: Using matrix interpretations over the reals in proofs of termination. In: Proceedings of PROLE’09, pp. 255–264 (2009)Albert, E., Genaim, S., Gutiérrez, R.: A Transformational Approach to Resource Analysis with Typed-Norms. Revised Selected Papers from LOPSTR’13. LNCS, vol. 8901, pp 38–53 (2013)de Angelis, E., Fioravante, F., Pettorossi, A., Proietti, M.: Proving correctness of imperative programs by linearizing constrained Horn clauses. Theory Pract. Log. 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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. 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infChecker. A Tool for Checking Infeasibility

[EN] Given a Conditional Term Rewriting System (CTRS) R and terms s and t, we say that the reachability condition s ->* t is *feasible* if there is a substitution \sigma instantiating the variables in s and t such that the *reachability test* \sigma(s)->* \sigma(t) succeeds; otherwise, we call it *infeasible*. Checking infeasibility of such (sequences of) reachability conditions is important in the analysis of computational properties of CTRSs, like confluence or operational termination. Recently, a logic-based approach to prove and disprove infeasibility has been introduced. In this paper we present infChecker, a new tool for checking infeasibility which is based on such an approach.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32, PROMETEO/2019/098, and SP20180225. Raul Gutierrez was also supported by INCIBE program Ayudas para la excelencia de los equipos de investigacion avanzada en ciberseguridad.Gutiérrez Gil, R.; Lucas Alba, S. (2019). infChecker. A Tool for Checking Infeasibility. Universidade de Brasilia. 38-42. http://hdl.handle.net/10251/181069S384

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. 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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. 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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

Towards an Incremental and Modular Termination Analysis of Context-Sensitive Rewriting Systems (Work in Progress)

Modularity is essential in software development, where a piece of software is often designed and implemented as a composition of simpler modules. So, if we want to prove that a program satisfies a given property, a modular approach becomes natural. With the development and successful use of the Dependency Pair Framework, which rather focuses on the decomposition of termination problems, less attention has been payed to modularity issues (which rather require the opposite approach). But the modular analysis of termination is still paramount for software developers. In this paper, we analyze modularity of context-sensitive rewrite systems. A modularity analysis was carried out by Gramlich and Lucas in 2002, but a correct notion of context-sensitive dependency pair (CS-DP) was not obtained until 2006. In this paper, we analyze modularity using CS-DPs.Partially supported by the EU (FEDER), MINECO projects TIN2010-21062-C02-02 and TIN 2013-45732-C4-1-P, and project PROMETEO/2011/052. Salvador Lucas’ research was developed during a sabbatical year at the CS Dept. of the UIUC and was also partially supported by NSF grant CNS 13-19109, Spanish MECD grant PRX12/00214, and GV grant BEST/2014/026. Raúl Gutiérrez is also partially supported by a Juan de la Cierva Fellowship from the Spanish MINECO, ref. JCI-2012-13528.Gutiérrez Gil, R.; Lucas Alba, S. (2014). Towards an Incremental and Modular Termination Analysis of Context-Sensitive Rewriting Systems (Work in Progress). Sistedes. http://hdl.handle.net/10251/71781

Function Calls at Frozen Positions in Termination of Context-Sensitive Rewriting

Context-sensitive rewriting (CSR) is a variant of rewriting where only selected arguments of function symbols can be rewritten. Consequently, the subterm positions of a term are classified as either active, i.e., positions of subterms that can be rewritten; or frozen, i.e., positions that cannot. Frozen positions can be used to denote subexpressions whose evaluation is delayed or just forbidden. A typical example is the if-then-else operator whose second and third arguments are not evaluated until the evaluation of the first argument yields either true or false. Imposing replacement restrictions can improve the termination behavior of rewriting-based computational systems. Termination of CSR has been investigated by several authors and a number of automatic tools are able to prove it. In this paper, we analyze how frozen subterms affect termination of CSR. This analysis helps us to improve our context-sensitive dependency pair (CS-DP) framework for automatically proving termination of CSR. We have implemented these improvements in our tool MU-TERM. The experiments show the power of the improvements in practice.Gutiérrez Gil, R.; Lucas Alba, S. (2015). Function Calls at Frozen Positions in Termination of Context-Sensitive Rewriting. http://hdl.handle.net/10251/5075

La facilidad con la cual se incurre en una duplicación involuntaria. A propósito de un caso de “Autoplagio”

Mr. Editor.Using the definitions of the International Committee of Medical Journal Editors1 “a duplicate publication is the publication of a manuscript that substantially overlaps with one already published without clear, visible reference to the previous publication”. We wrote this letter to make public and transparent a case of unintentional duplicate publication, described below:Sr. Editor.Utilizando las definiciones del Comité Internacional de Editores de Revistas Biomédicas1, podemos señalar que una publicación duplicada es la publicación de un manuscrito que se superpone sustancialmente con uno ya publicado, sin la debida referencia a la publicación anterior. Escribimos esta carta para hacer público y transparentar un caso de publicación duplicada involuntaria, que a continuación procedemos a describir

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