Extending Context-Sensitivity in Term Rewriting

We propose a generalized version of context-sensitivity in term rewriting based on the notion of "forbidden patterns". The basic idea is that a rewrite step should be forbidden if the redex to be contracted has a certain shape and appears in a certain context. This shape and context is expressed through forbidden patterns. In particular we analyze the relationships among this novel approach and the commonly used notion of context-sensitivity in term rewriting, as well as the feasibility of rewriting with forbidden patterns from a computational point of view. The latter feasibility is characterized by demanding that restricting a rewrite relation yields an improved termination behaviour while still being powerful enough to compute meaningful results. Sufficient criteria for both kinds of properties in certain classes of rewrite systems with forbidden patterns are presented

Two Decades of Maude

This paper is a tribute to José Meseguer, from the rest of us in the Maude team, reviewing the past, the present, and the future of the language and system with which we have been working for around two decades under his leadership. After reviewing the origins and the language's main features, we present the latest additions to the language and some features currently under development. This paper is not an introduction to Maude, and some familiarity with it and with rewriting logic are indeed assumed.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

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 rewriting strategies: a generic approach

We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modeled by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction principle is applied through the abstraction mechanism, where terms are replaced by variables representing any of their normal forms. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. Abstraction constraints can be used to control the narrowing mechanism, well known to easily diverge. The generic method is then instantiated for the innermost, outermost and local strategies.Comment: 49 page

Correct and Complete (Positive) Strategy Annotations for OBJ

AbstractStrategy annotations are used in several rewriting-based programming languages to introduce replacement restrictions aimed at improving efficiency and/or reducing the risk of nontermination. Unfortunately, rewriting restrictions can have a negative impact on the ability to compute normal forms. In this paper, we first ascertain/clarify the conditions ensuring correctness and completeness (regarding normalization) of computing with strategy annotations. Then, we define a program transformation methodology for (correct and) complete evaluations which applies to OBJ-like languages

Conditional Complexity

We propose a notion of complexity for oriented conditional term rewrite systems. This notion is realistic in the sense that it measures not only successful computations but also partial computations that result in a failed rule application. A transformation to unconditional context-sensitive rewrite systems is presented which reflects this complexity notion, as well as a technique to derive runtime and derivational complexity bounds for the latter

Methods for Proving Termination of Rewriting-based Programming Languages by Transformation

AbstractDespite the remarkable development of the theory of termination of rewriting, its application to high-level (rewriting-based) programming languages is far from being optimal. This is due to the need for features such as conditional equations and rules, types and subtypes, (possibly programmable) strategies for controlling the execution, matching modulo axioms, and so on, that are used in many programs and tend to place such programs outside the scope of current termination tools. The operational meaning of such features is often formalized in a proof theoretic manner by means of an inference system rather than just by a rewriting relation. The corresponding termination notions can also differ from the standard ones. During the last years we have introduced and implemented different notions and transformation techniques which have been proved useful for proving and disproving termination of such programs by using existing tools for proving termination of (variants of) rewriting. In this paper we provide an overview of our main contributions

Applications and extensions of context-sensitive rewriting

[EN] Context-sensitive rewriting is a restriction of term rewriting which is obtained by imposing replacement restrictions on the arguments of function symbols. It has proven useful to analyze computational properties of programs written in sophisticated rewriting-based programming languages such asCafeOBJ, Haskell, Maude, OBJ*, etc. Also, a number of extensions(e.g., to conditional rewritingor constrained equational systems) and generalizations(e.g., controlled rewritingor forbidden patterns) of context-sensitive rewriting have been proposed. In this paper, we provide an overview of these applications and related issues. (C) 2021 Elsevier Inc. All rights reserved.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32 and PROMETEO/2019/098.Lucas Alba, S. (2021). Applications and extensions of context-sensitive rewriting. Journal of Logical and Algebraic Methods in Programming. 121:1-33. https://doi.org/10.1016/j.jlamp.2021.10068013312

Completeness of Context-Sensitive Rewriting

Restrictions of rewriting may turn normal forms of some terms unreachable, leading to incomplete computations. Context-sensitive rewriting (csr) is the restriction of rewriting that only permits reductions on arguments selected by a replacement map μ, which associates a subset μ(f ) of argument indices with each function symbol f . Hendrix and Meseguer defined an algebraic semantics for Term Rewriting Systems (TRSs) executing csr that can be used to reason about programs written in programming languages like CafeOBJ and Maude, where such replacement restrictions can be specified in programs. Semantic completeness of csr was also defined. In this paper we show that canonical replacement maps, which play a prominent role in simulating rewriting computations with csr, are necessary for completeness in important classes of TRSs. © 2014 Elsevier B.V. All rights reserved.Supported by NSF CNS 13-19109, MINECO project TIN2010-21062-C02-02, GV (Generalitat Valenciana) Grants BEST/2014/026 and PROMETEO/2011/052.Lucas Alba, S. (2015). Completeness of Context-Sensitive Rewriting. Information Processing Letters. 115(2):87-92. https://doi.org/10.1016/j.ipl.2014.07.004S8792115