Using Well-Founded Relations for Proving Operational Termination

[EN] In this paper, we study operational termination, a proof theoretical notion for capturing the termination behavior of computational systems. We prove that operational termination can be characterized at different levels by means of well- founded relations on specific formulas which can be obtained from the considered system. We show how to obtain such well-founded relations from logical models which can be automatically generated using existing tools.Partially supported by the EU (FEDER), Projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/013.Lucas Alba, S. (2020). Using Well-Founded Relations for Proving Operational Termination. Journal of Automated Reasoning. 64(2):167-195. https://doi.org/10.1007/s10817-019-09514-2S167195642Alarcón, 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, Springer (2011)Aguirre, L., Martí-Oliet, N., Palomino, M., Pita, I.: Sentence-normalized conditional narrowing modulo in rewriting logic and Maude. J. Automat. Reason. 60(4), 421–463 (2018)Arts, T., Giesl, J.: Proving innermost normalisation automatically. In: Proceedings of RTA’97, LNCS, vol. 1232, pp. 157–171, Springer, Berlin (1997)Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000)Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.: All About Maude—A High-Performance Logical Framework. LNCS, vol. 4350, Springer (2007)Durán, F., Lucas, S., Meseguer, J.: Methods for proving termination of rewriting-based programming languages by transformation. Electron. Notes Theor. Comput. Sci. 248, 93–113 (2009)Durán, F., Lucas, S., Marché, C., Meseguer, J., Urbain, X.: Proving operational termination of membership equational programs. High. Order Symb. Comput. 21(1–2), 59–88 (2008)Falke, S., Kapur, D.: Operational termination of conditional rewriting with built-in numbers and semantic data structures. Electron. Notes Theor. Comput. Sci. 237, 75–90 (2009)Floyd, R.W.: Assigning meanings to programs. Math. Asp. Comput. Sci. 19, 19–32 (1967)Giesl, J., Arts, T.: Verification of Erlang processes by dependency pairs. Appl. Algebra Eng. Commun. Comput. 12, 39–72 (2001)Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reason. 37(3), 155–203 (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., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: automatic termination proofs in the dependency pair framework. In: Proceedings of IJCAR’06, LNAI, vol. 4130, pp. 281–286 (2006)Goguen, J., Meseguer, J.: Models and equality for logical programming. In: Proceedings of TAPSOFT’87, LNCS, vol. 250, pp. 1–22 (1987)Goguen, J., Meseguer, J.: Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci. 105, 217–273 (1992)Gutiérrez, R., Lucas, S., Reinoso, P.: A tool for the automatic generation of logical models of order-sorted first-order theories. In: Proceedings of PROLE’16, pp. 215–230 (2016)Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean termination tool 2. In: Proceedings of RTA 2009, LNCS, vol. 5595, pp. 295–304 (2009)Lalement, R.: Computation as Logic. Masson-Prentice Hall International, Paris (1993)Lucas, S.: Context-sensitive rewriting strategies. Inf. Comput. 178(1), 294–343 (2002)Lucas, S.: Use of logical models for proving operational termination in general logics. In: Selected Papers from WRLA’16, LNCS, vol. 9942, pp. 1–21 (2016)Lucas, S.: Directions of operational termination. In: Proceedings of PROLE’18. http://hdl.handle.net/11705/PROLE/2018/009 (2018). Accessed 9 Feb 2019Lucas, 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.: Dependency pairs for proving termination properties of conditional term rewriting systems. J. Log. Algebr. Methods Program. 86, 236–268 (2017)Lucas, S., Meseguer, J.: Proving operational termination of declarative programs in general logics. In: Proceedings of PPDP’14, pp. 111–122. ACM Digital Library (2014)McCune, W.: Prover9 & Mace4. http://www.cs.unm.edu/~mccune/prover9/ (2005–2010). Accessed 9 Feb 2019Mendelson, E.: Introduction to Mathematical Logic, 4th edn. Chapman & Hall, London (1997)Meseguer, J.: General logics. In: Logic Colloquium’87, pp. 275–329 (1989)O’Donnell, M.J.: Equational Logic as a Programming Language. The MIT Press, Cambridge (1985)Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, Berlin (2002)Prawitz, D.: Natural Deduction. A Proof Theoretical Study. Almqvist & Wiksell, 1965. Reprinted by Dover Publications (2006)Rosu, G., Stefanescu, A., Ciobaca, S., Moore, B.M.: One-path reachability logic. In: Proceedings of LICS 2013, pp. 358–367. IEEE Press (2013)Shapiro, S.: Foundations Without Foundationalism: A Case for Second-Order Logic. Clarendon Press, Oxford (1991)Schernhammer, F., Gramlich, B.: Characterizing and proving operational termination of deterministic conditional term rewriting systems. J. Log. Algebr. Program. 79, 659–688 (2010)Serbanuta, T., Rosu, G.: Computationally equivalent elimination of conditions. In: Proceedings of RTA’06, LNCS, vol. 4098, pp. 19–34. Springer, Berlin (2006)Turing, A.M.: Checking a large routine. In: Report of a Conference on High Speed Automatic Calculating Machines, Univ. Math. Lab., Cambridge, pp. 67–69 (1949

Termination of Priority Rewriting - Extended version

Introducing priorities in rewriting increases the expressive power of rules and helps to limit computations. Priority rewriting is used in rule-based programming as well as in functional programming. Termination of priority rewriting is then important to guarantee that programs give a result. We describe an inductive proof method for termination of priority rewriting, relying on an explicit induction on the termination property and working by generating proof trees, which model the rewriting relation by using abstraction and narrowing

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

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We present a technique to prove innermost normalisation of term rewriting systems (TRSs) automatically. In contrast to previous methods, our technique is able to prove innermost normalisation of TRSs that are not terminating. Our technique can also be used for termination proofs of all TRSs where innermost normalisation implies termination, such as non-overlapping TRSs or locally confluent overlay systems. In this way, termination of many (also non-simply terminating) TRSs can be verified automatically. 1. Introduction Innermost rewriting, i.e. rewriting where only innermost redexes are contracted, can be used to model call-by-value computation semantics. For that reason, there has been an increasing interest in innermost normalisation (also called innermost termination), i.e. in proving that the length of every innermost reduction is finite. Techniques for proving innermost normalisation can for example be utilized for termination proofs of functional programs (modelled by TRSs) or o..