Goal-Directed and Relative Dependency Pairs for Proving the Termination of Narrowing

Abstract. In this work, we first consider a goal-oriented extension of the dependency pair framework for proving termination w.r.t. a given set of initial terms. Then, we introduce a new result for proving relative termination in terms of a dependency pair problem. Both contributions put together allow us to define a simple and powerful approach to analyzing the termination of narrowing, an extension of rewriting that replaces matching with unification in order to deal with logic variables. Our approach could also be useful in other contexts where considering termination w.r.t. a given set of terms is also natural (e.g., proving the termination of functional programs).

Reducing relative termination to dependency pair problems

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-21401-6_11Relative termination, a generalized notion of termination, has been used in a number of different contexts like proving the confluence of rewrite systems or analyzing the termination of narrowing. In this paper, we introduce a new 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. The practical significance of our method is illustrated by means of an experimental evaluation.Germán Vidal is partially supported by the EU (FEDER) and the Spanish Ministerio de Economía y Competitividad under grant TIN2013-44742-C4-R and by the Generalitat Valenciana under grant PROMETEOII201/013. Akihisa Yamadais supported by the Austrian Science Fund (FWF): Y757Iborra, J.; Nishida, N.; Vidal Oriola, GF.; Yamada, A. (2015). Reducing relative termination to dependency pair problems. En Automated Deduction - CADE-25. Springer. 163-178. https://doi.org/10.1007/978-3-319-21401-6_11S163178Alarcón, 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)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)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. Reasoning 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: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 93–107. Springer, Heidelberg (2001)Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: automatic termination proofs in the dependency pair framework. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 281–286. Springer, Heidelberg (2006)Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reasoning 37(3), 155–203 (2006)Hirokawa, N., Middeldorp, A.: Polynomial interpretations with negative coefficients. In: Buchberger, B., Campbell, J. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 185–198. Springer, Heidelberg (2004)Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 249–268. Springer, Heidelberg (2004)Hirokawa, N., Middeldorp, A.: Decreasing diagrams and relative termination. J. Autom. Reasoning 47(4), 481–501 (2011)Hullot, J.M.: Canonical forms and unification. CADE-5. LNCS, vol. 87, pp. 318–334. Springer, Heidelberg (1980)Iborra, J., Nishida, N., Vidal, G.: Goal-directed and relative dependency pairs for proving the termination of narrowing. In: De Schreye, D. (ed.) LOPSTR 2009. LNCS, vol. 6037, pp. 52–66. Springer, Heidelberg (2010)Kamin, S., Lévy, J.J.: Two generalizations of the recursive path ordering (1980, unpublished note)Klop, J.W.: Term rewriting systems: a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. 32, 143–183 (1987)Koprowski, A., Zantema, H.: Proving liveness with fairness using rewriting. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 232–247. Springer, Heidelberg (2005)Koprowski, A.: TPA: termination proved automatically. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 257–266. Springer, Heidelberg (2006)Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean termination tool 2. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 295–304. Springer, Heidelberg (2009)Lankford, D.: Canonical algebraic simplification in computational logic. Technical report ATP-25, University of Texas (1975)Liu, J., Dershowitz, N., Jouannaud, J.-P.: Confluence by critical pair analysis. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 287–302. Springer, Heidelberg (2014)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-Verlag, London (2002)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: Garrigue, J., Hermenegildo, M.V. (eds.) FLOPS 2008. LNCS, vol. 4989, pp. 113–129. Springer, Heidelberg (2008)Yamada, A., Kusakari, K., Sakabe, T.: Nagoya termination tool. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 466–475. Springer, Heidelberg (2014)Yamada, A., Kusakari, K., Sakabe, T.: A unified ordering for termination proving. Sci. Comput. Program. (2014). doi: 10.1016/j.scico.2014.07.009Zantema, H.: Termination of term rewriting by semantic labelling. Fundamenta Informaticae 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, vol. 55, pp. 181–259. Cambridge University Press, Cambridge (2003

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

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