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

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

Models for logics and conditional constraints in automated proofs of termination

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-13770-4_3Reasoning about termination of declarative programs, which are described by means of a computational logic, requires the definition of appropriate abstractions as semantic models of the logic, and also handling the conditional constraints which are often obtained. The formal treatment of such constraints in automated proofs, often using numeric interpretations and (arithmetic) constraint solving can greatly benefit from appropriate techniques to deal with the conditional (in)equations at stake. Existing results from linear algebra or real algebraic geometry are useful to deal with them but have received only scant attention to date. We investigate the definition and use of numeric models for logics and the resolution of linear and algebraic conditional constraints as unifying techniques for proving termination of declarative programs.Developed during a sabbatical year at UIUC. Supported by projects NSF CNS13-19109, MINECO TIN2010-21062-C02-02 and TIN2013-45732-C4-1-P, and GV BEST/2014/026 and PROMETEO/2011/052.Lucas Alba, S.; Meseguer, J. (2014). Models for logics and conditional constraints in automated proofs of termination. En Artificial Intelligence and Symbolic Computation. Springer Verlag (Germany). 9-20. https://doi.org/10.1007/978-3-319-13770-4_3S920Alarcón, 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)Alarcón, B., Lucas, S., Navarro-Marset, R.: Using Matrix Interpretations over the Reals in Proofs of Termination. In: Proc. of PROLE 2009, pp. 255–264 (2009)Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C. (eds.): All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007)Contejean, E., Marché, C., Tomás, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. J. of Aut. Reas. 34(4), 325–363 (2006)Endrullis, J., Waldmann, J., Zantema, H.: Matrix Interpretations for Proving Termination of Term Rewriting. J. of Aut. Reas. 40(2-3), 195–220 (2008)Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: Maximal Termination. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 110–125. Springer, Heidelberg (2008)Futatsugi, K., Diaconescu, R.: CafeOBJ Report. AMAST Series. World Scientific (1998)Hudak, P., Peyton-Jones, S.J., Wadler, P.: Report on the Functional Programming Language Haskell: a non–strict, purely functional language. Sigplan Notices 27(5), 1–164 (1992)Lucas, S.: Context-sensitive computations in functional and functional logic programs. Journal of Functional and Logic Programming 1998(1), 1–61 (1998)Lucas, S.: Polynomials over the reals in proofs of termination: from theory to practice. RAIRO Theoretical Informatics and Applications 39(3), 547–586 (2005)Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Information Processing Letters 95, 446–453 (2005)Lucas, S., Meseguer, J.: Proving Operational Termination of Declarative Programs in General Logics. In: Proc. of PPDP 2014, pp. 111–122. ACM Digital Library (2014)Lucas, S., Meseguer, J.: 2D Dependency Pairs for Proving Operational Termination of CTRSs. In: Proc. of WRLA 2014. LNCS, vol. 8663 (to appear, 2014)Lucas, S., Meseguer, J., Gutiérrez, R.: Extending the 2D DP Framework for CTRSs. In: Selected papers of LOPSTR 2014. LNCS (to appear, 2015)Meseguer, J.: General Logics. In: Ebbinghaus, H.-D., et al. (eds.) Logic Colloquium 1987, pp. 275–329. North-Holland (1989)Nguyen, M.T., de Schreye, D., Giesl, J., Schneider-Kamp, P.: Polytool: Polynomial interpretations as a basis for termination of logic programs. Theory and Practice of Logic Programming 11(1), 33–63 (2011)Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer (April 2002)Prestel, A., Delzell, C.N.: Positive Polynomials. In: From Hilbert’s 17th Problem to Real Algebra. Springer, Berlin (2001)Podelski, A., Rybalchenko, A.: A Complete Method for the Synthesis of Linear Ranking Functions. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 239–251. Springer, Heidelberg (2004)Schrijver, A.: Theory of linear and integer programming. John Wiley & Sons (1986)Zantema, H.: Termination of Context-Sensitive Rewriting. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 172–186. Springer, Heidelberg (1997

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

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. Program. 15(4–5), 635–650 (2015)de Angelis, E., Fioravante, F., Pettorossi, A., Proietti, M.: Semantics-based generation of verification conditions by program specialization. In: Proceedings of PPDP’15, pp. 91–102. ACM Press, New York (2015)Aoto, T.: Solution to the problem of zantema on a persistent property of term rewriting systems. J. Funct. Log. Program. 2001(11), 1–20 (2001)Barwise, J.: An Introduction to First-Order Logic. In: Barwise, J. (ed.) Handbook of Mathematical Logic. North-Holland, Amsterdam (1977)Barwise, J.: Axioms for Abstract Model Theory. Ann. Math. Log. 7, 221–265 (1974)Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)Birkhoff, G., Lipson, J.D.: Heterogeneous algebras. J. Comb. Theory 8, 115–133 (1970)Bofill, M., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E., Rubio, A.: The Barcelogic SMT Solver. In: Proceedings of CAV’08. 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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. 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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

CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates

Termination is an important property of programs; notably required for programs formulated in proof assistants. It is a very active subject of research in the Turing-complete formalism of term rewriting systems, where many methods and tools have been developed over the years to address this problem. Ensuring reliability of those tools is therefore an important issue. In this paper we present a library formalizing important results of the theory of well-founded (rewrite) relations in the proof assistant Coq. We also present its application to the automated verification of termination certificates, as produced by termination tools

Proving Termination of Graph Transformation Systems using Weighted Type Graphs over Semirings

We introduce techniques for proving uniform termination of graph transformation systems, based on matrix interpretations for string rewriting. We generalize this technique by adapting it to graph rewriting instead of string rewriting and by generalizing to ordered semirings. In this way we obtain a framework which includes the tropical and arctic type graphs introduced in a previous paper and a new variant of arithmetic type graphs. These type graphs can be used to assign weights to graphs and to show that these weights decrease in every rewriting step in order to prove termination. We present an example involving counters and discuss the implementation in the tool Grez