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Reducing Relative Termination to Dependency Pair Problems

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Automated Deduction - CADE-25 (CADE 2015)

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Abstract

Relative 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-C4-1-R and by the Generalitat Valenciana under grant PROMETEOII2015/013. Akihisa Yamada is supported by the Austrian Science Fund (FWF): Y757.

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Notes

  1. 1.

    Available from URL http://www.termination-portal.org/.

  2. 2.

    http://www.win.tue.nl/rtaloop/.

  3. 3.

    Available at http://www.trs.cm.is.nagoya-u.ac.jp/NaTT/.

  4. 4.

    Details are available at http://www.trs.cm.is.nagoya-u.ac.jp/papers/CADE2015.

  5. 5.

    Available at http://z3.codeplex.com/.

  6. 6.

    Available at http://termination-portal.org/wiki/TPDB.

  7. 7.

    For one of the two problems, the union is terminating.

  8. 8.

    For four examples, AProVE proved relative termination but NaTT failed. There AProVE used semantic labeling [30], which is currently not implemented in NaTT.

References

  1. Alarcón, 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)

    Chapter  Google Scholar 

  2. Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000)

    Article  MathSciNet  Google Scholar 

  3. Arts, T., Giesl, J.: A collection of examples for termination of term rewriting using dependency pairs. Technical report AIB-2001-09, RWTH Aachen (2001)

    Google Scholar 

  4. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  5. Dershowitz, N.: Termination of rewriting. J. Symb. Comput. 3(1&2), 69–115 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reasoning 40(2–3), 195–220 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Geser, A.: Relative termination. Dissertation, Fakultät für Mathematik und Informatik, Universität Passau, Germany (1990)

    Google Scholar 

  8. Giesl, J., Kapur, D.: Dependency pairs for equational rewriting. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 93–107. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  9. 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)

    Chapter  Google Scholar 

  10. Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reasoning 37(3), 155–203 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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)

    Chapter  Google Scholar 

  12. Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 249–268. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  13. Hirokawa, N., Middeldorp, A.: Decreasing diagrams and relative termination. J. Autom. Reasoning 47(4), 481–501 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hullot, J.M.: Canonical forms and unification. CADE-5. LNCS, vol. 87, pp. 318–334. Springer, Heidelberg (1980)

    Google Scholar 

  15. 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)

    Chapter  Google Scholar 

  16. Kamin, S., Lévy, J.J.: Two generalizations of the recursive path ordering (1980, unpublished note)

    Google Scholar 

  17. Klop, J.W.: Term rewriting systems: a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. 32, 143–183 (1987)

    MATH  Google Scholar 

  18. 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)

    Chapter  Google Scholar 

  19. Koprowski, A.: TPA: termination proved automatically. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 257–266. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  20. 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)

    Chapter  Google Scholar 

  21. Lankford, D.: Canonical algebraic simplification in computational logic. Technical report ATP-25, University of Texas (1975)

    Google Scholar 

  22. 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)

    Google Scholar 

  23. Nishida, N., Sakai, M., Sakabe, T.: Narrowing-based simulation of term rewriting systems with extra variables. ENTCS 86(3), 52–69 (2003)

    Google Scholar 

  24. Nishida, N., Vidal, G.: Termination of narrowing via termination of rewriting. Appl. Algebra Eng. Commun. Comput. 21(3), 177–225 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer-Verlag, London (2002)

    Book  MATH  Google Scholar 

  26. 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)

    Google Scholar 

  27. 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)

    Chapter  Google Scholar 

  28. 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)

    Google Scholar 

  29. Yamada, A., Kusakari, K., Sakabe, T.: A unified ordering for termination proving. Sci. Comput. Program. (2014). doi:10.1016/j.scico.2014.07.009

  30. Zantema, H.: Termination of term rewriting by semantic labelling. Fundamenta Informaticae 24(1/2), 89–105 (1995)

    MathSciNet  MATH  Google Scholar 

  31. 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)

    Google Scholar 

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Acknowledgement

We would like to thank Nao Hirokawa and the anonymous reviewers for their helpful comments and suggestions in early stages of this work.

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Correspondence to Akihisa Yamada .

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Iborra, J., Nishida, N., Vidal, G., Yamada, A. (2015). Reducing Relative Termination to Dependency Pair Problems. In: Felty, A., Middeldorp, A. (eds) Automated Deduction - CADE-25. CADE 2015. Lecture Notes in Computer Science(), vol 9195. Springer, Cham. https://doi.org/10.1007/978-3-319-21401-6_11

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  • DOI: https://doi.org/10.1007/978-3-319-21401-6_11

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