Journal of Automated Reasoning

, Volume 50, Issue 1, pp 51–98 | Cite as

Paramodulation with Non-Monotonic Orderings and Simplification

Article

Abstract

Ordered paramodulation and Knuth-Bendix completion are known to remain complete when using non-monotonic orderings. However, these results do not imply the compatibility of the calculus with essential redundancy elimination techniques such as demodulation, i.e., simplification by rewriting, which constitute the primary mode of computation in most successful automated theorem provers. In this paper we present a complete ordered paramodulation calculus for non-monotonic orderings which is compatible with powerful redundancy notions including demodulation, hence strictly improving the previous results and making the calculus more likely to be used in practice. As a side effect, we obtain a Knuth-Bendix completion procedure compatible with simplification techniques, which can be used for finding, whenever it exists, a convergent term rewrite system for a given set of equations and a (possibly non-totalizable) reduction ordering.

Keywords

Automated theorem proving Equational reasoning Ordered paramodulation Knuth-Bendix completion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comp. Sci. 236, 133–178 (2000)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, New York, NY, USA (1998)Google Scholar
  3. 3.
    Bachmair, L., Dershowitz, N.: Equational inference, canonical proofs, and proof orderings. J. ACM 41(2), 236–276 (1994)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. J. Log. Comput. 4(3), 217–247 (1994)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bachmair, L., Ganzinger, H.: Equational reasoning in saturation-based theorem proving. In: Bibel, W., Schmitt, P. (eds.) Automated Deduction—A Basis for Applications, vol. I, chap 11, pp. 353–397. Kluwer, Dordrecht, The Netherlands (1998)Google Scholar
  6. 6.
    Bachmair, L., Dershowitz, N., Hsiang, J.: Orderings for equational proofs. In: First IEEE Symposium on Logic in Computer Science (LICS), IEEE Computer Society Press, pp. 346–357. Los Alamitos, CA, USA, Cambridge, MA, USA (1986)Google Scholar
  7. 7.
    Bachmair, L., Dershowitz, N., Plaisted, D.: Completion without failure. In: Aït-Kaci, H., Nivat, M. (eds.) Resolution of Equations in Algebraic Structures, vol. 2: Rewriting Techniques, chap 1, pp 1–30. Academic Press, New York (1989)Google Scholar
  8. 8.
    Bachmair, L., Ganzinger, H., Lynch, C., Snyder, W.: Basic paramodulation. Inf. Comput. 121(2), 172–192 (1995)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bofill, M., Rubio, A.: Well-foundedness is sufficient for completeness of ordered paramodulation. In: 18th International Conference on Automated Deduction (CADE), vol. 2392, pp. 456–470. Springer, Berlin Heidelberg, Germany, Copenhagen, Denmark, LNAI (2002)Google Scholar
  10. 10.
    Bofill, M., Rubio, A.: Redundancy notions for paramodulation with non-monotonic orderings. In: 2nd International Joint Conference on Automated Reasoning (IJCAR), vol. 3097, pp. 107–121. Springer-Verlag, Berlin Heidelberg, Germany, Cork, Ireland, LNAI (2004)Google Scholar
  11. 11.
    Bofill, M., Rubio, A.: Paramodulation with well-founded orderings. J. Log. Comput. 19(2), 263–302 (2009)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Bofill, M., Godoy, G., Nieuwenhuis, R., Rubio, A.: Paramodulation with non-monotonic orderings. In: 14th IEEE Symposium on Logic in Computer Science (LICS), pp. 225–233. IEEE Computer Society Press, Los Alamitos, CA, USA, Trento, Italy (1999)Google Scholar
  13. 13.
    Bofill, M., Godoy, G., Nieuwenhuis, R., Rubio, A.: Paramodulation and Knuth-Bendix completion with non-total and non-monotonic orderings. J. Autom. Reason. 30(1), 99–120 (2003)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Borralleras, C., Ferreira, M., Rubio, A.: Complete monotonic semantic path orderings. In: McAllester, D. (ed.) Proceedings of the 17th International Conference on Automated Deduction (CADE-17), vol. 1831, pp. 346–364. Springer, Pittsburgh, USA, LNAI (2000)Google Scholar
  15. 15.
    Dershowitz, N.: Orderings for term-rewriting systems. Theor. Comp. Sci. 17(3), 279–301 (1982)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Dershowitz, N., Jouannaud, J.P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B: Formal Models and Semantics, chap. 6, pp. 244–320. Elsevier Science B.V, Amsterdan, The Netherlands and The MIT Press, Cambridge, MA, USA (1990)Google Scholar
  17. 17.
    Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Commun. ACM 22(8), 465–476 (1979)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Devie, H.: Linear completion. In: Kaplan, S., Okada, M. (eds.) Conditional and Typed Rewriting Systems, 2nd International Workshop, Springer, Montreal, Canada, LNCS 516, pp. 233–245 (1990)Google Scholar
  19. 19.
    Hsiang, J., Rusinowitch, M.: Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. J. Artists’ Choice Mus. 38(3), 559–587 (1991)MathSciNetMATHGoogle Scholar
  20. 20.
    Kamin, S., Levy, J.J.: Two generalizations of the recursive path ordering. Unpublished note, Dept. of Computer Science, Univ. of Illinois, Urbana, IL (1980)Google Scholar
  21. 21.
    Kusakari, K., Nakamura, M., Toyama, Y.: Argument filtering transformation. In: Proceedings of the International Conference on Principles and Practice of Declarative Programming (PPDP’99), 1702, pp. 47–61. Springer, LNCS (1999)Google Scholar
  22. 22.
    Nieuwenhuis, R., Rubio, A.: Theorem proving with ordering and equality constrained clauses. J. Symb. Comput. 19(4), 321–351 (1995)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, J., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 1, chap. 7, pp. 372–444. Elsevier Science B.V, Amsterdan, The Netherlands and The MIT Press, Cambridge, MA, USA (2001)Google Scholar
  24. 24.
    Wechler, W.: Universal Algebra for Computer Scientists, EATCS Monographs on Theoretical Computer Science, vol. 25. Springer, New York (1992)Google Scholar
  25. 25.
    Wehrman, I., Stump, A., Westbrook, E.M.: Slothrop: Knuth-Bendix completion with a modern termination checker. In: Pfenning, F. (ed.) RTA, vol. 4098, pp. 287–296. Springer, Lecture Notes in Computer Science (2006)Google Scholar
  26. 26.
    Winkler, S., Middeldorp, A.: Termination tools in ordered completion. In: Proceedings of the 5th International Joint Conference on Automated Reasoning, Lecture Notes in Artificial Intelligence, vol. 6173, pp. 518–532. Springer-Verlag (2010)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Dept. Informàtica i Matemàtica AplicadaUniversitat de GironaGironaSpain
  2. 2.Dept. Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations