Obtaining Finite Local Theory Axiomatizations via Saturation

  • Matthias Horbach
  • Viorica Sofronie-Stokkermans
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8152)

Abstract

In this paper we present a method for obtaining local sets of clauses from possibly non-local ones. For this, we follow the work of Basin and Ganzinger and use saturation under a version of ordered resolution. In order to address the fact that saturation can generate infinite sets of clauses, we use constrained clauses and show that a link can be established between saturation and locality also for constrained clauses: This often allows us to give a finite representation of possibly infinite saturated sets of clauses.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: New results on rewrite-based satisfiability procedures. ACM Trans. Comput. Log. 10(1) (2009)Google Scholar
  2. 2.
    Armando, A., Ranise, S., Rusinowitch, M.: A rewriting approach to satisfiability procedures. Inf. Comput. 183(2), 140–164 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. J. of Logic and Computation 4(3), 217–247 (1994)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Basin, D., Ganzinger, H.: Complexity analysis based on ordered resolution. In: Proc. 11th IEEE Symposium on Logic in Computer Science (LICS 1996), pp. 456–465. IEEE Computer Society Press (1996)Google Scholar
  5. 5.
    Basin, D.A., Ganzinger, H.: Automated complexity analysis based on ordered resolution. Journal of the ACM 48(1), 70–109 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ganzinger, H.: Relating semantic and proof-theoretic concepts for polynomial time decidability of uniform word problems. In: Proc. 16th IEEE Symposium on Logic in Computer Science (LICS 2001), pp. 81–92. IEEE Computer Society Press (2001)Google Scholar
  7. 7.
    Horbach, M.: Superposition-based Decision Procedures for Fixed Domain and Minimal Model Semantics. PhD thesis, Max Planck Institute for Computer Science and Saarland University (2010)Google Scholar
  8. 8.
    Horbach, M., Sofronie-Stokkermans, V.: Obtaining finite local theory axiomatizations via saturation. Technical Report ATR 93, Sonderforschungsbereich/Transregio 14 AVACS (2013)Google Scholar
  9. 9.
    Horbach, M., Weidenbach, C.: Superposition for fixed domains. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 293–307. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Horbach, M., Weidenbach, C.: Decidability results for saturation-based model building. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 404–420. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Horbach, M., Weidenbach, C.: Deciding the inductive validity of ∀∃* queries. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 332–347. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Ihlemann, C., Jacobs, S., Sofronie-Stokkermans, V.: On local reasoning in verification. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 265–281. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Ihlemann, C., Sofronie-Stokkermans, V.: On hierarchical reasoning in combinations of theories. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS (LNAI), vol. 6173, pp. 30–45. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Kirchner, H., Ranise, S., Ringeissen, C., Tran, D.-K.: On superposition-based satisfiability procedures and their combination. In: Van Hung, D., Wirsing, M. (eds.) ICTAC 2005. LNCS, vol. 3722, pp. 594–608. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    Lynch, C., Morawska, B.: Automatic decidability. In: 17th IEEE Symposium on Logic in Computer Science (LICS 2002), pp. 7–16. IEEE Comp. Soc. (2002)Google Scholar
  16. 16.
    Lynch, C., Ranise, S., Ringeissen, C., Tran, D.-K.: Automatic decidability and combinability. Inf. Comput. 209(7), 1026–1047 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Nieuwenhuis, R., Rubio, A.: Theorem proving with ordering constrained clauses. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 477–491. Springer, Heidelberg (1992)Google Scholar
  18. 18.
    Sofronie-Stokkermans, V.: Hierarchic reasoning in local theory extensions. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 219–234. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Sofronie-Stokkermans, V.: Locality results for certain extensions of theories with bridging functions. In: Schmidt, R.A. (ed.) CADE-22. LNCS (LNAI), vol. 5663, pp. 67–83. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Tushkanova, E., Ringeissen, C., Giorgetti, A., Kouchnarenko, O.: Automatic decidability: A schematic calculus for theories with counting operators. In: Proceedings the RTA 2013 (to appear, 2013)Google Scholar
  21. 21.
    Weidenbach, C.: Combining superposition, sorts and splitting. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 2, ch. 27, pp. 1965–2012. Elsevier (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Matthias Horbach
    • 1
  • Viorica Sofronie-Stokkermans
    • 1
  1. 1.University Koblenz-Landau and Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations