Advertisement

Combinable Extensions of Abelian Groups

  • Enrica Nicolini
  • Christophe Ringeissen
  • Michaël Rusinowitch
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5663)

Abstract

The design of decision procedures for combinations of theories sharing some arithmetic fragment is a challenging problem in verification. One possible solution is to apply a combination method à la Nelson-Oppen, like the one developed by Ghilardi for unions of non-disjoint theories. We show how to apply this non-disjoint combination method with the theory of abelian groups as shared theory. We consider the completeness and the effectiveness of this non-disjoint combination method. For the completeness, we show that the theory of abelian groups can be embedded into a theory admitting quantifier elimination. For achieving effectiveness, we rely on a superposition calculus modulo abelian groups that is shown complete for theories of practical interest in verification.

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 Transactions on Computational Logic 10(1) (2009)Google Scholar
  2. 2.
    Armando, A., Ranise, S., Rusinowitch, M.: A rewriting approach to satisfiability procedures. Information and Computation 183(2), 140–164 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baader, F., Schulz, K.U.: Unification in the union of disjoint equational theories: Combining decision procedures. Journal of Symbolic Computation 21(2), 211–243 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonacina, M.P., Echenim, M.: T-decision by decomposition. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603, pp. 199–214. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Bonacina, M.P., Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Decidability and undecidability results for Nelson-Oppen and rewrite-based decision procedures. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS, vol. 4130, pp. 513–527. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Boudet, A., Jouannaud, J.-P., Schmidt-Schauß, M.: Unification in boolean rings and abelian groups. In: Kirchner, C. (ed.) Unification, pp. 267–296. Academic Press, London (1990)Google Scholar
  7. 7.
    Chenadec, P.L.: Canonical Forms in Finitely Presented Algebras. Research Notes in Theoretical Computer Science. Pitman-Wiley, Chichester (1986)zbMATHGoogle Scholar
  8. 8.
    de Moura, L.M., Bjørner, N.: Engineering DPLL(T) + saturation. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS, vol. 5195, pp. 475–490. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Dershowitz, N.: Orderings for term-rewriting systems. Theoretical Computer Science 17(3), 279–301 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eklof, P.C., Sabbagh, G.: Model-completions and modules. Annals of Mathematical Logic 2, 251–295 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ghilardi, S., Nicolini, E., Zucchelli, D.: A comprehensive combination framework. ACM Transactions on Computational Logic 9(2), 1–54 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Godoy, G., Nieuwenhuis, R.: Superposition with completely built-in abelian groups. Journal of Symbolic Computation 37(1), 1–33 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hodges, W.: Model Theory. Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press, Cambridge (1993)CrossRefzbMATHGoogle Scholar
  14. 14.
    Korovin, K., Voronkov, A.: Integrating linear arithmetic into superposition calculus. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 223–237. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Lynch, C., Tran, D.-K.: Automatic Decidability and Combinability Revisited. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603, pp. 328–344. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Transaction on Programming Languages and Systems 1(2), 245–257 (1979)CrossRefzbMATHGoogle Scholar
  17. 17.
    Nicolini, E., Ringeissen, C., Rusinowitch, M.: Combinable Extensions of Abelian Groups. Research Report, INRIA, RR-6920 (2009)Google Scholar
  18. 18.
    Nicolini, E., Ringeissen, C., Rusinowitch, M.: Satisfiability procedures for combination of theories sharing integer offsets. In: TACAS 2009. LNCS, vol. 5505, pp. 428–442. Springer, Heidelberg (2009)Google Scholar
  19. 19.
    Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch.7, vol. I, pp. 371–443. Elsevier Science, Amsterdam (2001)CrossRefGoogle Scholar
  20. 20.
    Peterson, G.E., Stickel, M.E.: Complete sets of reductions for some equational theories. J. ACM 28(2), 233–264 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Plotkin, G.: Building-in equational theories. Machine Intelligence 7, 73–90 (1972)zbMATHGoogle Scholar
  22. 22.
    Stuber, J.: Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science 208(1-2), 149–177 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Waldmann, U.: Superposition and chaining for totally ordered divisible abelian groups. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS, vol. 2083, pp. 226–241. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  24. 24.
    Waldmann, U.: Cancellative abelian monoids and related structures in refutational theorem proving (Part I,II). Journal of Symbolic Computation 33(6), 777–829 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, T.: Arithmetic integration of decision procedures. PhD thesis, Department of Computer Science, Stanford University, Stanford, US (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Enrica Nicolini
    • 1
  • Christophe Ringeissen
    • 1
  • Michaël Rusinowitch
    • 1
  1. 1.LORIA & INRIA Nancy Grand EstFrance

Personalised recommendations