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Generalizing DPLL to Richer Logics

  • Kenneth L. McMillan
  • Andreas Kuehlmann
  • Mooly Sagiv
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5643)

Abstract

The DPLL approach to the Boolean satisfiability problem (SAT) is a combination of search for a satisfying assignment and logical deduction, in which each process guides the other. We show that this approach can be generalized to a richer class of theories. In particular, we present an alternative to lazy SMT solvers, in which DPLL is used only to find propositionally satisfying assignments, whose feasibility is checked by a separate theory solver. Here, DPLL is applied directly to the theory. We search in the space of theory structures (for example, numerical assignments) rather than propositional assignments. This makes it possible to use conflict in model search to guide deduction in the theory, much in the way that it guides propositional resolution in DPLL. Some experiments using linear rational arithmetic demonstrate the potential advantages of the approach.

Keywords

Model Search Satisfying Assignment Compatible Pair Linear Arithmetic Theory Solver 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Barrett, C., Deters, M., Oliveras, A., Stump, A.: Design and results of the 4th annual satisfiability modulo theories competition, SMT-COMP 2008 (2008) (to appear)Google Scholar
  2. 2.
    Barrett, C., Sebastiani, R., Seshia, S., Tinelli, C.: Satisfiability modulo theories. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, ch. 8. IOS Press, Amsterdam (2009)Google Scholar
  3. 3.
    Bjørner, N., Dutertre, B., de Moura, L.: Accelerating lemma learning using joins - DPPL(⊔). In: LPAR (2008)Google Scholar
  4. 4.
    Bozzano, M., Bruttomesso, R., Cimatti, A., Junttila, T.A., Ranise, S., van Rossum, P., Sebastiani, R.: Efficient satisfiability modulo theories via delayed theory combination. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 335–349. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Cooper, D.C.: Theorem proving in arithmetic without multiplication. Machine Intelligence 7, 91–99 (1972)zbMATHGoogle Scholar
  6. 6.
    Cotton, S.: Algebraic satisfiability solving. Personal communication (2009)Google Scholar
  7. 7.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Communications of the ACM 5(7), 394–397 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. Journal of the ACM 7(3), 201–215 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Een, N., Biere, A.: Effective preprocessing in SAT through variable and clause elimination. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 61–75. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Flanagan, C., Joshi, R., Ou, X., Saxe, J.B.: Theorem proving using lazy proof explication. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 355–367. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Ganzinger, H., Hagen, G., Nieuwenhuis, R., Oliveras, A., Tinelli, C.: DPLL(T): Fast decision procedures. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 175–188. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Goldwasser, D., Strichman, O., Fine, S.: A theory-based decision heuristic for DPLL(T). In: FMCAD, pp. 1–8 (2008)Google Scholar
  13. 13.
    Koppensteiner, P., Veith, H.: A novel SAT procedure for linear real arithmetic. In: PDPAR (2005)Google 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.
    Strichman, O., Seshia, S.A., Bryant, R.E.: Deciding separation formulas with sat. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 209–222. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Wang, C., Gupta, A., Ganai, M.K.: Predicate learning and selective theory deduction for a difference logic solver. In: DAC, pp. 235–240 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Kenneth L. McMillan
    • 1
  • Andreas Kuehlmann
    • 1
  • Mooly Sagiv
    • 2
  1. 1.Cadence Research LabsUSA
  2. 2.Tel Aviv UniversityIsrael

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