Skip to main content

Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis

  • Conference paper
Logic for Programming, Artificial Intelligence, and Reasoning (LPAR 2006)

Abstract

Many approaches for Satisfiability Modulo Theory (SMT \({\mathcal({T})})\) rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory \({\mathcal{T}} ({\mathcal{T}}-solver\)). When \({\mathcal{T}}\) is the combination \({{\mathcal{T}}_1\cup{\mathcal{T}}_2}\) of two simpler theories, the approach is typically handled by means of Nelson-Oppen’s (NO) theory combination schema in which two specific \({\mathcal{T}}\)-solver deduce and exchange (disjunctions of) interface equalities.

In recent papers we have proposed a new approach to \(({{\mathcal{T}}_1\cup{\mathcal{T}}_2})\), called Delayed Theory Combination (Dtc). Here part or all the (possibly very expensive) task of deducing interface equalities is played by the SAT solver itself, at the potential cost of an enlargement of the boolean search space. In principle this enlargement could be up to exponential in the number of interface equalities generated.

In this paper we show that this estimate was too pessimistic. We present a comparative analysis of Dtc vs. NO for SMT \(({{\mathcal{T}}_1\cup{\mathcal{T}}_2})\), which shows that, using state-of-the-art SAT-solving techniques, the amount of boolean branches performed by Dtc can be upper bounded by the number of deductions and boolean branches performed by NO on the same problem. We prove the result for different deduction capabilities of the \({\mathcal({T}-solver)}\) and for both convex and non-convex theories.

This work has been partly supported by ISAAC, an European sponsored project, contract no. AST3-CT-2003-501848, by ORCHID, a project sponsored by Provincia Autonoma di Trento, and by a grant from Intel Corporation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Armando, A., Castellini, C., Giunchiglia, E., Maratea, M.: A SAT-Based Decision Procedure for the Boolean Combination of Difference Constraints. In: H. Hoos, H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 16–29. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  2. Barrett, C.W., Berezin, S.: CVC Lite: A New Implementation of the Cooperating Validity Checker Category B. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 515–518. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  3. 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)

    Chapter  Google Scholar 

  4. Bozzano, M., Bruttomesso, R., Cimatti, A., Junttila, T.A., van Rossum, P., Schulz, S., Sebastiani, R.: An Incremental and Layered Procedure for the Satisfiability of Linear Arithmetic Logic. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 317–333. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  5. Bozzano, M., Bruttomesso, R., Cimatti, A., Junttila, T., van Rossum, P., Schulz, S., Sebastiani, R.: MathSAT: A Tight Integration of SAT and Mathematical Decision Procedure. Journal of Automated Reasoning (to appear, 2005)

    Google Scholar 

  6. Bozzano, M., Bruttomesso, R., Cimatti, A., Junttila, T., van Rossum, P., Ranise, S., Sebastiani, R.: Efficient Theory Combination via Boolean Search. Information and Computation (to appear, 2005)

    Google Scholar 

  7. Bruttomesso, R., Cimatti, A., Franzén, A., Griggio, A., Sebastiani, R.: Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: a Comparative Analysis. Technical Report DIT-06-032, DIT, University of Trento (2006), Available at: http://dit.unitn.it/~rseba/papers/lpar06_dtc_extended.pdf

  8. Cotton, S., Asarin, E., Maler, O., Niebert, P.: Some Progress in Satisfiability Checking for Difference Logic. In: Proc. FORMATS-FTRTFT 2004 (2004)

    Google Scholar 

  9. Filliâtre, J.-C., Owre, S., Rueß, H., Shankar, N.: ICS: Integrated Canonizer and Solver. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 246–249. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  10. 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)

    Chapter  Google Scholar 

  11. Nelson, G., Oppen, D.C.: Simplification by Cooperating Decision Procedures. ACM Trans. on Programming Languages and Systems 1(2), 245–257 (1979)

    Article  MATH  Google Scholar 

  12. Nieuwenhuis, R., Oliveras, A.: Congruence Closure with Integer Offsets. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 77–89. Springer, Heidelberg (2003)

    Google Scholar 

  13. Shostak, R.E.: Deciding Combinations of Theories. Journal of the ACM 31, 1–12 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  14. Zhang, L., Madigan, C.F., Moskewicz, M.H., Malik, S.: Efficient conflict driven learning in a boolean satisfiability solver. In: Proc. ICCAD 2001. IEEE Press, Los Alamitos (2001)

    Google Scholar 

  15. Zhang, L., Malik, S.: The quest for efficient boolean satisfiability solvers. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 17–36. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bruttomesso, R., Cimatti, A., Franzén, A., Griggio, A., Sebastiani, R. (2006). Delayed Theory Combination vs. Nelson-Oppen for Satisfiability Modulo Theories: A Comparative Analysis. In: Hermann, M., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2006. Lecture Notes in Computer Science(), vol 4246. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11916277_36

Download citation

  • DOI: https://doi.org/10.1007/11916277_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-48281-9

  • Online ISBN: 978-3-540-48282-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics