Abstract
We extend clause learning as performed by most modern SAT Solvers by integrating the computation of Boolean Gröbner bases into the conflict learning process. Instead of learning only one clause per conflict, we compute and learn additional binary clauses from a Gröbner basis of the current conflict. We used the Gröbner basis engine of the logic package Redlog contained in the computer algebra system Reduce to extend the SAT solver MiniSAT with Gröbner basis learning. Our approach shows a significant reduction of conflicts and a reduction of restarts and computation time on many hard problems from the SAT 2009 competition.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Lynce, I., Marques da Silva, J.P.: SAT in bioinformatics: Making the case with haplotype inference. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 136–141. Springer, Heidelberg (2006)
Bonet, M.L., John, K.S.: Efficiently calculating evolutionary tree measures using SAT. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 4–17. Springer, Heidelberg (2009)
Clarke, E.M., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Formal Methods in System Design 19(1), 7–34 (2001)
Biere, A., Cimatti, A., Clarke, E.M., Strichman, O., Zhu, Y.: Bounded model checking. In: Zelkowitz, M. (ed.) Highly Dependable Software. Advances in Computers, vol. 58. Academic Press, San Diego (2003)
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Communications of the ACM 5(7), 394–397 (1962)
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, Universität Innsbruck (1965)
Kapur, D., Narendran, P.: An equational approach to theorem proving in first-order predicate calculus. ACM SIGSOFT Notes 10(4), 63–66 (1985)
Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009)
Van Gelder, A.: Combining preorder and postorder resolution in a satisfiability solver. Electronic Notes in Discrete Mathematics 9, 115–128 (2001)
Bacchus, F.: Enhancing davis putnam with extended binary clause reasoning. In: 18th National Conference on Artificial Intelligence, pp. 613–619. AAAI Press, Menlo Park (2002)
Condrat, C., Kalla, P.: A gröbner basis approach to CNF-formulae preprocessing. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 618–631. Springer, Heidelberg (2007)
Dershowitz, N., Hsiang, J., Huang, G.S., Kaiss, D.: Boolean rings for intersection-based satisfiability. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 482–496. Springer, Heidelberg (2006)
Seidl, A.M., Sturm, T.: Boolean quantification in a first-order context. In: Proceedings of the CASC 2003. Institut für Informatik, Technische Universität München, Garching, pp. 329–345 (2003)
Beame, P., Kautz, H., Sabharwal, A.: Towards understanding and harnessing the potential of clause learning. JAIR 22(1), 319–351 (2004)
Stone, M.H.: The theory of representations for boolean algebras. Transactions of the American Mathematical Society 40, 37–111 (1936)
Brickenstein, M., Dreyer, A., Greuel, G.M., Wedler, M.: New developments in the theory of gröbner bases and applications to formal verification. Journal of Pure and Applied Algebra 213, 1612–1635 (2009)
Küchlin, W.: Canonical hardware representation using Gröbner bases. In: Proceedings of the A3L 2005, Passau, Germany, pp. 147–154 (2005)
Becker, T., Weispfenning, V.: Gröbner Bases: A Computational Approach to Commutative Algebra. Springer, New York (1993)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Springer, New York (2007)
Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)
Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)
Sturm, T., Zengler, C.: Parametric quantified SAT solving. In: Proceedings of the ISSAC 2010. ACM, New York (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zengler, C., Küchlin, W. (2010). Extending Clause Learning of SAT Solvers with Boolean Gröbner Bases. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2010. Lecture Notes in Computer Science, vol 6244. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15274-0_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-15274-0_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15273-3
Online ISBN: 978-3-642-15274-0
eBook Packages: Computer ScienceComputer Science (R0)