Abstract
Finding small unsatisfiable subformulas (unsat cores) of infeasible propositional SAT problems is an active area of research. Analogous investigations in the polynomial algebra domain are, however, somewhat lacking. This paper investigates an algorithmic approach to identify a small unsatisfiable core of a set of polynomials, where the corresponding polynomial ideal is found to have an empty variety. We show that such a core can be identified by employing extensions of the Buchberger’s algorithm. By further analyzing S-polynomial reductions, we identify certain conditions that are helpful in ascertaining whether or not a polynomial from the given generating set is a part of the unsat core. Our algorithm cannot guarantee a minimal unsat core; the paper describes an approach to refine the identified core. Experiments are performed on a variety of instances using a computer-algebra implementation of our algorithm.
This research is funded in part by the US National Science Foundation grants CCF-1320335 and CCF-1320385.
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References
de Siqueira, J.L., Puget, J.-F.: Explanation-based generalization of failures. In: Proceedings of European Conference Artificial Intelligence, pp. 339–344 (1988)
Jiang, J.-H.R., Lee, C.-C., Mishchenko, A., Huang, C.-Y.: To SAT or Not to SAT: scalable exploration of functional dependency. IEEE Trans. Comp. 59(4), 457–466 (2010)
Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. J. ACM 50(5), 752–794 (2003)
Marques-Silva, J.: Minimal unsatisfiability: models, algorithms and applications. In: IEEE International Symposium on Multi-Valued Logic, pp. 9–14 (2010)
Nadel, A., Ryvchin, V., Strichman, O.: Accelerated deletion-based extraction of minimal unsatisfiable cores. J. Satisfiability Boolean Model. Comput. 9, 27–51 (2014)
Belov, A., Lynce, I., Marques-Silva, J.: Towards efficient MUS extraction. AI Commun. 25(2), 97–116 (2012)
Adams, W.W., Loustaunau, P.: An Introduction to Gröbner Bases. American Mathematical Society, Providence (1994)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, New York (2007)
Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Rest-klassen-ringes nach einem nulldimensionalen Polynomideal, Ph.D. Dissertation. Philosophiesche Fakultat an der Leopold-Franzens-Universitat, Austria (1965)
Clegg, M., Edmonds, J., Impagliazzo, R.: Using the Gröbner basis algorithm to find proofs of unsatisfiability. In: ACM Symposium on Theory of Computing, pp. 174–183 (1996)
Beame, P., Impagliazzo, R., Krajiček, J., Pitassi, T., Pudlák, P.: Lower bounds on Hilbert’s Nullstellensatz and propositional proofs. Proc. Lond. Math. Soc. 73, 1–26 (1996)
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)
Zengler, C., Küchlin, W.: Extending clause learning of SAT solvers with Boolean Gröbner bases. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2010. LNCS, vol. 6244, pp. 293–302. Springer, Heidelberg (2010)
Brickenstein, M., Dreyer, A.: Polybori: a framework for Gröbner basis computations with Boolean polynomials. J. Symbolic Comput. 44(9), 1326–1345 (2009)
Vardi, M.Y., Tran, Q.: Groebner bases computation in Boolean rings for symbolic model checking. In: IASTED (2007)
van Loon, J.N.M.: Irreducibly inconsistent systems of linear inequalities. Eur. J. Oper. Res. 8(3), 283–288 (1981)
Chinneck, J.W., Dravnieks, E.: Locating minimal infeasible constraint sets in linear programs. INFORMS J. Comput. 3(2), 157–168 (1991)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3–1-3 — A computer algebra system for polynomial computations (2011). http://www.singular.uni-kl.de
Agnew, G.B., Mullin, R.C., Onyszchuk, I., Vanstone, S.A.: An implementation for a fast public-key cryptosystem. J. Cryptology 3(2), 63–79 (1991)
Reyhani-Masoleh, A., Hasan, M.A.: Low complexity word-level sequential normal basis multipliers. IEEE Trans. Comput. 54(2), 98–110 (2005)
Lv, J., Kalla, P., Enescu, F.: Efficient Gröbner basis reductions for formal verification of Galois field arithmetic circuits. IEEE Trans. CAD 32(9), 1409–1420 (2013)
Gao, S., Platzer, A., Clarke, E.M.: Quantifier elimination over finite fields using Gröbner bases. In: Winkler, F. (ed.) CAI 2011. LNCS, vol. 6742, pp. 140–157. Springer, Heidelberg (2011)
Kalla, P.: Formal verification of arithmetic datapaths using algebraic geometry and symbolic computation. In: Invited Tutorial, Proceedings of FMCAD, p. 2 (2015). http://www.cs.utexas.edu/users/hunt/FMCAD/FMCAD15/slides/fmcad15-tutorial-kalla.pdf
Sayed-Ahmed, A., Große, D., Kühne, U., Soeken, M., Drechsler, R.: Formal verification of integer multipliers by combining Gröbner basis with logic reduction. In: Proceedings of Design Automation and Test in Europe, pp. 1048–1053 (2016)
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Sun, X., Ilioaea, I., Kalla, P., Enescu, F. (2016). Finding Unsatisfiable Cores of a Set of Polynomials Using the Gröbner Basis Algorithm. In: Rueher, M. (eds) Principles and Practice of Constraint Programming. CP 2016. Lecture Notes in Computer Science(), vol 9892. Springer, Cham. https://doi.org/10.1007/978-3-319-44953-1_54
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