Abstract
Acyclicity is a recurring property of solutions to many important combinatorial problems. In this work we study embeddings of specialized acyclicity constraints in the satisfiability problem of the classical propositional logic (SAT). We propose an embedding of directed graphs in SAT, with arcs labelled with propositional variables, and an extended SAT problem in which all clauses have to be satisfied and the subgraph consisting of arcs labelled true is acyclic. We devise a constraint propagator for the acyclicity constraint and show how it can be incorporated in off-the-shelf SAT solvers. We show that all existing encodings of acyclicity constraints in SAT are either prohibitively large or do not sanction all inferences made by the constraint propagator. Our experiments demonstrate the advantages of our solver over other approaches for handling acyclicity.
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References
Audemard, G., Bertoli, P.G., Cimatti, A., Kornilowicz, A., Sebastiani, R.: A SAT based approach for solving formulas over Boolean and linear mathematical propositions. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 195–210. Springer, Heidelberg (2002)
Corander, J., Janhunen, T., Rintanen, J., Nyman, H., Pensar, J.: Learning chordal Markov networks by constraint satisfaction. In: Burges, C.J.C., Bottou, L., Welling, M., Ghahramani, Z., Weinberger, K. (eds.) Advances in Neural Information Processing Systems 26, pp. 1349–1357 (2014)
Cotton, S., Maler, O.: Fast and flexible difference constraint propagation for DPLL(T). In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 170–183. Springer, Heidelberg (2006)
Cussens, J.: Bayesian network learning by compiling to weighted MAX-SAT. In: Proceedings of the Conference on Uncertainty in Artificial Intelligence, pp. 105–112. AUAI Press (2008)
Denecker, M., Ternovska, E.: A logic of nonmonotone inductive definitions. ACM Transactions on Computational Logic 9(2), 14:1–14:52 (2008)
Dooms, G., Deville, Y., Dupont, P.E.: Cp(graph): Introducing a graph computation domain in constraint programming. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 211–225. Springer, Heidelberg (2005)
Dooms, G., Katriel, I.: The minimum spanning tree constraint. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 152–166. Springer, Heidelberg (2006)
Gebser, M., Janhunen, T., Rintanen, J.: Answer set programming as SAT modulo acyclicity. In: Proceedings of the 21st European Conference on Artificial Intelligence, ECAI 2014. IOS Press (2014)
Heljanko, K., Keinänen, M., Lange, M., Niemelä, I.: Solving parity games by a reduction to SAT. Journal for Computer and System Sciences 78(2), 430–440 (2012)
Hoffmann, H.F., van Beek, P.: A global acyclicity constraint for Bayesian network structure learning (September 2013) (unpublished manuscript in the Doctoral Program of the International Conference on Principles and Practice of Constraint Programming)
Janhunen, T.: Evaluating the effect of semi-normality on the expressiveness of defaults. Artificial Intelligence 144(1-2), 233–250 (2003)
Janhunen, T.: Some (in)translatability results for normal logic programs and propositional theories. Journal of Applied Non-Classical Logics 16(1-2), 35–86 (2006)
Kautz, H., Selman, B.: Pushing the envelope: planning, propositional logic, and stochastic search. In: Proceedings of the 13th National Conference on Artificial Intelligence and the 8th Innovative Applications of Artificial Intelligence Conference, pp. 1194–1201. AAAI Press (1996)
Lin, F., Zhao, Y.: ASSAT: Computing answer sets of a logic program by SAT solvers. Artificial Intelligence Journal 157(1), 115–137 (2004)
Mahfoudh, M., Niebert, P., Asarin, E., Maler, O.: A satisfiability checker for difference logic. In: Proceedings of SAT 2002 – Theory and Applications of Satisfiability Testing, vol. 2, pp. 222–230 (2002)
Marques-Silva, J.P., Sakallah, K.A.: GRASP: A new search algorithm for satisfiability. In: 1996 IEEE/ACM International Conference on Computer-Aided Design, ICCAD 1996. Digest of Technical Papers, pp. 220–227 (1996)
Mitchell, D.G.: A SAT solver primer. EATCS Bulletin 85, 112–133 (2005)
Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of the 38th ACM/IEEE Design Automation Conference (DAC 2001), pp. 530–535. ACM Press (2001)
Niemelä, I.: Stable models and difference logic. Annals of Mathematics and Artificial Intelligence 53(1-4), 313–329 (2008)
Nieuwenhuis, R., Oliveras, A.: DPLL(T) with exhaustive theory propagation and its application to difference logic. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 321–334. Springer, Heidelberg (2005)
Rintanen, J., Heljanko, K., Niemelä, I.: Parallel encodings of classical planning as satisfiability. In: Alferes, J.J., Leite, J. (eds.) JELIA 2004. LNCS (LNAI), vol. 3229, pp. 307–319. Springer, Heidelberg (2004)
Subbarayan, S., Pradhan, D.K.: NiVER: Non-increasing variable elimination resolution for preprocessing SAT instances. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 276–291. Springer, Heidelberg (2005)
Tamura, N., Taga, A., Kitagawa, S., Banbara, M.: Compiling finite linear CSP into SAT. Constraints 14(2), 254–272 (2009)
Tarjan, R.E.: Depth first search and linear graph algorithms. SIAM Journal on Computing 1(2), 146–160 (1972)
Wolfman, S.A., Weld, D.S.: The LPSAT engine & its application to resource planning. In: Proceedings of the 16th International Joint Conference on Artificial Intelligence, pp. 310–315. Morgan Kaufmann Publishers (1999)
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Gebser, M., Janhunen, T., Rintanen, J. (2014). SAT Modulo Graphs: Acyclicity. In: Fermé, E., Leite, J. (eds) Logics in Artificial Intelligence. JELIA 2014. Lecture Notes in Computer Science(), vol 8761. Springer, Cham. https://doi.org/10.1007/978-3-319-11558-0_10
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DOI: https://doi.org/10.1007/978-3-319-11558-0_10
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