Advertisement

Computing with SAT Oracles: Past, Present and Future

  • Joao Marques-Silva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10936)

Abstract

Boolean Satisfiability (SAT) epitomizes NP-completeness, and so what is arguably the best known class of intractable problems. NP-complete decision problems are pervasive in all areas of computing, with literally thousands of well-known examples. Nevertheless, SAT solvers routinely challenge the problem’s intractability by solving propositional formulas with millions of variables, many representing the translation from some other NP-complete or NP-hard problem. The practical effectiveness of SAT solvers has motivated their use as oracles for NP, enabling new algorithms that solve an ever-increasing range of hard computational problems. This paper provides a brief overview of this ongoing effort, summarizing some of the recent past and present main successes, and highlighting directions for future research.

References

  1. 1.
    Abío, I., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E., Mayer-Eichberger, V.: A new look at BDDs for pseudo-boolean constraints. J. Artif. Intell. Res. 45, 443–480 (2012)MathSciNetMATHGoogle Scholar
  2. 2.
    Achá, R.J.A., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Practical algorithms for unsatisfiability proof and core generation in SAT solvers. AI Commun. 23(2–3), 145–157 (2010)MathSciNetMATHGoogle Scholar
  3. 3.
    Angelino, E., Larus-Stone, N., Alabi, D., Seltzer, M., Rudin, C.: Learning certifiably optimal rule lists. In: KDD, pp. 35–44 (2017)Google Scholar
  4. 4.
    Arif, M.F., Mencía, C., Marques-Silva, J.: Efficient MUS enumeration of horn formulae with applications to axiom pinpointing. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 324–342. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-24318-4_24CrossRefMATHGoogle Scholar
  5. 5.
    Asín, R., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Cardinality networks: a theoretical and empirical study. Constraints 16(2), 195–221 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P.F. (eds.): The Description Logic Handbook: Theory, Implementation, and Applications. Cambridge University Press, Cambridge (2007)MATHGoogle Scholar
  7. 7.
    Baader, F., Horrocks, I., Lutz, C., Sattler, U.: An Introduction to Description Logic. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  8. 8.
    Bacchus, F., Davies, J., Tsimpoukelli, M., Katsirelos, G.: Relaxation search: a simple way of managing optional clauses. In: AAAI, pp. 835–841 (2014)Google Scholar
  9. 9.
    Bacchus, F., Katsirelos, G.: Using minimal correction sets to more efficiently compute minimal unsatisfiable sets. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9207, pp. 70–86. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21668-3_5CrossRefGoogle Scholar
  10. 10.
    Bailleux, O., Boufkhad, Y., Roussel, O.: New encodings of Pseudo-Boolean constraints into CNF. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 181–194. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-02777-2_19CrossRefMATHGoogle Scholar
  11. 11.
    Belov, A., Lynce, I., Marques-Silva, J.: Towards efficient MUS extraction. AI Commun. 25(2), 97–116 (2012)MathSciNetMATHGoogle Scholar
  12. 12.
    Beyersdorff, O., Chew, L., Schmidt, R.A., Suda, M.: Lifting QBF resolution calculi to DQBF. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 490–499. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40970-2_30CrossRefGoogle Scholar
  13. 13.
    Beyersdorff, O., Pich, J.: Understanding Gentzen and Frege systems for QBF. In: LICS, pp. 146–155 (2016)Google Scholar
  14. 14.
    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)MATHGoogle Scholar
  15. 15.
    Buss, S., Bonet, M.L., Ignatiev, A., Marques-Silva, J., Morgado, A.: MaxSAT resolution with the dual rail encoding. In: AAAI, February 2018Google Scholar
  16. 16.
    Cadoli, M., Schaerf, A.: Compiling problem specifications into SAT. Artif. Intell. 162(1–2), 89–120 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chakraborty, S., Meel, K.S., Vardi, M.Y.: Algorithmic improvements in approximate counting for probabilistic inference: from linear to logarithmic SAT calls. In: IJCAI, pp. 3569–3576 (2016)Google Scholar
  18. 18.
    Cimatti, A., Griggio, A.: Software model checking via IC3. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 277–293. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31424-7_23CrossRefGoogle Scholar
  19. 19.
    Cook, S.A.: The complexity of theorem-proving procedures. In: STOC, pp. 151–158 (1971)Google Scholar
  20. 20.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3CrossRefMATHGoogle Scholar
  21. 21.
    Davies, J., Bacchus, F.: Solving MAXSAT by solving a sequence of simpler SAT instances. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 225–239. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-23786-7_19CrossRefGoogle Scholar
  22. 22.
    de Haan, R., Szeider, S.: The parameterized complexity of reasoning problems beyond NP. In: KR (2014)Google Scholar
  23. 23.
    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).  https://doi.org/10.1007/978-3-540-24605-3_37. MiniSat 2.2. https://github.com/niklasso/minisat.gitCrossRefGoogle Scholar
  24. 24.
    Eén, N., Sörensson, N.: Translating pseudo-Boolean constraints into SAT. JSAT 2(1–4), 1–26 (2006)MATHGoogle Scholar
  25. 25.
    Fomin, F.V., Kaski, P.: Exact exponential algorithms. Commun. ACM 56(3), 80–88 (2013)CrossRefGoogle Scholar
  26. 26.
    Fu, Z., Malik, S.: On solving the partial MAX-SAT problem. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 252–265. Springer, Heidelberg (2006).  https://doi.org/10.1007/11814948_25CrossRefGoogle Scholar
  27. 27.
    Ganzinger, H., Korovin, K.: New directions in instantiation-based theorem proving. In: LICS, pp. 55–64 (2003)Google Scholar
  28. 28.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)MATHGoogle Scholar
  29. 29.
    Gebser, M., Kaminski, R., Kaufmann, B., Schaub, T.: Answer Set Solving in Practice. Morgan & Claypool Publishers, San Rafael (2012)MATHGoogle Scholar
  30. 30.
    Grégoire, É., Lagniez, J., Mazure, B.: An experimentally efficient method for (MSS, CoMSS) partitioning. In: AAAI, pp. 2666–2673 (2014)Google Scholar
  31. 31.
    Helmert, M.: The fast downward planning system. J. Artif. Intell. Res. 26, 191–246 (2006)CrossRefGoogle Scholar
  32. 32.
    Heras, F., Morgado, A., Marques-Silva, J.: Core-guided binary search algorithms for maximum satisfiability. In: AAAI (2011)Google Scholar
  33. 33.
    Heras, F., Morgado, A., Planes, J., Silva, J.P.M.: Iterative SAT solving for minimum satisfiability. In: ICTAI, pp. 922–927 (2012)Google Scholar
  34. 34.
    Heule, M.J.H., Kullmann, O.: The science of brute force. Commun. ACM 60(8), 70–79 (2017)CrossRefGoogle Scholar
  35. 35.
    Heule, M.J.H., Kullmann, O., Marek, V.W.: Solving very hard problems: cube-and-conquer, a hybrid SAT solving method. In: IJCAI, pp. 4864–4868 (2017)Google Scholar
  36. 36.
    Ignatiev, A., Morgado, A., Marques-Silva, J.: Propositional abduction with implicit hitting sets. In: ECAI, pp. 1327–1335 (2016)Google Scholar
  37. 37.
    Ignatiev, A., Morgado, A., Marques-Silva, J.: On tackling the limits of resolution in SAT solving. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 164–183. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-66263-3_11CrossRefMATHGoogle Scholar
  38. 38.
    Ignatiev, A., Pereira, F., Narodytska, N., Marques-Silva, J.: A SAT-based approach to learn explainable decision sets. In: IJCAR (2018)CrossRefGoogle Scholar
  39. 39.
    Ignatiev, A., Previti, A., Marques-Silva, J.: SAT-based formula simplification. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 287–298. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-24318-4_21CrossRefMATHGoogle Scholar
  40. 40.
    Janhunen, T., Niemelä, I.: Compact translations of non-disjunctive answer set programs to propositional clauses. In: Balduccini, M., Son, T.C. (eds.) Logic Programming, Knowledge Representation, and Nonmonotonic Reasoning. LNCS (LNAI), vol. 6565, pp. 111–130. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-20832-4_8CrossRefMATHGoogle Scholar
  41. 41.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.M.: Solving QBF with counterexample guided refinement. Artif. Intell. 234, 1–25 (2016)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Janota, M., Lynce, I., Marques-Silva, J.: Algorithms for computing backbones of propositional formulae. AI Commun. 28(2), 161–177 (2015)MathSciNetMATHGoogle Scholar
  43. 43.
    Janota, M., Marques-Silva, J.: Abstraction-based algorithm for 2QBF. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 230–244. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-21581-0_19CrossRefGoogle Scholar
  44. 44.
    Janota, M., Marques-Silva, J.: Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577, 25–42 (2015)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Janota, M., Marques-Silva, J.: Solving QBF by clause selection. In: IJCAI, pp. 325–331 (2015)Google Scholar
  46. 46.
    Janota, M., Marques-Silva, J.: On the query complexity of selecting minimal sets for monotone predicates. Artif. Intell. 233, 73–83 (2016)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Kautz, H.A., Selman, B.: Unifying SAT-based and graph-based planning. In: IJCAI, pp. 318–325 (1999)Google Scholar
  48. 48.
    Korovin, K.: Inst-Gen – a modular approach to instantiation-based automated reasoning. In: Voronkov, A., Weidenbach, C. (eds.) Programming Logics: Essays in Memory of Harald Ganzinger. LNCS, vol. 7797, pp. 239–270. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-37651-1_10CrossRefGoogle Scholar
  49. 49.
    Kroening, D., Strichman, O.: Decision Procedures - An Algorithmic Point of View, 2nd edn. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-50497-0CrossRefMATHGoogle Scholar
  50. 50.
    Kullmann, O., Marques-Silva, J.: Computing maximal autarkies with few and simple oracle queries. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 138–155. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-24318-4_11CrossRefMATHGoogle Scholar
  51. 51.
    Lagniez, J., Berre, D.L., de Lima, T., Montmirail, V.: A recursive shortcut for CEGAR: application to the modal logic K satisfiability problem. In: IJCAI, pp. 674–680 (2017)Google Scholar
  52. 52.
    Lakkaraju, H., Bach, S.H., Leskovec, J.: Interpretable decision sets: a joint framework for description and prediction. In: KDD, pp. 1675–1684 (2016)Google Scholar
  53. 53.
    Li, O., Liu, H., Chen, C., Rudin, C.: Deep learning for case-based reasoning through prototypes: a neural network that explains its predictions. In: AAAI, February 2018Google Scholar
  54. 54.
    Liang, J.H., Ganesh, V., Poupart, P., Czarnecki, K.: Exponential recency weighted average branching heuristic for SAT solvers. In: AAAI, pp. 3434–3440 (2016)Google Scholar
  55. 55.
    Liffiton, M.H., Previti, A., Malik, A., Marques-Silva, J.: Fast, flexible MUS enumeration. Constraints 21(2), 223–250 (2016)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Luo, M., Li, C., Xiao, F., Manyà, F., Lü, Z.: An effective learnt clause minimization approach for CDCL SAT solvers. In: IJCAI, pp. 703–711 (2017)Google Scholar
  57. 57.
    Marques-Silva, J., Heras, F., Janota, M., Previti, A., Belov, A.: On computing minimal correction subsets. In: IJCAI, pp. 615–622 (2013)Google Scholar
  58. 58.
    Marques-Silva, J., Janota, M., Belov, A.: Minimal sets over monotone predicates in Boolean formulae. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 592–607. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39799-8_39CrossRefGoogle Scholar
  59. 59.
    Marques-Silva, J., Janota, M., Ignatiev, A., Morgado, A.: Efficient model based diagnosis with maximum satisfiability. In: IJCAI, pp. 1966–1972 (2015)Google Scholar
  60. 60.
    Marques-Silva, J., Janota, M., Mencía, C.: Minimal sets on propositional formulae. problems and reductions. Artif. Intell. 252, 22–50 (2017)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Marques-Silva, J., Planes, J.: On using unsatisfiability for solving maximum satisfiability. CoRR, abs/0712.1097 (2007)Google Scholar
  62. 62.
    Marques-Silva, J., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Martins, R., Joshi, S., Manquinho, V., Lynce, I.: Incremental cardinality constraints for MaxSAT. In: O’Sullivan, B. (ed.) CP 2014. LNCS, vol. 8656, pp. 531–548. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-10428-7_39CrossRefGoogle Scholar
  64. 64.
    McMillan, K.L.: Interpolation and SAT-based model checking. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-45069-6_1CrossRefGoogle Scholar
  65. 65.
    Mencía, C., Ignatiev, A., Previti, A., Marques-Silva, J.: MCS extraction with sublinear oracle queries. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 342–360. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40970-2_21CrossRefGoogle Scholar
  66. 66.
    Mencía, C., Previti, A., Marques-Silva, J.: Literal-based MCS extraction. In: IJCAI, pp. 1973–1979 (2015)Google Scholar
  67. 67.
    Morgado, A., Dodaro, C., Marques-Silva, J.: Core-guided MaxSAT with soft cardinality constraints. In: O’Sullivan, B. (ed.) CP 2014. LNCS, vol. 8656, pp. 564–573. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-10428-7_41CrossRefGoogle Scholar
  68. 68.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: DAC, pp. 530–535 (2001)Google Scholar
  69. 69.
    Nadel, A., Ryvchin, V., Strichman, O.: Efficient MUS extraction with resolution. In: FMCAD, pp. 197–200 (2013)Google Scholar
  70. 70.
    Narodytska, N., Bacchus, F.: Maximum satisfiability using core-guided MaxSAT resolution. In: AAAI, pp. 2717–2723 (2014)Google Scholar
  71. 71.
    Narodytska, N., Ignatiev, A., Pereira, F., Marques-Silva, J.: Learning optimal decision trees with SAT. In: IJCAI (2018)Google Scholar
  72. 72.
    Niemetz, A., Preiner, M., Biere, A.: Propagation based local search for bit-precise reasoning. Form. Methods Syst. Des. 51(3), 608–636 (2017)CrossRefGoogle Scholar
  73. 73.
    Nöhrer, A., Biere, A., Egyed, A.: Managing SAT inconsistencies with HUMUS. In: VaMoS, pp. 83–91 (2012)Google Scholar
  74. 74.
    Ohrimenko, O., Stuckey, P.J., Codish, M.: Propagation via lazy clause generation. Constraints 14(3), 357–391 (2009)MathSciNetCrossRefGoogle Scholar
  75. 75.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1993)MATHGoogle Scholar
  76. 76.
    Piskac, R., de Moura, L.M., Bjørner, N.: Deciding effectively propositional logic using DPLL and substitution sets. J. Autom. Reason. 44(4), 401–424 (2010)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Previti, A., Ignatiev, A., Morgado, A., Marques-Silva, J.: Prime compilation of non-clausal formulae. In: IJCAI, pp. 1980–1988 (2015)Google Scholar
  78. 78.
    Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symb. Log. 62(3), 981–998 (1997)MathSciNetCrossRefGoogle Scholar
  79. 79.
    Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987)MathSciNetCrossRefGoogle Scholar
  80. 80.
    Sebastiani, R.: Lazy satisfiability modulo theories. JSAT 3(3–4), 141–224 (2007)MathSciNetMATHGoogle Scholar
  81. 81.
    Sinz, C.: Towards an optimal CNF encoding of Boolean cardinality constraints. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 827–831. Springer, Heidelberg (2005).  https://doi.org/10.1007/11564751_73CrossRefMATHGoogle Scholar
  82. 82.
    Suda, M.: Property directed reachability for automated planning. J. Artif. Intell. Res. 50, 265–319 (2014)MathSciNetMATHGoogle Scholar
  83. 83.
    Torlak, E., Jackson, D.: Kodkod: a relational model finder. In: TACAS, pp. 632–647 (2007)Google Scholar
  84. 84.
    Voronkov, A.: AVATAR: the architecture for first-order theorem provers. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 696–710. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-08867-9_46CrossRefGoogle Scholar
  85. 85.
    Walsh, T.: SAT v CSP. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 441–456. Springer, Heidelberg (2000).  https://doi.org/10.1007/3-540-45349-0_32CrossRefGoogle Scholar
  86. 86.
    Warners, J.P.: A linear-time transformation of linear inequalities into conjunctive normal form. Inf. Process. Lett. 68(2), 63–69 (1998)MathSciNetCrossRefGoogle Scholar
  87. 87.
    Woeginger, G.J.: Exact algorithms for NP-hard problems: a survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization — Eureka, You Shrink!. LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-36478-1_17CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LASIGE, Faculty of ScienceUniversity of LisbonLisbonPortugal

Personalised recommendations