Skip to main content
SpringerLink
Log in

Probabilistic satisfiability: algorithms with the presence and absence of a phase transition

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

We study algorithms for probabilistic satisfiability (PSAT), an NP-complete problem that is central to logic-probabilisti reasoning, focusing on the presence and absence of a phase transition phenomenon for each algorithm. Our study starts by defining a PSAT normal form, on which all algorithms are based. The proposed algorithms consist of several forms of reductions of PSAT to classical propositional satisfiability (SAT). The first algorithm is a canonical reduction of PSAT instances to SAT instances; three other algorithms are reductions to linear optimization with distinct column generation procedures, namely on auxiliary calls to SAT, weighted MAXSAT or SMT solvers. Theoretical and practical limitations of each algorithm are discussed. Several implementations were developed and compared by means of experiments using randomly generated input problems. Some of the implementations are shown to present a phase transition behavior. We show that variations of these algorithms may lead to the partial occlusion of the phase transition phenomenon and discuss the reasons for this change ofc practical behavior.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andersen, K., Pretolani, D.: Easy cases of probabilistic satisfiability. Ann. Math. Artif. Intell. 33(1), 69–91 (2001)

    Article  MathSciNet  Google Scholar 

  2. Andrade, P.S.S., Rocha, J., Couto, D.P., da Costa Teves, A., Cozman, F.: A toolset for propositional probabilistic logic. In: Dutra, I. de C. (ed.) ENIA 2007 (2007)

  3. Baioletti, M., Capotorti, A., Tulipani, S., Vantaggi, B.: Elimination of boolean variables for probabilistic coherence. Soft Computing-A Fusion of Foundations. Methodologies Appl. 4(2), 81–88 (2000)

    Google Scholar 

  4. Baioletti, M., Capotorti, A., Tulipani, S., Vantaggi, B.: Simplification rules for the coherent probability assessment problem. Ann. Math. Artif. Intell. 35(1), 11–28 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barrett, C.W., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Handbook of Satisfiability, pp. 825–885 (2009)

  6. Bertsimas, D., Tsitsiklis, J.N.: Introduction to linear optimization. Athena Scientific (1997)

  7. Boole, G.: An Investigation on the Laws of Thought. Macmillan. Available on project Gutemberg at, London (1854). http://www.gutenberg.org/etext/15114

    Google Scholar 

  8. Borgward, K.H.: The Simplex Method: A Probabilistic Analysis. Algorithms and Combinatorics 1. Springer (1986)

  9. Cheeseman, P., Kanefsky, B., Taylor, W.M.: Where the really hard problems are. In: 12th IJCAI, pp. 331–337. Morgan Kaufmann 1991. http://portal.acm.org/citation.cfm?id=1631171.1631221

  10. Coletti, G., Scozzafava, R.: Probabilistic Logic in a Coherent Setting. Trends in Logic, Vol. 15: Studia Logica Library. Kluwer Academic Publishers (2002)

  11. Cook, S.A.: The complexity of theorem-proving procedures. In: Conference Record of Third Annual ACM Symposium on Theory of Computing STOC, pp. 151–158. ACM (1971)

  12. Cozman, F.G., di Ianni, L.F.: Probabilistic satisfiability and coherence checking through integer programming. Int. J. Approx. Reason. 58, 57–70 (2014)

    Article  Google Scholar 

  13. Finetti, de B.: Theory of Probability, vol. 1. Wiley, London (1974)

    Google Scholar 

  14. Eckhoff, J., Helly, Radon: Caratheodory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, pp. 389–448. Elsevier Science Publishers (1993)

  15. Eén, N., Sörensson, N.: An extensible SAT-solver (2003)

  16. Eén, N., Sörensson, N.: Translating pseudo-boolean constraints into SAT. JSAT 2(1-4), 1–26 (2006)

    MATH  Google Scholar 

  17. Finger, M., De Bona, G.: A refuted conjecture on probabilistic satisfiability. In: SBIA, pp. 293–302 (2010)

  18. Finger, M., De Bona, G.: Probabilistic satisfiability: Logic-based algorithms and phase transition (2011)

  19. Finger, M., Le Bras, R., Gomes, C.P., Selman, B.: Solutions for hard and soft constraints using optimized probabilistic satisfiability. In: Theory and Applications of Satisfiability Testing–SAT 2013, pp. 233–249. Springer (2013)

  20. Garling, D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press (2007)

  21. Gent, I.P., Walsh, T.: The SAT phase transition. In: ECAI94 – Proceedings of the Eleventh European Conference on Artificial Intelligence, pp. 105–109. John Wiley & Sons (1994)

  22. Georgakopoulos, G., Kavvadias, D., Papadimitriou, C.H.: Probabilistic satisfiability. J. Complex. 41, 1–11 (1988)

    Article  MathSciNet  Google Scholar 

  23. Hailperin, T.: Probability logic Notre Dame . J. Form. Log. 25(3), 198–212 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hailperin, T.: Boole’s Logic and Probability, Studies in Logic and the Foundations of Mathematics, vol. 85, second enlarged edn. North-Holland, Amsterdam (1986)

    Google Scholar 

  25. Hansen, P., Jaumard, B.: Algorithms for the maximum satisfiability problem. Comput. 44, 279–303 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hansen, P., Jaumard, B.: Probabilistic satisfiability. In: Handbook of Defeasible Reasoning and Uncertainty Management Systems. Vol.5, p. 321. Springer Netherlands (2000)

  27. Hansen, P., Jaumard, B., Aragão, de M.P., Chauny, F., Perron, S.: Probabilistic satisfiability with imprecise probabilities. In: 1st International Symposium on Imprecise Probabilities and Their Applications, pp165–174 (1999)

  28. Hansen, P., Jaumard, B., Nguetsé, G.B.D., Aragão, de M.P.: Models and algorithms for probabilistic and bayesian logic. In: IJCAI, pp 1862–1868 (1995)

  29. Hansen, P., Perron, S.: Merging the local and global approaches to probabilistic satisfiability. Int. J. Approx. Reason. 47(2), 125–140 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Heule, M., van Maaren, H.: march hi: solver description. In: SAT Competition, pp. 27–28 (2009)

  31. Jaumard, B., Hansen, P., Aragão, de M.P.: Column generation methods for probabilistic logic. INFORMS J. Comput. 3(2), 135–148 (1991)

    Article  MATH  Google Scholar 

  32. Kavvadias, D., Papadimitriou, C.H.: A linear programming approach to reasoning about probabilities. Ann. Math. Artif. Intell. 1, 189–205 (1990)

    Article  MATH  Google Scholar 

  33. Klinov, P., Parsia, B.: A hybrid method for probabilistic satisfiability. In: Automated Deduction–CADE-23, pp. 354–368. Springer (2011)

  34. Klinov, P., Parsia, B.: Pronto: A practical probabilistic description logic reasoner. In: Uncertainty Reasoning for the Semantic Web II (URSW), LNCS, vol. 7123, pp. 59–79. Springer (2013)

  35. Klinov, P., Parsia, B., Muiño, D.P.: The consistency of the medical expert system cadiag-2: A probabilistic approach. Interdiscip. Adv.Inf. Technol. Res., 1 (2013)

  36. Levin, L.A.: Universal search problems. Probl. Inf. Transm. 9(3), 265–266 (1973)

    Google Scholar 

  37. Li, C., Manya, F., Mohamedou, N., Planes, J.: Exploiting cycle structures in Max-SAT. In: Theory and Applications of Satisfiability Testing — SAT, pp. 467–480. Springer (2009)

  38. Marques-Silva, J.: The MSUNCORE MAXSAT solver. In: SAT competition, pp. 151–152 (2009)

  39. Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an Efficient SAT Solver. In: Proceedings of the 38th Design Automation Conference DAC 01, pp. 530–535 (2001)

  40. de Moura, L.M.: Z3: An efficient SMT solver. In: TACAS, pp. 337–340 (2008)

  41. Nilsson, N.: Probabilistic logic. Artif. Intell. 28(1), 71–87 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  42. Nilsson, N.: Probabilistic logic revisited. Artif. Intell. 59(1–2), 39–42 (1993)

    Article  MathSciNet  Google Scholar 

  43. Odifreddi, P.: Classical Recursion Theory, Studies in Logic and the Foundations of Mathematics, vol. 125. Elsevier (1989)

  44. Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity Dover (1998)

  45. Prestwich, S.D.: CNF encodings. In: Handbook of Satisfiability, pp. 75–97 (2009)

  46. Pretolani, D.: Probability Logic and Optimization SAT: The PSAT and CPA Models. Ann. Math. Artif. Intell. 43 (1-4), 211–221 (2005). doi:10.1007/s10472-004-9430-3

    Article  MATH  MathSciNet  Google Scholar 

  47. Savage, J.E.: The Complexity of Computing. Krieger Publishing Co., Inc., Melbourne, FL USA (1976)

  48. Sebastiani, R.: Lazy satisfiability modulo theories. JSAT 3(3-4), 141–224 (2007)

    MATH  MathSciNet  Google Scholar 

  49. Todd, M.J.: The many facets of linear programming. Math. Program. 91, 417–436 (2002). doi:10.1007/s101070100261

    Article  MATH  MathSciNet  Google Scholar 

  50. Walley, P., Pelessoni, R., Vicig, P.: Direct algorithms for checking consistency and making inferences from conditional probability assessments. J. Stat. Plan. Infer. 126 (1), 119–151 (2004). doi:10.1016/j.jspi.2003.09.005 http://www.sciencedirect.com/science/article/B6V0M-49TJMSP-1/2/ fd1ab649db4753c886c2ad0535b889ea

    Article  MATH  MathSciNet  Google Scholar 

  51. Warners, J.P.: A linear-time transformation of linear inequalities into conjunctive normal form. Inf. Process. Lett. 68(2), 63–69 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of Sao Paulo, Sao Paulo, Brazil

    Marcelo Finger & Glauber De Bona

Authors
  1. Marcelo Finger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Glauber De Bona
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Glauber De Bona.

Additional information

This work was partially supported by CNPq grant PQ 302553/2010-0.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Finger, M., De Bona, G. Probabilistic satisfiability: algorithms with the presence and absence of a phase transition. Ann Math Artif Intell 75, 351–389 (2015). https://doi.org/10.1007/s10472-015-9466-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-015-9466-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation