Skip to main content
Springer Nature Link
Log in

Probability logic and optimization SAT: The PSAT and CPA models

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

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Both probabilistic satisfiability (PSAT) and the check of coherence of probability assessment (CPA) can be considered as probabilistic counterparts of the classical propositional satisfiability problem (SAT). Actually, CPA turns out to be a particular case of PSAT; in this paper, we compare the computational complexity of these two problems for some classes of instances. First, we point out the relations between these probabilistic problems and two well known optimization counterparts of SAT, namely Max SAT and Min SAT. We then prove that Max SAT with unrestricted weights is NP-hard for the class of graph formulas, where Min SAT can be solved in polynomial time. In light of the aforementioned relations, we conclude that PSAT is NP-complete for ideal formulas, where CPA can be solved in linear time.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. F.A. Aloul, A. Ramani, I.L. Markov and K.A. Sakallah, PBS: A backtrack-search pseudo-Boolean solver and optimizer, presented at SAT02 — Fifth Int. Symp. on Theory and Applications of Satisfiability Testing, Cincinnati, OH (2002).

  2. K.A. Andersen and D. Pretolani, Easy cases of probabilistic satisfiability, Annals of Mathematics and Artificial Intelligence 33(1) (2001) 69–91.

    Article  MathSciNet  Google Scholar 

  3. M. Baioletti, A. Capotorti, S. Tulipani and B. Vantaggi, Elimination of Boolean variables for probabilistic coherence, Software Computing 4(2) (2000) 81–88.

    Article  Google Scholar 

  4. M. Baioletti, A. Capotorti, S. Tulipani and B. Vantaggi, Simplification rules for the coherent probability assessment problem, Annals of Mathematics and Artificial Intelligence 35 (2002) 11–28.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Boole, An Investigation of the Laws of Thought (Walton and Maberley, London, 1854).

    Google Scholar 

  6. G. Coletti, Coherence principles for handling qualitative and quantitative partial probabilistic assessment, Mathware and Software Computing 3 (1996) 159–172.

    MATH  MathSciNet  Google Scholar 

  7. M. Conforti and G. Cornuéjols, A class of logic problems solvable by Linear Programming, Journal of the Association for Computing Machinery 42 (1995) 1107–1113.

    Article  MATH  MathSciNet  Google Scholar 

  8. N. Creignou, Complexity versus stability for classes of propositional formulas, Information Processing Letters 68 (1998) 161–165.

    Article  MathSciNet  Google Scholar 

  9. B. de Finetti, Teoria della probabilità, Vols. I, II (Einaudi, Torino, 1970). English translation: Theory of Probability, Vols. I, II (Wiley, New York, 1974).

    Google Scholar 

  10. G. Felici and K. Truemper, A MINSAT approach for learning in logic domains, INFORMS Journal on Computing 14 (2002) 20–36.

    Article  MATH  MathSciNet  Google Scholar 

  11. M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, New York, 1979).

    MATH  Google Scholar 

  12. G. Georgakopoulos, D. Kavvadias and C.H. Papadimitriou, Probabilistic satisfiability, Journal of Complexity 4 (1988) 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Hailperin, Boole’s Logic and Probability, 2nd ed., Studies in Logic and the Foundations of Mathematics, Vol. 85 (North-Holland, Amsterdam, 1986).

    MATH  Google Scholar 

  14. P. Hansen, B. Jaumard, M. Poggi de Aragão, F. Chauny and S. Perron, Probabilistic satisfiability with imprecise probabilities, International Journal of Approximate Reasoning 24 (2000) 171–189.

    Article  MATH  MathSciNet  Google Scholar 

  15. B. Jaumard, P. Hansen and M. Poggi de Aragão, Column generation methods for probabilistic logic, ORSA Journal on Computing 3 (1991) 135–148.

    Article  MATH  Google Scholar 

  16. B. Jaumard and B. Simeone, On the complexity of the Maximum Satisfiability Problem for Horn formulas, Information Processing Letters 26 (1987/88) 1–4.

    Article  MATH  MathSciNet  Google Scholar 

  17. G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization (Wiley, New York, 1988).

    MATH  Google Scholar 

  18. N.J. Nilsson, Probabilistic logic, Artificial Intelligence 28 (1986) 71–87.

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Slaney and T. Walsh, Phase transition behavior: From decision to optimization, presented at SAT02 — Fifth Int. Symp. on Theory and Applications of Satisfiability Testing, Cincinnati, OH (2002).

  20. K. Truemper, Effective Logic Computation (Wiley, New York, 1998).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Camerino, via Madonna delle Carceri, I-62032, Camerino, Italia

    Daniele Pretolani

Authors
  1. Daniele Pretolani
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pretolani, D. Probability logic and optimization SAT: The PSAT and CPA models. Ann Math Artif Intell 43, 211–221 (2005). https://doi.org/10.1007/s10472-005-0430-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-005-0430-8

Keywords