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Probably Half True: Probabilistic Satisfiability over Łukasiewicz Infinitely-Valued Logic

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Automated Reasoning (IJCAR 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10900))

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Abstract

We study probabilistic-logic reasoning in a context that allows for “partial truths”, focusing on computational and algorithmic properties of non-classical Łukasiewicz Infinitely-valued Probabilistic Logic. In particular, we study the satisfiability of joint probabilistic assignments, which we call LIPSAT. Although the search space is initially infinite, we provide linear algebraic methods that guarantee polynomial size witnesses, placing LIPSAT complexity in the NP-complete class. An exact satisfiability decision algorithm is presented which employs, as a subroutine, the decision problem for Łukasiewicz Infinitely-valued (non probabilistic) logic, that is also an NP-complete problem. We develop implementations of the algorithms described and discuss the empirical presence of a phase transition behavior for those implementations.

M. Finger—Partially supported by Fapesp projs. 2015/21880-4 and 2014/12236-1 and CNPq grant 306582/2014-7.

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Notes

  1. 1.

    Thus C is more restrictive than the full class of states of an MV-algebra, in the sense of [24], which will not be discussed here.

  2. 2.

    The source code for all experiments under license GPLv3 are publicly available at http://lipsat.sourceforge.net.

References

  1. Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009). http://scip.zib.de/

    Article  MathSciNet  Google Scholar 

  2. Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization. Athena Scientific, Belmont (1997)

    Google Scholar 

  3. Bofill, M., Manya, F., Vidal, A., Villaret, M.: Finding hard instances of satisfiability in Łukasiewicz logics. In: ISMVL, pp. 30–35. IEEE (2015)

    Google Scholar 

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

  5. Borgward, K.H.: The Simplex Method: A Probabilistic Analysis. Algorithms and Combinatorics, vol. 1. Springer, Heidelberg (1986). https://doi.org/10.1007/978-3-642-61578-8

    Book  Google Scholar 

  6. Bova, S., Flaminio, T.: The coherence of Łukasiewicz assessments is NP-complete. Int. J. Approx. Reason. 51(3), 294–304 (2010)

    Article  Google Scholar 

  7. Cheeseman, P., Kanefsky, B., Taylor, W.M.: Where the really hard problems are. In: 12th IJCAI, pp. 331–337. Morgan Kaufmann (1991)

    Google Scholar 

  8. Cignoli, R., d’Ottaviano, I., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic. Springer, Heidelberg (2000). https://doi.org/10.1007/978-94-015-9480-6

    Book  MATH  Google Scholar 

  9. de Finetti, B.: Sul significato soggettivo della probabilità. Fundamenta Mathematicae 17(1), 298–329 (1931)

    Article  Google Scholar 

  10. de Finetti, B.: La prévision: Ses lois logiques, ses sources subjectives (1937)

    Google Scholar 

  11. de Finetti, B.: Theory of Probability: A Critical Introductory Treatment. Wiley, Hoboken (2017). Translated by Antonio Machí and Adrian Smith

    Book  Google Scholar 

  12. Dutertre, B.: Yices 2.2. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 737–744. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08867-9_49

    Chapter  Google Scholar 

  13. Eckhoff, J.: Helly, Radon, and Caratheodory type theorems. In: Handbook of Convex Geometry, pp. 389–448. Elsevier Science Publishers (1993)

    Chapter  Google Scholar 

  14. Finger, M., Bona, G.D.: Probabilistic satisfiability: logic-based algorithms and phase transition. In: Walsh, T. (ed.) IJCAI, IJCAI/AAAI, pp. 528–533 (2011)

    Google Scholar 

  15. Finger, M., De Bona, G.: Probabilistic satisfiability: algorithms with the presence and absence of a phase transition. AMAI 75(3), 351–379 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Finger, M., De Bona, G.: Probabilistic satisfiability: algorithms with the presence and absence of a phase transition. Ann. Math. Artif. Intell. 75(3), 351–379 (2015)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  18. Georgakopoulos, G., Kavvadias, D., Papadimitriou, C.H.: Probabilistic satisfiability. J. Complex. 4(1), 1–11 (1988)

    Article  MathSciNet  Google Scholar 

  19. Hähnle, R.: Towards an efficient tableau proof procedure for multiple-valued logics. In: Börger, E., Kleine Büning, H., Richter, M.M., Schönfeld, W. (eds.) CSL 1990. LNCS, vol. 533, pp. 248–260. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54487-9_62

    Chapter  Google Scholar 

  20. Hansen, P., Jaumard, B.: Probabilistic satisfiability. In: Kohlas, J., Moral, S. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems. HAND, vol. 5, pp. 321–367. Springer, Dordrecht (2000). https://doi.org/10.1007/978-94-017-1737-3_8

    Chapter  Google Scholar 

  21. Hansen, P., Jaumard, B.: Algorithms for the maximum satisfiability problem. Computing 44, 279–303 (1990). https://doi.org/10.1007/BF02241270

    Article  MathSciNet  MATH  Google Scholar 

  22. Kavvadias, D., Papadimitriou, C.H.: A linear programming approach to reasoning about probabilities. AMAI 1, 189–205 (1990)

    MATH  Google Scholar 

  23. McNaughton, R.: A theorem about infinite-valued sentential logic. J. Symb. Log. 16, 1–13 (1951)

    Article  MathSciNet  Google Scholar 

  24. Mundici, D.: Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic. Springer, Heidelberg (2011). https://doi.org/10.1007/978-94-007-0840-2

    Book  MATH  Google Scholar 

  25. Mundici, D.: Satisfiability in many-valued sentential logic is NP-complete. Theor. Comput. Sci. 52(1–2), 145–153 (1987)

    Article  MathSciNet  Google Scholar 

  26. Mundici, D.: A constructive proof of McNaughton’s theorem in infinite-valued logic. J. Symb. Log. 59(2), 596–602 (1994)

    Article  MathSciNet  Google Scholar 

  27. Mundici, D.: Bookmaking over infinite-valued events. Int. J. Approx. Reason. 43(3), 223–240 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of São Paulo, São Paulo, Brazil

    Marcelo Finger & Sandro Preto

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  1. Marcelo Finger
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  2. Sandro Preto
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Correspondence to Marcelo Finger .

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Editors and Affiliations

  1. Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Didier Galmiche

  2. Baden-Wuerttemberg Cooperative State University, Stuttgart, Germany

    Stephan Schulz

  3. University of Trento, Trento, Italy

    Roberto Sebastiani

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Finger, M., Preto, S. (2018). Probably Half True: Probabilistic Satisfiability over Łukasiewicz Infinitely-Valued Logic. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds) Automated Reasoning. IJCAR 2018. Lecture Notes in Computer Science(), vol 10900. Springer, Cham. https://doi.org/10.1007/978-3-319-94205-6_14

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  • DOI: https://doi.org/10.1007/978-3-319-94205-6_14

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