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.
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.
The source code for all experiments under license GPLv3 are publicly available at http://lipsat.sourceforge.net.
References
Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 1–41 (2009). http://scip.zib.de/
Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization. Athena Scientific, Belmont (1997)
Bofill, M., Manya, F., Vidal, A., Villaret, M.: Finding hard instances of satisfiability in Łukasiewicz logics. In: ISMVL, pp. 30–35. IEEE (2015)
Boole, G.: An Investigation on the Laws of Thought. Macmillan, London (1854). Available on project Gutemberg at http://www.gutenberg.org/etext/15114
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
Bova, S., Flaminio, T.: The coherence of Łukasiewicz assessments is NP-complete. Int. J. Approx. Reason. 51(3), 294–304 (2010)
Cheeseman, P., Kanefsky, B., Taylor, W.M.: Where the really hard problems are. In: 12th IJCAI, pp. 331–337. Morgan Kaufmann (1991)
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
de Finetti, B.: Sul significato soggettivo della probabilità. Fundamenta Mathematicae 17(1), 298–329 (1931)
de Finetti, B.: La prévision: Ses lois logiques, ses sources subjectives (1937)
de Finetti, B.: Theory of Probability: A Critical Introductory Treatment. Wiley, Hoboken (2017). Translated by Antonio Machí and Adrian Smith
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
Eckhoff, J.: Helly, Radon, and Caratheodory type theorems. In: Handbook of Convex Geometry, pp. 389–448. Elsevier Science Publishers (1993)
Finger, M., Bona, G.D.: Probabilistic satisfiability: logic-based algorithms and phase transition. In: Walsh, T. (ed.) IJCAI, IJCAI/AAAI, pp. 528–533 (2011)
Finger, M., De Bona, G.: Probabilistic satisfiability: algorithms with the presence and absence of a phase transition. AMAI 75(3), 351–379 (2015)
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)
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)
Georgakopoulos, G., Kavvadias, D., Papadimitriou, C.H.: Probabilistic satisfiability. J. Complex. 4(1), 1–11 (1988)
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
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
Hansen, P., Jaumard, B.: Algorithms for the maximum satisfiability problem. Computing 44, 279–303 (1990). https://doi.org/10.1007/BF02241270
Kavvadias, D., Papadimitriou, C.H.: A linear programming approach to reasoning about probabilities. AMAI 1, 189–205 (1990)
McNaughton, R.: A theorem about infinite-valued sentential logic. J. Symb. Log. 16, 1–13 (1951)
Mundici, D.: Advanced Łukasiewicz calculus and MV-algebras. Trends in Logic. Springer, Heidelberg (2011). https://doi.org/10.1007/978-94-007-0840-2
Mundici, D.: Satisfiability in many-valued sentential logic is NP-complete. Theor. Comput. Sci. 52(1–2), 145–153 (1987)
Mundici, D.: A constructive proof of McNaughton’s theorem in infinite-valued logic. J. Symb. Log. 59(2), 596–602 (1994)
Mundici, D.: Bookmaking over infinite-valued events. Int. J. Approx. Reason. 43(3), 223–240 (2006)
Nilsson, N.: Probabilistic logic. Artif. Intell. 28(1), 71–87 (1986)
Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover, Mineola (1998)
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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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