Quantitative Logic Reasoning
In this paper we show several similarities among logic systems that deal simultaneously with deductive and quantitative inference. We claim it is appropriate to call the tasks those systems perform as Quantitative Logic Reasoning. Analogous properties hold throughout that class, for whose members there exists a set of linear algebraic techniques applicable in the study of satisfiability decision problems. In this presentation, we consider as Quantitative Logic Reasoning the tasks performed by propositional Probabilistic Logic; first-order logic with counting quantifiers over a fragment containing unary and limited binary predicates; and propositional Łukasiewicz Infinitely-valued Probabilistic Logic.
We are very thankful to Daniele Mundici for several discussions on multi-valued logics; we would also like to thank two reviewers for their very detailed comments. This work was supported by Fapesp projects 2015/21880-4 and 2014/12236-1 and CNPq grant PQ 306582/2014-7.
- 3.Baader, F., S. Brandt, and C. Lutz. 2005. Pushing the EL envelope. In IJCAI05, 19th International Joint Conference on Artificial Intelligence, pp. 364–369.Google Scholar
- 4.Bertsimas, D., and J.N. Tsitsiklis. 1997. Introduction to Linear Optimization. Athena Scientific.Google Scholar
- 5.Biere, A. 2014. Lingeling essentials, a tutorial on design and implementation aspects of the the sat solver lingeling. In POS@ SAT, pp. 88. Citeseer.Google Scholar
- 8.Boole, G. 1854. An Investigation on the Laws of Thought. London: Macmillan. Available on project Gutemberg at http://www.gutenberg.org/etext/15114.
- 10.Bulatov, A.A., and A. Hedayaty. 2015. Galois correspondence for counting quantifiers. Multiple-Valued Logic and Soft Computing 24 (5–6): 405–424.Google Scholar
- 11.Calvanese, D., G. De Giacomo, D. Lembo, M. Lenzerini, and R. Rosati (2005). DL-Lite: Tractable description logics for ontologies. In Proceedings of the 20th National Conference on Artificial Intelligence (AAAI 2005), vol. 5, pp. 602–607.Google Scholar
- 13.de Finetti, B. 1931. Sul significato soggettivo della probabilità. Fundamenta Mathematicae 17(1): 298–329. Translated into English as “On the Subjective Meaning of Probability”. In Probabilitàe Induzione, ed. P. Monari, and D. Cocchi, 291–321. Bologna: Clueb (1993).Google Scholar
- 14.de Finetti, B. 1937. La prévision: Ses lois logiques, ses sources subjectives. In Annales de l’institut Henri Poincaré, vol. 7:1, pp. 1–68. English translation by Henry E. Kyburg Jr., as “Foresight: Its Logical Laws, its Subjective Sources.” In H. E. Kyburg Jr., and H. E. Smokler, “Studies in Subjective Probability”, J. Wiley, New York, pp. 93–158, 1964. Second edition published by Krieger, New York, pp. 53–118, 1980.Google Scholar
- 15.de Finetti, B. 2017. Theory of probability: A critical introductory treatment. Translated by Antonio Machí and Adrian Smith. Wiley.Google Scholar
- 16.Eckhoff, J. 1993. Helly, Radon, and Carathéodory type theorems. In Handbook of Convex Geometry, Edited by P.M. Gruber, and J.M. Wills, pp. 389–448. Elsevier Science Publishers.Google Scholar
- 17.Eén, N. and N. Sörensson. 2003. An extensible SAT-solver. In SAT 2003, vol. 2919 LNCS, pp. 502–518. Springer.Google Scholar
- 18.Eén, N., and N. Sörensson. 2006. Translating pseudo-boolean constraints into sat. Journal on Satisfiability, Boolean Modeling and Computation 2 (1–4): 1–26.Google Scholar
- 20.Finger, M., and G.D. Bona. 2011. Probabilistic satisfiability: Logic-based algorithms and phase transition. In Internatioinal Joint Congerence on Artificial Intelligence (IJCAI), Edited by T. Walsh, pp. 528–533. IJCAI/AAAI Press.Google Scholar
- 21.Finger, M., and G.D. Bona. 2017. Algorithms for deciding counting quantifiers over unary predicates. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, February 4–9, 2017, San Francisco, California, USA., Edited by S.P. Singh, and S. Markovitch, pp. 3878–3884. AAAI Press.Google Scholar
- 23.Finger, M., and S. Preto. 2018. Probably half true: Probabilistic satisfiability over Łukasiewicz infinitely-valued logic. In preparation.Google Scholar
- 27.Hailperin, T. 1986. Boole’s Logic and Probability (Second enlarged edition ed.), vol. 85 Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland.Google Scholar
- 28.Hansen, P., and B. Jaumard. 2000. Probabilistic satisfiability. In Handbook of Defeasible Reasoning and Uncertainty Management Systems, Edited by J. Kohlas, and S. Moral, vol.5, pp. 321–367. Springer.Google Scholar
- 29.Hansen, P., B. Jaumard, G.-B.D. Nguetsé, and M.P. de Aragão. 1995. Models and algorithms for probabilistic and bayesian logic. In IJCAI, pp. 1862–1868.Google Scholar
- 33.Lindström, P. 1966. First order predicate logic with generalized quantifiers. Theoria 32 (3): 186–195.Google Scholar
- 39.Papadimitriou, C., and K. Steiglitz. 1998. Combinatorial Optimization: Algorithms and Complexity. Dover.Google Scholar
- 42.Schrijver, A. 1986. Theory of Linear and Integer Programming. New York: Wiley.Google Scholar