Generalized Probabilistic Satisfiability

This paper studies the Generalized Probabilistic Satisfiability (GPSAT) problem, where the probabilistic satisfiability problem is extended by allowing Boolean combinations of probabilistic assertions and nested probabilistic formulas. We introduce a normal form for this problem and show that nesting of probabilities does not increase the expressivity in GPSAT. An algorithm to solve GPSAT problems via mixed integer programming is proposed, and its implementation shows evidence of phase-transition phenomena.

[1]  Theodore Hailperin.  Boole's logic and probability - a critical exposition from the standpoint of contemporary algebra, logic and probability theory (2. ed.) , 1986, Studies in logic and the foundations of mathematics.

[2]  Marcelo Finger,et al.  Probabilistic Satisfiability: Logic-Based Algorithms and Phase Transition , 2011, IJCAI.

[3]  Daniele Pretolani,et al.  Easy Cases of Probabilistic Satisfiability , 2001, Annals of Mathematics and Artificial Intelligence.

[4]  Ronald Fagin,et al.  Reasoning about knowledge and probability , 1988, JACM.

[5]  V. S. Subrahmanian,et al.  Probabilistic Logic Programming , 1992, Inf. Comput..

[6]  Soshichi Uchii.  Higher Order Probabilities and Coherence , 1973, Philosophy of Science.

[7]  George Boole.  The laws of thought (1854) , 1916 .

[8]  Nils J. Nilsson,et al.  Probabilistic Logic * , 2022 .

[9]  Pierre Hansen,et al.  Merging the local and global approaches to probabilistic satisfiability , 2004, Int. J. Approx. Reason..

[10]  Vijay Chandru,et al.  Optimization Methods for Logical Inference: Chandru/Optimization , 1999 .

[11]  Joseph Y. Halpern.  Reasoning about uncertainty , 2003 .

[12]  Christos H. Papadimitriou,et al.  Probabilistic satisfiability , 1988, J. Complex..

[13]  Pierre Hansen,et al.  Column Generation Methods for Probabilistic Logic , 1989, INFORMS J. Comput..

[14]  T. Hailperin,et al.  Best Possible Inequalities for the Probability of a Logical Function of Events , 1965 .

[15]  V. Chandru,et al.  Optimization Methods for Logical Inference , 1999 .

[16]  Gai CarSO.  A Logic for Reasoning about Probabilities * , 2004 .

[17]  Thomas Lukasiewicz,et al.  Expressive probabilistic description logics , 2008, Artif. Intell..

[18]  Fabio Gagliardi Cozman,et al.  Probabilistic satisfiability and coherence checking through integer programming , 2013, Int. J. Approx. Reason..

[19]  Toby Walsh,et al.  The SAT Phase Transition , 1994, ECAI.