Generalized probabilistic satisfiability through integer programming

BackgroundThis paper studies the generalized probabilistic satisfiability (GPSAT) problem, where the probabilistic satisfiability (PSAT) problem is extended by allowing Boolean combinations of probabilistic assertions and nested probabilistic formulas.MethodsWe introduce a normal form for this problem and show that both nesting of probabilities and multi-agent probabilities do not increase the expressivity of GPSAT. An algorithm to solve GPSAT instances in the normal form via mixed integer linear programming is proposed.ResultsThe implementation of the algorithm is used to explore the complexity profile of GPSAT, and it shows evidence of phase-transition phenomena.ConclusionsEven though GPSAT is considerably more expressive than PSAT, it can be handled using integer linear programming techniques.

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

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

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

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

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

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

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

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

[9]  George Boole,et al.  An Investigation of the Laws of Thought: Frontmatter , 2009 .

[10]  Fabio Gagliardi Cozman,et al.  Towards classifying propositional probabilistic logics , 2014, J. Appl. Log..

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

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

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

[14]  J. Eckhoff.  Helly, Radon, and Carathéodory Type Theorems , 1993 .

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

[16]  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.

[17]  H. Paul Williams,et al.  Logic and Integer Programming , 2009, Logic and Integer Programming.

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

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

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

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

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