Probabilistic satisfiability: algorithms with the presence and absence of a phase transition

We study algorithms for probabilistic satisfiability (PSAT), an NP-complete problem that is central to logic-probabilisti reasoning, focusing on the presence and absence of a phase transition phenomenon for each algorithm. Our study starts by defining a PSAT normal form, on which all algorithms are based. The proposed algorithms consist of several forms of reductions of PSAT to classical propositional satisfiability (SAT). The first algorithm is a canonical reduction of PSAT instances to SAT instances; three other algorithms are reductions to linear optimization with distinct column generation procedures, namely on auxiliary calls to SAT, weighted MAXSAT or SMT solvers. Theoretical and practical limitations of each algorithm are discussed. Several implementations were developed and compared by means of experiments using randomly generated input problems. Some of the implementations are shown to present a phase transition behavior. We show that variations of these algorithms may lead to the partial occlusion of the phase transition phenomenon and discuss the reasons for this change ofc practical behavior.

[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]  D. Garling,et al.  Inequalities: A Journey into Linear Analysis , 2007 .

[3]  P. Walley,et al.  Direct algorithms for checking consistency and making inferences from conditional probability assessments , 2004 .

[4]  Dov M. Gabbay,et al.  Handbook of defeasible reasoning and uncertainty management systems: volume 2: reasoning with actual and potential contradictions , 1998 .

[5]  Pavel Klinov,et al.  Pronto: A Practical Probabilistic Description Logic Reasoner , 2010, UniDL.

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

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

[8]  Theodore Hailperin,et al.  Probability logic , 1984, Notre Dame J. Formal Log..

[9]  Steven David Prestwich,et al.  CNF Encodings , 2021, Handbook of Satisfiability.

[10]  Pierre Hansen,et al.  Algorithms for the maximum satisfiability problem , 1987, Computing.

[11]  Nikolaj Bjørner,et al.  Z3: An Efficient SMT Solver , 2008, TACAS.

[12]  Ronan Le Bras,et al.  Solutions for Hard and Soft Constraints Using Optimized Probabilistic Satisfiability , 2013, SAT.

[13]  D. V. Gokhale,et al.  Theory of Probability, Vol. I , 1975 .

[14]  Andrea Capotorti,et al.  Simplification Rules for the Coherent Probability Assessment Problem , 2004, Annals of Mathematics and Artificial Intelligence.

[15]  K. Borgwardt.  The Simplex Method: A Probabilistic Analysis , 1986 .

[16]  P. Odifreddi.  Classical recursion theory , 1989 .

[17]  Pavel Klinov,et al.  A Hybrid Method for Probabilistic Satisfiability , 2011, CADE.

[18]  Pierre Hansen,et al.  Models and Algorithms for Probabilistic and Bayesian Logic , 1995, IJCAI.

[19]  Peter C. Cheeseman,et al.  Where the Really Hard Problems Are , 1991, IJCAI.

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

[21]  W. Spears.  Probabilistic Satisfiability , 1992 .

[22]  Felip Manyà,et al.  Exploiting Cycle Structures in Max-SAT , 2009, SAT.

[23]  Andrea Capotorti,et al.  Elimination of Boolean variables for probabilistic coherence , 2000, Soft Comput..

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

[25]  Jürgen Bokowski,et al.  Carathéodory-type Theorems à la Bárány , 2011, Discret. Comput. Geom..

[26]  Daniele Pretolani,et al.  Probability logic and optimization SAT: The PSAT and CPA models , 2005, Annals of Mathematics and Artificial Intelligence.

[27]  Pavel Klinov,et al.  The Consistency of the Medical Expert System CADIAG-2: A Probabilistic Approach , 2011, J. Inf. Technol. Res..

[28]  Christos H. Papadimitriou,et al.  A linear programming approach to reasoning about probabilities , 1990, Annals of Mathematics and Artificial Intelligence.

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

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

[31]  Niklas Sörensson,et al.  Translating Pseudo-Boolean Constraints into SAT , 2006, J. Satisf. Boolean Model. Comput..

[32]  Jürg Kohlas,et al.  Handbook of Defeasible Reasoning and Uncertainty Management Systems , 2000 .

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

[34]  K. Borgwardt.  The Simplex Method: A Probabilistic Analysis , 1986 .

[35]  H. Jeffreys,et al.  Theory of probability , 1896 .

[36]  Joost P. Warners,et al.  A Linear-Time Transformation of Linear Inequalities into Conjunctive Normal Form , 1998, Inf. Process. Lett..

[37]  Pierre Hansen,et al.  Probabilistic satisfiability with imprecise probabilities , 1999, Int. J. Approx. Reason..

[38]  Fabio Gagliardi Cozman,et al.  A Toolset for Propositional Probabilistic Logic , 2022 .

[39]  Guillermo Matera,et al.  On the computation of rational points of a hypersurface over a finite field , 2015, J. Complex..

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

[41]  Sharad Malik,et al.  Chaff: engineering an efficient SAT solver , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[42]  C. Papadimitriou,et al.  The Complexity of Computing a , 2009 .

[43]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[44]  Nils J. Nilsson,et al.  Probabilistic Logic Revisited , 1993, Artif. Intell..

[45]  Theodore Hailperin,et al.  Boole's logic and probability , 1976 .

[46]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[47]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

[48]  R. Scozzafava,et al.  Probabilistic Logic in a Coherent Setting , 2002 .

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

[50]  Michael J. Todd,et al.  The many facets of linear programming , 2002, Math. Program..

[51]  B. M. Hill,et al.  Theory of Probability , 1990 .

[52]  Marcelo Finger,et al.  A Refuted Conjecture on Probabilistic Satisfiability , 2010, SBIA.

[53]  Cesare Tinelli,et al.  Satisfiability Modulo Theories , 2021, Handbook of Satisfiability.

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

[55]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .