Algorithms for Deciding Counting Quantifiers over Unary Predicates

We study algorithms for fragments of first order logic extended with counting quantifiers, which are known to be highly complex in general. We propose a fragment over unary predicates that is NP-complete and for which there is a normal form where Counting Quantification sentences have a single Unary predicate, thus call it the CQU fragment. We provide an algebraic formulation of the CQU satisfiability problem in terms of Integer Linear Programming based on which two algorithms are proposed, a direct reduction to SAT instances and an Integer Linear Programming version extended with a column generation mechanism. The latter is shown to lead to a viable implementation and experiments shows this algorithm presents a phase transition behavior.

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

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

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

[4]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[5]  Friedrich Eisenbrand,et al.  Carathéodory bounds for integer cones , 2006, Oper. Res. Lett..

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

[7]  Martin Otto,et al.  On Logics with Two Variables , 1999, Theor. Comput. Sci..

[8]  Ian Pratt-Hartmann.  Complexity of the Two-Variable Fragment with Counting Quantifiers , 2005, J. Log. Lang. Inf..

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

[10]  Armin Biere.  Lingeling Essentials, A Tutorial on Design and Implementation Aspects of the the SAT Solver Lingeling , 2014, POS@SAT.

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

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

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

[14]  Barnaby Martin,et al.  Constraint Satisfaction with Counting Quantifiers , 2012, CSR.

[15]  Perlindström.  First Order Predicate Logic with Generalized Quantifiers , 1966 .

[16]  Hector J. Levesque,et al.  Hard and Easy Distributions of SAT Problems , 1992, AAAI.

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

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

[19]  Ndrei,et al.  Galois correspondence for counting quantifiers , 2013 .

[20]  Franz Baader,et al.  Cardinality Restrictions on Concepts , 1994, KI.

[21]  Ian Pratt-Hartmann.  On the Computational Complexity of the Numerically Definite Syllogistic and Related Logics , 2008, Bull. Symb. Log..

[22]  Marcelo Finger,et al.  Probabilistic satisfiability: algorithms with the presence and absence of a phase transition , 2015, Annals of Mathematics and Artificial Intelligence.

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