On the Boolean connectivity problem for Horn relations

Gopalan et al. studied in [P. Gopalan, P.G. Kolaitis, E.N. Maneva, C.H. Papadimitriou, The connectivity of Boolean satisfiability: computational and structural dichotomies, in: Proceedings of the 33rd International Colloquium on Automata, Languages and Programming, ICALP 2006, 2006, pp. 346-357] and [P. Gopalan, P.G. Kolaitis, E.N. Maneva, C.H. Papadimitriou, The connectivity of Boolean satisfiability: computational and structural dichotomies, SIAM J. Comput. 38 (6) (2009) 2330-2355] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on the connectivity problems in Schaefer's framework. A set S of logical relations is Schaefer if all relations in S are either bijunctive, Horn, dual Horn, or affine. They first conjectured that the connectivity problem for Schaefer is in P. We disprove their conjecture by showing that there exists a set S of Horn relations such that the connectivity problem for S is coNP-complete. We also investigate a tractable aspect of Horn and dual Horn relations with respect to characteristic sets.

[1]  Nadia Creignou,et al.  A Complete Classification of the Complexity of Propositional Abduction , 2006, SIAM J. Comput..

[2]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[3]  Martha Sideri,et al.  The Inverse Satisfiability Problem , 1996, SIAM J. Comput..

[4]  Rina Dechter,et al.  Structure Identification in Relational Data , 1992, Artif. Intell..

[5]  Luca Trevisan,et al.  The Approximability of Constraint Satisfaction Problems , 2001, SIAM J. Comput..

[6]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas , 1979, Inf. Process. Lett..

[7]  Nadia Creignou,et al.  Complexity of Generalized Satisfiability Counting Problems , 1996, Inf. Comput..

[8]  Federico Ricci-Tersenghi,et al.  On the solution-space geometry of random constraint satisfaction problems , 2006, STOC '06.

[9]  Christos H. Papadimitriou,et al.  The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies , 2006, SIAM J. Comput..

[10]  Georg Gottlob,et al.  Computational aspects of monotone dualization: A brief survey , 2008, Discret. Appl. Math..

[11]  Dan Roth,et al.  Reasoning with Models , 1994, Artif. Intell..

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[13]  Nadia Creignou,et al.  A Dichotomy Theorem for Maximum Generalized Satisfiability Problems , 1995, J. Comput. Syst. Sci..

[14]  Laurent Juban,et al.  Dichotomy Theorem for the Generalized Unique Satisfiability Problem , 1999, FCT.

[15]  Phokion G. Kolaitis,et al.  The complexity of minimal satisfiability problems , 2003, Inf. Comput..

[16]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[17]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraint satisfaction problems on a 3-element set , 2006, JACM.

[18]  Dan Roth,et al.  Defaults and Relevance in Model-Based Reasoning , 1997, Artif. Intell..

[19]  Bart Selman,et al.  Horn Approximations of Empirical Data , 1995, Artif. Intell..

[20]  J. C. C. McKinsey,et al.  The decision problem for some classes of sentences without quantifiers , 1943, Journal of Symbolic Logic.

[21]  Sanjeev Khanna,et al.  Complexity classifications of Boolean constraint satisfaction problems , 2001, SIAM monographs on discrete mathematics and applications.

[22]  Kazuhisa Makino,et al.  Generating All Abductive Explanations for Queries on Propositional Horn Theories , 2003, CSL.

[23]  Roni Khardon.  Translating between Horn Representations and their Characteristic Models , 1995, J. Artif. Intell. Res..

[24]  John N. Hooker,et al.  Extended Horn sets in propositional logic , 1991, JACM.

[25]  Toshihide Ibaraki,et al.  Computing Intersections of Horn Theories for Reasoning with Models , 1998, Artif. Intell..

[26]  Christos H. Papadimitriou,et al.  On Horn Envelopes and Hypergraph Transversals , 1993, ISAAC.

[27]  Andrei A. Bulatov,et al.  A dichotomy theorem for constraints on a three-element set , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[28]  Michael E. Saks,et al.  A Complexity Index for Satisfiability Problems , 1994, IPCO.

[29]  Leonid Khachiyan,et al.  On the Complexity of Dualization of Monotone Disjunctive Normal Forms , 1996, J. Algorithms.

[30]  Bart Selman,et al.  Reasoning With Characteristic Models , 1993, AAAI.

[31]  Peter L. Hammer,et al.  On Connected Boolean Functions , 1999, Discret. Appl. Math..

[32]  Harry R. Lewis,et al.  Renaming a Set of Clauses as a Horn Set , 1978, JACM.

[33]  Marco Cadoli,et al.  The Complexity of Model Checking for Circumscriptive Formulae , 1992, Inf. Process. Lett..