Redundancy in logic II: 2CNF and Horn propositional formulae

We report results about the redundancy of formulae in 2CNF form. In particular, we give a slight improvement over the trivial redundancy algorithm and give some complexity results about some problems related to finding Irredundant Equivalent Subsets (i.e.s.) of 2CNF formulae. The problems of checking whether a 2CNF formula has a unique i.e.s. and checking whether a clause in is all its i.e.s.'s are polynomial. Checking whether a 2CNF formula has an i.e.s. of a given size and checking whether a clause is in some i.e.s.'s of a 2CNF formula are polynomial or NP-complete depending on whether the formula is cyclic. Some results about Horn formulae are also reported.

[1]  Alvaro del Val.  On 2-SAT and Renamable Horn , 2000, AAAI/IAAI.

[2]  Giorgio Ausiello,et al.  Minimal Representation of Directed Hypergraphs , 1986, SIAM J. Comput..

[3]  Christopher Umans.  The minimum equivalent DNF problem and shortest implicants , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[4]  Edith Hemaspaandra,et al.  The minimization problem for Boolean formulas , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[5]  Martha Sideri,et al.  Generating all maximal models of a Boolean expression , 2000, Inf. Process. Lett..

[6]  Georg Gottlob,et al.  Removing Redundancy from a Clause , 1993, Artif. Intell..

[7]  Stefan Szeider,et al.  Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference , 2002, Theor. Comput. Sci..

[8]  Hans Kleine Büning,et al.  Extension and equivalence problems for clause minimal formulae , 2005, Annals of Mathematics and Artificial Intelligence.

[9]  Renato Bruni,et al.  Approximating minimal unsatisfiable subformulae by means of adaptive core search , 2003, Discret. Appl. Math..

[10]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[11]  Lei Zheng,et al.  Improving GSAT Using 2SAT , 2002, CP.

[12]  Paolo Liberatore.  Merging Locally Correct Knowledge Bases: A Preliminary Report , 2002, ArXiv.

[13]  W. Quine.  On Cores and Prime Implicants of Truth Functions , 1959 .

[14]  Peter L. Hammer,et al.  Optimal Compression of Propositional Horn Knowledge Bases: Complexity and Approximation , 1993, Artif. Intell..

[15]  Wayne Snyder,et al.  Detecting Redundant Production Rules , 1997, AAAI/IAAI.

[16]  Christos H. Papadimitriou,et al.  The complexity of facets resolved , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[17]  David G. Mitchell,et al.  Minimum Witnesses for Unsatisfiable 2CNFs , 2006, SAT.

[18]  Paul D. Seymour,et al.  Graph Minors. XX. Wagner's conjecture , 2004, J. Comb. Theory B.

[19]  Sartaj Sahni,et al.  Computationally Related Problems , 1974, SIAM J. Comput..

[20]  Christos H. Papadimitriou,et al.  The even-path problem for graphs and digraphs , 1984, Networks.

[21]  John E. Hopcroft,et al.  The Directed Subgraph Homeomorphism Problem , 1978, Theor. Comput. Sci..

[22]  Alon Itai,et al.  On the Complexity of Timetable and Multicommodity Flow Problems , 1976, SIAM J. Comput..

[23]  David Maier.  Minimum Covers in Relational Database Model , 1980, JACM.

[24]  Lei Zheng,et al.  Improving SAT Using 2SAT , 2002, ACSC.

[25]  Xishun Zhao,et al.  Complexity Results for 2CNF Default Theories , 2001, Fundam. Informaticae.

[26]  Hantao Zhang,et al.  Improving exact algorithms for MAX-2-SAT , 2005, Annals of Mathematics and Artificial Intelligence.

[27]  Samir Khuller,et al.  On Strongly Connected Digraphs with Bounded Cycle Length , 1996, Discret. Appl. Math..

[28]  Paolo Liberatore,et al.  Redundancy in logic III: Non-monotonic reasoning , 2005, Artif. Intell..

[29]  Samir Khuller,et al.  Approximating the minimum equivalent digraph , 1994, SODA '94.

[30]  Paolo Liberatore,et al.  Redundancy in logic I: CNF propositional formulae , 2002, Artif. Intell..

[31]  Allen Ginsberg.  Knowledge-Base Reduction: A New Approach to Checking knowledge Bases for Inconsistency and Redundancy , 1988, AAAI.