NP-Completeness of Refutability by Literal-Once Resolution

A boolean formula in conjunctive normal form (CNF) F is refuted by literal-once resolution if the empty clause is inferred from F by resolving on each literal of F at most once. Literal-once resolution refutations can be found nondeterministically in polynomial time, though this restricted system is not complete. We show that despite of the weakness of literal-once resolution, the recognition of CNF-formulas which are refutable by literal-once resolution is NP-complete. We study the relationship between literal-once resolution and read-once resolution (introduced by Iwama and Miyano). Further we answer a question posed by Kullmann related to minimal unsatisfiability.

[1]  Oliver Kullmann,et al.  An application of matroid theory to the SAT problem , 2000, Proceedings 15th Annual IEEE Conference on Computational Complexity.

[2]  Zvi Galil,et al.  On the Complexity of Regular Resolution and the Davis-Putnam Procedure , 1977, Theor. Comput. Sci..

[3]  Kazuo Iwama,et al.  Intractability of read-once resolution , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

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

[5]  Hans Kleine Büning,et al.  On subclasses of minimal unsatisfiable formulas , 2000, Discret. Appl. Math..

[6]  Wolfgang Bibel,et al.  On Matrices with Connections , 1981, JACM.

[7]  J. A. Robinson,et al.  A Machine-Oriented Logic Based on the Resolution Principle , 1965, JACM.

[8]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .

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

[10]  Armin Haken,et al.  The Intractability of Resolution , 1985, Theor. Comput. Sci..

[11]  Nathan Linial,et al.  Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas , 1986, J. Comb. Theory, Ser. A.

[12]  Hans K. Buning,et al.  Propositional Logic: Deduction and Algorithms , 1999 .