The Complexity of Finding Read-Once NAE-Resolution Refutations

Abstract

In this paper, we analyze boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation. A read-once (resolution) refutation is one in which each input clause is used at most once. It is well-known that read-once resolution is not complete, i.e., there exist unsatisfiable formulas for which no read-once resolution exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called Not-All-Equal Satisfiability (NAE-Satisfiability). NAE-Satisfiability is the problem of checking whether an arbitrary CNF formula has a satisfying assignment in which at least one literal in each clause is set to false. It is well-known that NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of the class of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution, which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. We focus on a variant of NAE-resolution called read-once NAE-resolution, in which each input clause can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for checking the NAE-satisfiability of 2CNF formulas; we also provide a polynomial time algorithm to determine the shortest read-once NAE-resolution of a 2CNF formula. Finally, we establish that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

P. Wojciechowski—This research is supported in part by the National Science Foundation under Award CCF-0827397.

K. Subramani—This work was supported by the Air Force Research Laboratory under US Air Force contract FA8750-16-3-6003. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.