Properties of SLUR formulae

Abstract

Single look-ahead unit resolution (SLUR) algorithm is a nondeterministic polynomial time algorithm which for a given input formula in a conjunctive normal form (CNF) either outputs its satisfying assignment or gives up. A CNF formula belongs to the SLUR class if no sequence of nondeterministic choices causes the SLUR algorithm to give up on it. The SLUR class is reasonably large. It is known to properly contain the well studied classes of Horn CNFs, renamable Horn CNFs, extended Horn CNFs, and CC-balanced CNFs. In this paper we show that the SLUR class is considerably larger than the above mentioned classes of CNFs by proving that every Boolean function has at least one CNF representation which belongs to the SLUR class. On the other hand, we show, that given a CNF it is coNP-complete to decide whether it belongs to the SLUR class or not. Finally, we define a non-collapsing hierarchy of CNF classes SLUR(i) in such a way that for every fixed i there is a polynomial time satisfiability algorithm for the class SLUR(i), and that every CNF on n variables belongs to SLUR(i) for some i ≤ n. © 2012 Springer-Verlag.