On the occurrence of null clauses in random instances of satisfiability

Abstract

We analyze a popular probabilistic model for generating instances of Satisfiability. According to this model, each literal of a set L = {υ 1, υ 1, υ 2, υ 2, …, υ r, υ r} of literals appears independently in each of n clauses with probability p. This model allows null clauses and the frequency of occurrence of such clauses depends on the relationship between the parameters n, r, and p. If an instance contains a null clause it is trivially unsatisfiable. Several papers present polynomial average time results under this model when null clauses are numerous but, until now, not all such cases have been covered by average case efficient algorithms. In fact, a recent paper by Bugrara, Pan and Purdon shows that the average complexity of the pure literal rule is superpolynomial even when most random instances contain a null clause. We show here that a simple strategy based on locating null clauses in a given random input has polynomial average complexity if either n ≤ r 5, and pr < ln(n) 2 ; or n = r ε,ε ≠ 1, and pr < c(ε) ln(n) 2 ; or n = βr, β a positive constant, and 2.64(1 − e −2 βpr (1 + 2 βpr)) < βe −2 pr . These are essentially the conditions for which null clauses appear in random instances with probability tending to one; and the results presented here are an improvement over some results in the references cited above. The strategy is as follows. Search the input for a null clause. If one is found, immediately decide the instance is unsatisfiable. Otherwise, set variables appearing excactly once to satisfy the clauses they occupy and determine satisfiability by exhaustively trying all possible truth assignments on the remaining literals of the input. Because the good average case performance depends completely on the presence of null clauses, we see this work as illuminating properties of the probabilistic model which cause polynomial average time rather than presenting a new algorithm with improved average time behavior.