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The Probability of Pure Literals

Abstract : We describe an error in earlier probabilistic analyses of the pure literal heuristic as a procedure for solving k-SAT. All probabilistic analyses are in the constant degree model in which a random instance C of k-SAT consists of m clauses selected independently and uniformly (with replacement) from the set of all k-clauses over n variables. We provide a new analysis for k = 2. Specifically, we show with probability approaching 1 as m goes to infinity one can apply the pure literal rule repeatedly to a random instance of 2-SAT until the number of clauses is small provided n/m>lambda>1. But if n/m<lambda<1, with probability approaching 1 if the pure literal rule is applied as much as possible, then at least m(1/5) clauses will remain.