Dimension-dependent Behavior in the Satisfiability of Random K-horn Formulae

Abstract. We determine the asymptotical satisfiability probability of a random at-most-k-Horn formula, via a probabilistic analysis of a simple version, called PUR, of positive unit resolution. We show that for k = k(n) → ∞ the problem can be “reduced” to the case k(n) = n, that was solved in [17]. On the other hand, in the case k = constant the behavior of PUR is modeled by a simple queuing chain, leading to a closed-form solution when k = 2. Our analysis predicts an “easy-hard-easy” pattern in this latter case. Under a rescaled parameter, the graphs of satisfaction probability corresponding to finite values of k converge to the one for the uniform case, a “dimensiondependent behavior” similar to the one found experimentally in [20] for k-SAT. The phenomenon is qualitatively explained by a threshold property for the number of iterations of PUR makes on random satisfiable Horn formulas. Also, for k = 2 PUR has a peak in its average complexity at the critical point.