The phase transition in random Horn satis ability

This paper studies the phase transition in random Horn satissability: under the random model (n; m) in which a formula is obtained by choosing m clauses independently, uniformly at random, and with repetition from all Horn clauses in n variables, (n) = 2 n is the satissability threshold for the Horn satissability. The threshold is coarse since, if (n) = c 2 n then lim n!1 Pr 2(n;;(n)) is satissable] = 1 ? F(e ?c); where F(x) = (1 ?x)(1?x 2)(1 ?x 4)(1 ?x 8) 1. This resolves both of the two remaining cases of the problem of analyzng phase transitions of the six maximally tractable cases in Schaefer's Dichotomy Theorem.