A sharp threshold in proof complexity yields lower bounds for satisfiability search

Let F ( n; n) denote a random CNF formula consisting of n randomly chosen 2-clauses and n randomly chosen 3-clauses, for some arbitrary constants ; 0. It is well-known that, with probability 1 o(1), if > 1 then F ( n; n) has a linear-size resolution refutation. We prove that, with probability 1 o(1), if < 1 then F ( n; n) has no subexponential-size resolution refutation. Our result also yields the first proof that random 3-CNF formulas with densities well below the generally accepted range of the satisfiability threshold (and thus believed to be satisfiable) cause natural Davis-Putnam algorithms to take exponential time (to find a satisfying assignment).