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

A Sharp Threshold in Proof Complexity Yields Lower Bounds for Satisfiability Search

We give the first example of a sharp threshold in proof complexity. More precisely, we show that for any sufficiently small > 0 and > 2:28, random formulas consisting of (1 )n 2-clauses and n 3-clauses, which are known to be unsatisfiable almost certainly, almost certainly require resolution and Davis-Putnam proofs of unsatisfiability of exponential size, whereas it is easily seen that random formulas with (1 + )n 2-clauses (and n 3-clauses) have linear size proofs of unsatisfiability almost certainly. A consequence of our result also yields the first proof that typical random 3-CNF formulas at ratios below the generally accepted range of the satisfiability threshold (and thus expected to be satisfiable almost certainly) cause natural Davis-Putnam algorithms to take exponential time to find satisfying assignments.