A sharp threshold in proof complexity

Dimitris Achlioptas, Paul Beame, Michael Molloy
2001 Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01  
We give the first example of a sharp threshold in proof complexity. More precisely, we show that for any sufficiently small¯ ¼ and ¡ ¾ ¾ , random formulas consisting of´½ ¯µÒ 2-clauses and ¡Ò 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´½ ·¯µÒ 2-clauses (and ¡Ò 3-clauses) have linear size proofs of unsatisfiability almost
more » ... tainly. 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.
doi:10.1145/380752.380820 dblp:conf/stoc/AchlioptasBM01 fatcat:tonk7hhnbffmxa6w2qdgaescmy