A note on SAT algorithms and proof complexity

We apply classical proof complexity ideas to transfer lengths-of-proofs lower bounds for a propositional proof system P   into examples of hard unsatisfiable formulas for a class Alg(P)Alg(P) of SAT algorithms determined by P  . The class Alg(P)Alg(P) contains those algorithms M for which P proves in polynomial size tautologies expressing the soundness of M  . For example, the class Alg(Fd)Alg(Fd) determined by a depth d Frege system contains the commonly considered enhancements of DPLL (even for small d  ). Exponential lower bounds are known for all FdFd. Such results can be interpreted as a form of consistency of P≠NPP≠NP.Further we show how the soundness statements can be used to find hard satisfiable instances, if they exist.

► Lengths-of-proofs lower bounds yield formulas hard for SAT algorithms.
► Lower bounds for a proof system are a form of consistency of “P differs from NP”.
► The soundness statements for a SAT algorithm yield hard satisfiable formulas.