A note on SAT algorithms and proof complexity

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Abstract

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) of SAT algorithms determined by P. The class 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) 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 Fd. Such results can be interpreted as a form of consistency of PNP.

Further we show how the soundness statements can be used to find hard satisfiable instances, if they exist.

Highlights

► 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.

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Supported in part by grants IAA100190902 and MSM0021620839. Also partially affiliated with the Institute of Mathematics of the Academy of Sciences and grant AV0Z10190503.

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