On Proof Systems Behind Efficient SAT Solvers

Conventional algorithms for Boolean satisfiability (SAT) work within the framework of resolution as a proof system. However it has been known since the 1980s that every resolution proof of the pigeonhole principle (PHPm n ), suitably encoded as a CNF instance, includes exponentially many steps[4]. Therefore SAT solvers based upon the DLL procedure [1] or the DP procedure [2] must take exponential time. Polynomial-sized proofs of the PHP exist for more powerful proof systems, but general-purpose SAT solvers often remain confined to resolution. Our work seeks automatizable proof systems other than resolution. In an earlier work [9], an implementation of a CompressedBFS algorithm empirically solved PHPn+1 n instances in Θ(n4) time. Here, we add to this claim, and show analytically that these instances are solvable in polynomial time by Compressed-BFS. Thus the class of tautolgies efficiently provable by CompressedBFS is different than that of resolution. We hope that the details of our proof shed some light on the proof system implied by Compressed-BFS. Our proof focuses on structural invariants within the compressed data structure that stores collections of sets of open clauses during the CompressedBFS algorithm. We bound the size of this data structure, as well as the overall memory, by a polynomial. Because of the nature of ZDD operations, the overall runtime is bounded by a polynomial of the data structure’s size.