DRAT and Propagation Redundancy Proofs Without New Variables

Abstract

We study the proof complexity of RAT proofs and related systems including BC, SPR and PR which use blocked clauses and (subset) propagation redundancy. These systems arise in satisfiability (SAT) solving, and allow inferences which preserve satisfiability but not logical implication. We introduce a new inference SR using substitution redundancy. We consider systems both with and without deletion. With new variables allowed, the systems are known to have the same proof theoretic strength as extended resolution. We focus on the systems that do not allow new variables to be introduced.Our first main result is that the systems DRAT − , DSPR − and DPR − , which allow deletion but not new variables, are polynomially equivalent. By earlier work of Kiesl, Rebola-Pardo and Heule, they are also equivalent to DBC − . Without deletion and without new variables, we show that SPR − can polynomially simulate PR − provided only short clauses are inferred by SPR inferences. Our next main result is that many of the well-known "hard" principles have polynomial size SPR − refutations (without deletions or new variables). These include the pigeonhole principle, bit pigeonhole principle, parity principle, Tseitin tautologies and clique-coloring tautologies; SPR −