Sam Buss and Neil Thapen
    Reordering Rule Makes OBDD Proof Systems Stronger
    Preprint, March 2018.

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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 results are 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- can also handle or-fication and xor-ification. Our final result is an exponential size lower bound for RAT- refutations, giving exponential separations between RAT- and both DRAT- and SPR-.

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