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DRAT Proofs, Propagation Redundancy, and Extended Resolution

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Book cover Theory and Applications of Satisfiability Testing – SAT 2019 (SAT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11628))

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\({}^-\).

S. Buss—This work was initiated on a visit of the first author to the Czech Academy of Sciences in July 2018, supported by ERC advanced grant 339691 (FEALORA). Also supported by Simons Foundation grant 578919.

N. Thapen—Partially supported by GA ČR project 19-05497S and by ERC advanced grant 339691 (FEALORA) and RVO:67985840.

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Authors and Affiliations

  1. Department of Mathematics, U.C. San Diego, La Jolla, CA, USA

    Sam Buss

  2. Institute of Mathematics, Czech Academy of Sciences, Prague, Czech Republic

    Neil Thapen

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  1. Sam Buss
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Correspondence to Sam Buss .

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Editors and Affiliations

  1. University of Lisbon, Lisbon, Portugal

    Mikoláš Janota

  2. University of Lisbon, Lisbon, Portugal

    Inês Lynce

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Buss, S., Thapen, N. (2019). DRAT Proofs, Propagation Redundancy, and Extended Resolution. In: Janota, M., Lynce, I. (eds) Theory and Applications of Satisfiability Testing – SAT 2019. SAT 2019. Lecture Notes in Computer Science(), vol 11628. Springer, Cham. https://doi.org/10.1007/978-3-030-24258-9_5

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  • DOI: https://doi.org/10.1007/978-3-030-24258-9_5

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