Lower Bounds for Set-Blocked Clauses Proofs

Author Emre Yolcu

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Emre Yolcu
  • Carnegie Mellon University, Pittsburgh, PA, USA


I thank Sam Buss, Marijn Heule, Jakob Nordström, and Ryan O'Donnell for useful discussions. I thank Jeremy Avigad, Ryan O'Donnell, and Bernardo Subercaseaux for feedback on an earlier version of the paper. Finally, I thank the STACS reviewers for their highly detailed comments and suggestions.

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Emre Yolcu. Lower Bounds for Set-Blocked Clauses Proofs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 59:1-59:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We study propositional proof systems with inference rules that formalize restricted versions of the ability to make assumptions that hold without loss of generality, commonly used informally to shorten proofs. Each system we study is built on resolution. They are called BC⁻, RAT⁻, SBC⁻, and GER⁻, denoting respectively blocked clauses, resolution asymmetric tautologies, set-blocked clauses, and generalized extended resolution - all "without new variables." They may be viewed as weak versions of extended resolution (ER) since they are defined by first generalizing the extension rule and then taking away the ability to introduce new variables. Except for SBC⁻, they are known to be strictly between resolution and extended resolution. Several separations between these systems were proved earlier by exploiting the fact that they effectively simulate ER. We answer the questions left open: We prove exponential lower bounds for SBC⁻ proofs of a binary encoding of the pigeonhole principle, which separates ER from SBC⁻. Using this new separation, we prove that both RAT⁻ and GER⁻ are exponentially separated from SBC⁻. This completes the picture of their relative strengths.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • proof complexity
  • separations
  • resolution
  • extended resolution
  • blocked clauses


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