Hard QBFs for Merge Resolution

Authors Olaf Beyersdorff , Joshua Blinkhorn , Meena Mahajan , Tomáš Peitl , Gaurav Sood

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Author Details

Olaf Beyersdorff
  • Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany
Joshua Blinkhorn
  • Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany
Meena Mahajan
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Tomáš Peitl
  • Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany
Gaurav Sood
  • The Institute of Mathematical Sciences, HBNI, Chennai, India


Part of this work was done during the Dagstuhl Seminar "SAT and Interactions" (Seminar 20061).

Cite AsGet BibTex

Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan, Tomáš Peitl, and Gaurav Sood. Hard QBFs for Merge Resolution. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We prove the first proof size lower bounds for the proof system Merge Resolution (MRes [Olaf Beyersdorff et al., 2020]), a refutational proof system for prenex quantified Boolean formulas (QBF) with a CNF matrix. Unlike most QBF resolution systems in the literature, proofs in MRes consist of resolution steps together with information on countermodels, which are syntactically stored in the proofs as merge maps. As demonstrated in [Olaf Beyersdorff et al., 2020], this makes MRes quite powerful: it has strategy extraction by design and allows short proofs for formulas which are hard for classical QBF resolution systems. Here we show the first exponential lower bounds for MRes, thereby uncovering limitations of MRes. Technically, the results are either transferred from bounds from circuit complexity (for restricted versions of MRes) or directly obtained by combinatorial arguments (for full MRes). Our results imply that the MRes approach is largely orthogonal to other QBF resolution models such as the QCDCL resolution systems QRes and QURes and the expansion systems ∀Exp+Res and IR.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • QBF
  • resolution
  • proof complexity
  • lower bounds


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