Upward Translation of Optimal and P-Optimal Proof Systems in the Boolean Hierarchy over NP

Authors Fabian Egidy , Christian Glaßer, Martin Herold

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Fabian Egidy
  • Julius-Maximilians-Universität Würzburg, Germany
Christian Glaßer
  • Julius-Maximilians-Universität Würzburg, Germany
Martin Herold
  • Max-Planck-Institut für Informatik, Saarbrücken, Germany

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Fabian Egidy, Christian Glaßer, and Martin Herold. Upward Translation of Optimal and P-Optimal Proof Systems in the Boolean Hierarchy over NP. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We study the existence of optimal and p-optimal proof systems for classes in the Boolean hierarchy over NP. Our main results concern DP, i.e., the second level of this hierarchy: - If all sets in DP have p-optimal proof systems, then all sets in coDP have p-optimal proof systems. - The analogous implication for optimal proof systems fails relative to an oracle. As a consequence, we clarify such implications for all classes 𝒞 and 𝒟 in the Boolean hierarchy over NP: either we can prove the implication or show that it fails relative to an oracle. Furthermore, we show that the sets SAT and TAUT have p-optimal proof systems, if and only if all sets in the Boolean hierarchy over NP have p-optimal proof systems which is a new characterization of a conjecture studied by Pudlák.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Theory of computation → Oracles and decision trees
  • Computational Complexity
  • Boolean Hierarchy
  • Proof Complexity
  • Proof Systems
  • Oracle Construction


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