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