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

We study the existence of optimal and p-optimal proof systems for classes in the Boolean hierarchy over $\mathrm{N}\mathrm{P}$. Our main results concern $\mathrm{D}\mathrm{P}$, i.e., the second level of this hierarchy: If all sets in $\mathrm{D}\mathrm{P}$ have p-optimal proof systems, then all sets in $\mathrm{c}\mathrm{o}\mathrm{D}\mathrm{P}$ 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 $\mathcal{C}$ and $\mathcal{D}$ in the Boolean hierarchy over $\mathrm{N}\mathrm{P}$: either we can prove the implication or show that it fails relative to an oracle. Furthermore, we show that the sets $\mathrm{S}\mathrm{A}\mathrm{T}$ and $\mathrm{T}\mathrm{A}\mathrm{U}\mathrm{T}$ have p-optimal proof systems, if and only if all sets in the Boolean hierarchy over $\mathrm{N}\mathrm{P}$ have p-optimal proof systems which is a new characterization of a conjecture studied by Pudlák.