We study the existence of optimal and p-optimal proof systems for classes in the Boolean hierarchy over $\mathrm{NP}$. Our main results concern $\mathrm{DP}$, i.e., the second level of this hierarchy: If all sets in $\mathrm{DP}$ have p-optimal proof systems, then all sets in $\mathrm{coDP}$ 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 $\mathcal{C}$ and $\mathcal{D}$ in the Boolean hierarchy over $\mathrm{NP}$: either we can prove the implication or show that it fails relative to an oracle. Furthermore, we show that the sets $\mathrm{SAT}$ and $\mathrm{TAUT}$ have p-optimal proof systems, if and only if all sets in the Boolean hierarchy over $\mathrm{NP}$ have p-optimal proof systems which is a new characterization of a conjecture studied by Pudl\'ak.

翻译：我们研究了NP上布尔层级中各类别存在最优和p-最优证明系统的问题。主要结果涉及DP，即该层级的第二层：若DP中所有集合均具有p-最优证明系统，则coDP中所有集合也均具有p-最优证明系统。而最优证明系统的类似蕴含关系相对于一个谕示不成立。由此，我们厘清了NP上布尔层级中所有类别C和D的此类蕴含关系：要么能证明该蕴含关系成立，要么表明其相对于某个谕示不成立。此外，我们证明SAT集和TAUT集拥有p-最优证明系统当且仅当NP上布尔层级中的所有集合均拥有p-最优证明系统——这是对Pudlák所研究猜想的一个新刻画。