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 NP 上布尔层级中各类别存在最优与 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 中所有集合均存在 P-最优证明系统，则 coDP 中所有集合也存在 P-最优证明系统。对于最优证明系统，类似的蕴含关系相对于某个谕言机不成立。由此，我们阐明了 NP 上布尔层级中所有类别 C 与 D 的此类蕴含关系：要么可证明该蕴含成立，要么表明其相对于某个谕言机不成立。此外，我们证明：集合 SAT 与 TAUT 存在 P-最优证明系统，当且仅当 NP 上布尔层级中所有集合均存在 P-最优证明系统——这是对 Pudlák 所研究猜想的一个新刻画。