Unlike in TFNP, for which there is an abundance of problems capturing natural existence principles which are incomparable (in the black-box setting), Kleinberg et al. [KKMP21] observed that many of the natural problems considered so far in the second level of the total function polynomial hierarchy (TF$Σ_2$) reduce to the Strong Avoid problem. In this work, we prove that the Linear Ordering Principle does not reduce to Strong Avoid in the black-box setting, exhibiting the first TF$Σ_2$ problem that lies outside of the class of problems reducible to Strong Avoid. The proof of our separation exploits a connection between total search problems in the polynomial hierarchy and proof complexity, recently developed by Fleming, Imrek, and Marciot [FIM25]. In particular, this implies that to show our separation, it suffices to show that there is no small proof of the Linear Ordering Principle in a $Σ_2$-variant of the Sherali-Adams proof system. To do so, we extend the classical pseudo-expectation method to the $Σ_2$ setting, showing that the existence of a $Σ_2$ pseudo-expectation precludes a $Σ_2$ Sherali-Adams proof. The main technical challenge is in proving the existence of such a pseudo-expectation, we manage to do so by solving a combinatorial covering problem about permutations. We also show that the extended pseudo-expectation bound implies that the Linear Ordering Principle cannot be reduced to any problem admitting a low-degree Sherali-Adams refutation.

翻译：与TFNP中大量存在刻画自然存在性原理且（在黑盒设置下）不可比较的问题不同，Kleinberg等人[KKMP21]观察到，在完全函数多项式层级第二层（TF$Σ_2$）中迄今考虑的许多自然问题均可归约到Strong Avoid问题。本研究证明，线性排序原理（Linear Ordering Principle）在黑盒设置下并不能归约到Strong Avoid，从而首次展示了处于可归约到Strong Avoid的问题类之外的TF$Σ_2$问题。该分离性证明利用了Fleming、Imrek和Marciot[FIM25]近期发展的多项式层级中全搜索问题与证明复杂性之间的联系。特别地，这一联系意味着要证明该分离性，只需证明在$Σ_2$变体的Sherali-Adams证明系统中不存在线性排序原理的小规模证明。为此，我们将经典伪期望方法推广到$Σ_2$设置，证明$Σ_2$伪期望的存在性排除了$Σ_2$ Sherali-Adams证明。主要技术挑战在于证明此类伪期望的存在性——我们通过求解关于排列的组合覆盖问题成功实现了这一目标。我们还证明，该扩展伪期望下界意味着线性排序原理无法归约到任何容许低阶Sherali-Adams反驳的问题。