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不同（在TFNP中存在大量捕获自然存在性原理且彼此不可比较的问题（在黑盒设置下）），Kleinberg等人[KKMP21]观察到，迄今为止在全函数多项式层级第二级（TF$Σ_2$）中考虑的许多自然问题都可归约到强避免问题。在本工作中，我们证明线性序原理在黑盒设置下不能归约到强避免问题，从而展示了第一个位于可归约于强避免问题类之外的TF$Σ_2$问题。我们的分离证明利用了Fleming、Imrek和Marciot[FIM25]最近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，这意味着要证明我们的分离结果，只需证明在$Σ_2$变体的Sherali-Adams证明系统中不存在线性序原理的小证明。为此，我们将经典的伪期望方法扩展到$Σ_2$设置，证明$Σ_2$伪期望的存在排除了$Σ_2$ Sherali-Adams证明的可能性。主要的技术挑战在于证明此类伪期望的存在性，我们通过解决一个关于排列的组合覆盖问题成功地实现了这一点。我们还证明了扩展的伪期望界意味着线性序原理不能归约到任何允许低阶Sherali-Adams反驳的问题。