Korten and Pitassi (FOCS, 2024) defined a new complexity class $L_2^P$ as the polynomial-time Turing closure of the Linear Ordering Principle. They put it between $MA$ (Merlin--Arthur protocols) and $S_2^P$ (the second symmetric level of the polynomial hierarchy). In this paper we sandwich $L_2^P$ between $P^{prMA}$ and $P^{prSBP}$. (The oracles here are promise problems, and $SBP$ is the only known class between $MA$ and $AM$.) The containment in $P^{prSBP}$ is proved via an iterative process that uses a $prSBP$ oracle to estimate the average order rank of a subset and find the minimum of a linear order. Another containment result of this paper is $P^{prO_2^P} \subseteq O_2^P$ (where $O_2^P$ is the input-oblivious version of $S_2^P$). These containment results altogether have several byproducts: We give an affirmative answer to an open question posed by of Chakaravarthy and Roy (Computational Complexity, 2011) whether $P^{prMA} \subseteq S_2^P$, thereby settling the relative standing of the existing (non-oblivious) Karp-Lipton-style collapse results of Chakaravarthy and Roy (2011) and Cai (2007), We give an affirmative answer to an open question of Korten and Pitassi whether a Karp-Lipton-style collapse can be proven for $L_2^P$, We show that the Karp-Lipton-style collapse to $P^{prOMA}$ is actually better than both known collapses to $P^{prMA}$ due to Chakaravarthy and Roy (Computational Complexity, 2011) and to $O_2^P$ also due to Chakaravarthy and Roy (STACS, 2006). Thus we resolve the controversy between previously incomparable Karp-Lipton collapses stemming from these two lines of research.

翻译：Korten和Pitassi（FOCS, 2024）定义了新的复杂度类$L_2^P$作为线性排序原理的多项式时间图灵闭包。他们将其置于$MA$（梅林-亚瑟协议）与$S_2^P$（多项式层级第二对称层）之间。本文在$P^{prMA}$与$P^{prSBP}$之间夹逼$L_2^P$（其中的谕示为承诺问题，而$SBP$是唯一已知介于$MA$与$AM$之间的类）。通过使用$prSBP$谕示估计子集的平均序秩并寻找线性序的最小元，迭代过程证明了$L_2^P$包含于$P^{prSBP}$。本文的另一包含结果为$P^{prO_2^P} \subseteq O_2^P$（其中$O_2^P$是$S_2^P$的输入无关版本）。这些包含结果共同产生若干副产品：我们肯定地回答了Chakaravarthy与Roy（计算复杂性，2011）提出的开放问题——$P^{prMA} \subseteq S_2^P$是否成立，从而厘清了Chakaravarthy与Roy（2011）及Cai（2007）现有（非无关）Karp-Lipton型坍塌结果的相对地位；我们肯定地回答了Korten与Pitassi的开放问题——$L_2^P$能否证明Karp-Lipton型坍塌；我们表明$P^{prOMA}$的Karp-Lipton型坍塌实际上优于Chakaravarthy与Roy（计算复杂性，2011）的$P^{prMA}$坍塌以及同样由Chakaravarthy与Roy（STACS, 2006）得到的$O_2^P$坍塌。由此我们解决了此前这两条研究路线中不可比较的Karp-Lipton坍塌结果之间的争议。