Korten and Pitassi (FOCS, 2024) defined a new complexity class $L_2P$ as the polynomial-time Turing closure of the Linear Ordering Principle. They put it between $MA$ (Merlin--Arthur protocols) and $S_2P$ (the second symmetric level of the polynomial hierarchy). In this paper we sandwich $L_2P$ 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 $P^{prO_2P} \subseteq O_2P$ (where $O_2P$ is the input-oblivious version of $S_2P$). 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 $P^{prMA} \subseteq S_2P$, 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 $L_2P$, 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 $O_2P$ 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与Pitassi（FOCS，2024）将线性排序原理的多项式时间图灵闭包定义为新的复杂度类$L_2P$，并将其置于$MA$（Merlin-Arthur协议）与$S_2P$（多项式谱系的第二对称层级）之间。本文通过证明$L_2P$包含于$P^{prMA}$且包含于$P^{prSBP}$，从而给出该类的精确界定（此处预言机对应承诺问题，而$SBP$是当前已知唯一介于$MA$与$AM$之间的复杂度类）。其中$P^{prSBP}$包含性的证明通过迭代过程实现：利用$prSBP$预言机估计子集的平均序秩并寻找线性序的最小元。本文另一包含性结果为$P^{prO_2P} \subseteq O_2P$（其中$O_2P$为$S_2P$的输入不可知版本）。这些包含关系共同产生若干推论：我们肯定回答了Chakaravarthy与Roy（Computational Complexity，2011）提出的开放问题——是否$P^{prMA} \subseteq S_2P$成立，从而确立了Chakaravarthy与Roy（2011）和Cai（2007）现有（非不可知）Karp-Lipton式塌缩结果的相对关系；我们肯定回答了Korten与Pitassi关于$L_2P$是否存在Karp-Lipton式塌缩的开放问题；我们证明塌缩至$P^{prOMA}$的Karp-Lipton式结果实际上优于Chakaravarthy与Roy（Computational Complexity，2011）提出的$P^{prMA}$塌缩以及Chakaravarthy与Roy（STACS，2006）提出的$O_2P$塌缩，由此解决了源自这两条研究脉络、先前不可比较的Karp-Lipton塌缩结果之间的争议。