Upper and Lower Bounds for the Linear Ordering Principle

Korten and Pitassi (FOCS, 2024) defined a new complexity class $L_2P$ as the polynomial-time Turing closure of the Linear Ordering Principle. They asked whether a Karp--Lipton--style collapse can be proven for $L_2P$. We answer this question affirmatively by showing that $P^{prMA}\subseteq L_2P$. As a byproduct, we also answer an open question of Chakaravarthy and Roy (Computational Complexity, 2011) whether $P^{prMA}\subseteq S_2P$. We complement this result by providing a new upper bound for $L_2P$, namely $L_2P\subseteq P^{prSBP}$. Thus we are placing $L_2P$ between $P^{prMA}$ and $P^{prSBP}$. One technical ingredient of this result is an algorithm that approximates the number of satisfying assignments of a Boolean circuit using a $prSBP$ oracle (i.e. in $FP^{prSBP}$), which could be of independent interest. Finally, we prove that $P^{prO_2P}\subseteq O_2P$, which implies 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).