Hyperarithmetical complexity of infinitary action logic with multiplexing

In 2023, Kuznetsov and Speranski introduced infinitary action logic with multiplexing $!^{m}\nabla \textrm{ACT}_{\omega }$ and proved that the derivability problem for it lies between the $\omega $ level and the $\omega ^{\omega }$ level of the hyperarithmetical hierarchy. We prove that this problem is $\varDelta ^{0}_{\omega ^{\omega }}$-complete under Turing reductions. Namely, we show that it is recursively isomorphic to the satisfaction predicate for computable infinitary formulas of rank less than $\omega ^{\omega }$ in the language of arithmetic. As a consequence we prove that the closure ordinal for $!^{m}\nabla \textrm{ACT}_{\omega }$ equals $\omega ^{\omega }$. We also prove that the fragment of $!^{m}\nabla \textrm{ACT}_{\omega }$ where Kleene star is not allowed to be in the scope of the subexponential is $\varDelta ^{0}_{\omega ^{\omega }}$-complete. Finally, we present a family of logics, which are fragments of $!^{m}\nabla \textrm{ACT}_{\omega }$, such that the complexity of the $k$-th logic lies between $\varDelta ^{0}_{\omega ^{k}}$ and $\varDelta ^{0}_{\omega ^{k+1}}$.