Hyperarithmetical Complexity of Infinitary Action Logic with Multiplexing

Abstract: In 2023, Kuznetsov and Speranski introduced infinitary action logic with multiplexing $!^m\nabla \mathrm{ACT}<em>\omega$ and proved that the derivability problem for it lies between the $\omega$ and $\omega<sup>\omega$ levels of the hyperarithmetical hierarchy. We prove that this problem is $\Delta⁰</em>{\omega<sup>\omega}$-complete under Turing reductions. Namely, we show that it is recursively isomorphic to the satisfaction predicate for computable infinitary formulas of rank less than $\omega<sup>\omega$ in the language of arithmetic. As a consequence we prove that the closure ordinal for $!^m\nabla \mathrm{ACT}<em>\omega$ equals $\omega<sup>\omega$. We also prove that the fragment of $!^m\nabla \mathrm{ACT}</em>\omega$ where Kleene star is not allowed to be in the scope of the subexponential is $\Delta<sup>0_{\omega<sup>\omega}$-complete. Finally, we present a family of logics, which are fragments of $!^m\nabla \mathrm{ACT}<em>\omega$, such that the complexity of the $k$-th logic lies between $\Delta⁰</em>{\omega<sup>k}$ and $\Delta<sup>0_{\omega<sup>{k+1}}$.