In 2023, Kuznetsov and Speranski introduced infinitary action logic with multiplexing $!^m\nabla \mathrm{ACT}_\omega$ and proved that the derivability problem for it lies between the $\omega$ and $\omega^\omega$ levels of the hyperarithmetical hierarchy. We prove that this problem is $\Delta^0_{\omega^\omega}$-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 $!^m\nabla \mathrm{ACT}_\omega$ equals $\omega^\omega$. We also prove that the fragment of $!^m\nabla \mathrm{ACT}_\omega$ where Kleene star is not allowed to be in the scope of the subexponential is $\Delta^0_{\omega^\omega}$-complete. Finally, we present a family of logics, which are fragments of $!^m\nabla \mathrm{ACT}_\omega$, such that the complexity of the $k$-th logic lies between $\Delta^0_{\omega^k}$ and $\Delta^0_{\omega^{k+1}}$.

翻译：Kuznetsov和Speranski于2023年引入了带有复制$!^m\nabla \mathrm{ACT}_\omega$的无穷动作逻辑，并证明了其可推导性问题位于超算术层级中的$\omega$层与$\omega^\omega$层之间。我们证明该问题在图灵归约下是$\Delta^0_{\omega^\omega}$-完备的。具体而言，我们证明该问题与算术语言中秩小于$\omega^\omega$的可计算无穷公式的满足谓词递归同构。由此推论，$!^m\nabla \mathrm{ACT}_\omega$的闭包序数等于$\omega^\omega$。我们还证明了$!^m\nabla \mathrm{ACT}_\omega$中不允许Kleene星号处于子指数辖域内的片段是$\Delta^0_{\omega^\omega}$-完备的。最后，我们给出一个逻辑族（作为$!^m\nabla \mathrm{ACT}_\omega$的片段），其中第$k$个逻辑的复杂度介于$\Delta^0_{\omega^k}$与$\Delta^0_{\omega^{k+1}}$之间。