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 prove that it is recursively isomorphic to the satisfaction predicate for computable infinitary formulas of rank less than $\omega^\omega$ in the language of arithmetic. We also prove this result for the fragment of $!^m\nabla \mathrm{ACT}_\omega$ where Kleene star is not allowed to be in the scope of the subexponential. 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 is between $\Delta^0_{\omega^k}$ and $\Delta^0_{\omega^{k+1}}$.

翻译：库兹涅佐夫和斯佩兰斯基于2023年引入了具有复用功能的无穷行动逻辑$!^m\nabla \mathrm{ACT}_\omega$，并证明了其可推导性问题的复杂度介于超算术层级的$\omega$与$\omega^\omega$层之间。我们证明该问题在Turing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可满足谓词是递归同构的。我们同样证明了$!^m\nabla \mathrm{ACT}_\omega$中不允许Kleene星号出现在亚指数算子作用域内的片段也具有相同结果。最后，我们给出了一族作为$!^m\nabla \mathrm{ACT}_\omega$片段的逻辑系统，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。