Hyperarithmetical Complexity of Infinitary Action Logic with Multiplexing
(2312.04091)Abstract
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 $\Delta0{\omega\omega}$-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 $\Delta0_{\omega\omega}$-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 $\Delta0{\omegak}$ and $\Delta0_{\omega{k+1}}$.
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