带多路复用的无穷动作逻辑的超算术复杂度
Hyperarithmetical Complexity of Infinitary Action Logic with Multiplexing
AI总结：
本文针对带多路复用的无穷动作逻辑$!^m\nabla \mathrm{ACT}_\omega$的可推导性问题，证明其在图灵归约下为$\Delta^0_{\omega^\omega}$-完全，确定其闭包序数为$\omega^\omega$，并给出相关片段的复杂度结果。
摘要：
2023年，Kuznetsov和Speranski提出了带多路复用的无穷动作逻辑$!^m\nabla \mathrm{ACT}_\omega$，并证明其可推导性问题处于超算术分层的$\omega$层与$\omega^\omega$层之间。本文证明该问题在图灵归约下是$\Delta^0_{\omega^\omega}$-完全的：具体而言，它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。由此可得$!^m\nabla \mathrm{ACT}_\omega$的闭包序数等于$\omega^\omega$。本文还证明，Kleene星不允许出现在子指数作用域内的$!^m\nabla \mathrm{ACT}_\omega$片段也是$\Delta^0_{\omega^\omega}$-完全的。最后，本文提出一族作为$!^m\nabla \mathrm{ACT}_\omega$片段的逻辑，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。
英文摘要：
In 2023, Kuznetsov and Speranski introduced infinitary action logic with multiplexing $!^m\nabla \mathrm{ACT}_ω$ and proved that the derivability problem for it lies between the $ω$ and $ω^ω$ levels of the hyperarithmetical hierarchy. We prove that this problem is $Δ^0_{ω^ω}$-complete under Turing reductions. Namely, we show that it is recursively isomorphic to the satisfaction predicate for computable infinitary formulas of rank less than $ω^ω$ in the language of arithmetic. As a consequence we prove that the closure ordinal for $!^m\nabla \mathrm{ACT}_ω$ equals $ω^ω$. We also prove that the fragment of $!^m\nabla \mathrm{ACT}_ω$ where Kleene star is not allowed to be in the scope of the subexponential is $Δ^0_{ω^ω}$-complete. Finally, we present a family of logics, which are fragments of $!^m\nabla \mathrm{ACT}_ω$, such that the complexity of the $k$-th logic lies between $Δ^0_{ω^k}$ and $Δ^0_{ω^{k+1}}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
