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Hyperarithmetical Complexity of Infinitary Action Logic with Multiplexing

Tikhon Pshenitsyn

TL;DR

The paper determines the precise algorithmic complexity of derivability for the infinitary action logic with multiplexing $!^m\\nabla \\mathrm{ACT}_\\omega$, proving $\\Delta^0_{\\omega^\\omega}$-completeness under Turing reductions by establishing a computable isomorphism with the satisfaction predicate for computable infinitary formulas of rank $<\\omega^\\omega$. It shows the closure ordinal of this calculus is $\\omega^\\omega$ and extends the analysis to the $!^m\\nabla \\mathrm{ACT}_\\omega^-$ fragment and to a family of depth-bounded fragments with complexities between $\\Delta^0_{\\omega^k}$ and $\\Delta^0_{\\omega^{k+1}}$, converging to $\\Delta^0_{\\omega^\\omega}$ as $k$ grows. The core method encodes a low-level program inside the logic and uses a bottom–top analysis of derivations together with effective transfinite recursion to relate derivability to the hyperarithmetical hierarchy. The results illuminate a unique complexity level for a natural propositional logic with subexponentials and multiplexing, with potential implications for formal linguistics and computability theory.

Abstract

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 $Δ^0_{ω^{k+1}}$.

Hyperarithmetical Complexity of Infinitary Action Logic with Multiplexing

TL;DR

The paper determines the precise algorithmic complexity of derivability for the infinitary action logic with multiplexing , proving -completeness under Turing reductions by establishing a computable isomorphism with the satisfaction predicate for computable infinitary formulas of rank . It shows the closure ordinal of this calculus is and extends the analysis to the fragment and to a family of depth-bounded fragments with complexities between and , converging to as grows. The core method encodes a low-level program inside the logic and uses a bottom–top analysis of derivations together with effective transfinite recursion to relate derivability to the hyperarithmetical hierarchy. The results illuminate a unique complexity level for a natural propositional logic with subexponentials and multiplexing, with potential implications for formal linguistics and computability theory.

Abstract

In 2023, Kuznetsov and Speranski introduced infinitary action logic with multiplexing and proved that the derivability problem for it lies between the and levels of the hyperarithmetical hierarchy. We prove that this problem is -complete under Turing reductions. Namely, we show that it is recursively isomorphic to the satisfaction predicate for computable infinitary formulas of rank less than in the language of arithmetic. As a consequence we prove that the closure ordinal for equals . We also prove that the fragment of where Kleene star is not allowed to be in the scope of the subexponential is -complete. Finally, we present a family of logics, which are fragments of , such that the complexity of the -th logic lies between and .
Paper Structure (15 sections, 24 theorems, 57 equations)

This paper contains 15 sections, 24 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Infinitary Action Logic with Multiplexing
  4. Basic Derivations
  5. Hyperarithmetical Hierarchy
  6. Computable Functions as String Rewriting Systems
  7. Main Results
  8. Construction
  9. Bottom-Top Analysis
  10. Proof of Lemma
  11. Reduction of Derivability in $!^m\nabla \mathrm{ACT}_\omega$ to $\mathrm{SAT}_{<\omega^\omega}$
  12. Closure Ordinal for $!^m\nabla \mathrm{ACT}_\omega$
  13. Modification of the Construction for $!^m\nabla \mathrm{ACT}_\omega^-$
  14. Fragments of $!^m\nabla \mathrm{ACT}_\omega$ with Intermediate Complexities
  15. Conclusion

Key Result

Theorem 1

The derivability problem for $!^m\nabla \mathrm{ACT}_\omega$ is $\Delta^0_{\omega^\omega}$-complete under Turing reductions.

Theorems & Definitions (63)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Example 1
  • Lemma 2
  • Definition 4
  • ...and 53 more