Wider systems for linear logic with fixed points: proof theory and complexity

TL;DR

This paper develops infinitary proof systems for linear logic with fixed points, establishing their proof-theoretic properties and complexity bounds related to the hyperarithmetical hierarchy.

Contribution

It introduces a novel infinitary proof system with transfinite branching rules for linear logic with fixed points, and analyzes its proof-theoretic strength and complexity.

Findings

Provability aligns with the ^{\u03b1^} level of the hyperarithmetical hierarchy.

Develops cut elimination and focusing results for these systems.

Provides bounds on proof search space height using a formula rank measure.

Abstract

We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some computable ordinal $\alpha$ is complete for the $\omega^{\alpha^\omega}$ level of the hyperarithmetical hierarchy. To this end we first develop proof theoretic foundations, namely cut elimination and focussing results, to control both the upper and lower bound analysis. Our arguments employ a carefully calibrated notion of formula rank, calculating a tight bound on the height of the (cut-free) proof search space.