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

TL;DR

This paper extends linear logic with fixed points to infinitary, transfinite-branching systems $μ ext{MALL}_{α,α}$ indexed by a computable ordinal $α$, and proves that provability in these systems is complete for the level $ω^{α^ω}$ of the hyperarithmetical hierarchy. It develops cut-elimination and focusing via a carefully calibrated formula rank to tightly bound the height of cut-free proof-search and to enable precise upper and lower bound analyses. The upper bound follows from the rank-based bound on proof-search height, while the lower bound encodes true computable infinitary sentences complete for each hyperarithmetical level, using focusing to justify adequacy. The results provide a robust ordinal-analysis-style framework for fixed points in linear logic and broaden the understanding of provability with arbitrarily indexed infinite branching.

Abstract

We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $α$ for fixed points. Our main result is that provability in the system for some computable ordinal $α$ is complete for the $ω^{α^ω}$ 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.