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

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.