Taming Modal Impredicativity: Superlazy Reduction

Abstract. Pure, or type-free, Linear Logic proof nets are Turing com-plete once cut-elimination is considered as computation. We introduce modal impredicativity as a new form of impredicativity causing the com-plexity of cut-elimination to be problematic from a complexity point of view. Modal impredicativity occurs when, during reduction, the conclu-sion of a residual of a box b interacts with a node that belongs to the proof net inside another residual of b. Technically speaking, superlazy reduction is a new notion of reduction that allows to control modal im-predicativity. More specifically, superlazy reduction replicates a box only when all its copies are opened. This makes the overall cost of reducing a proof net finite and predictable. Specifically, superlazy reduction ap-plied to any pure proof nets takes primitive recursive time. Moreover, any primitive recursive function can be computed by a pure proof net via superlazy reduction