# Taming Modal Impredicativity: Superlazy Reduction

• Ugo Dal Lago
• Luca Roversi
• Luca Vercelli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5407)

## Abstract

Pure, or type-free, Linear Logic proof nets are Turing complete once cut-elimination is considered as computation. We introduce modal impredicativity as a new form of impredicativity causing cut-elimination to be problematic from a complexity point of view. Modal impredicativity occurs when, during reduction, the conclusion 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 impredicativity. 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 applied 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.

## Keywords

Linear logic implicit computational complexity proof theory

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## Authors and Affiliations

• Ugo Dal Lago
• 1
• Luca Roversi
• 2
• Luca Vercelli
• 3
1. 1.Dipartimento di InformaticaUniversità di BolognaItaly
2. 2.Dipartimento di InformaticaUniversità di TorinoItaly
3. 3.Dipartimento di MatematicaUniversità di TorinoItaly