Least and Greatest Fixed Points in Linear Logic

Abstract

The first-order theory of MALL (multiplicative, additive linear logic) over only equalities is a well-structured but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and ?), we add least and greatest fixed point operators. The resulting logic, which we call μ MALL, satisfies two fundamental proof theoretic properties: we establish weak normalization for it, and we design a focused proof system that we prove complete with respect to the initial system. That second result provides a strong normal form for cut-free proof structures that can be used, for example, to help automate proof search. We show how these foundations can be applied to intuitionistic logic.