Foundations of Substructural Dependent Type Theory

Abstract

This paper presents preliminary work on a general system for integrating dependent types into substructural type systems such as linear logic and linear type theory. Prior work on this front has generally managed to deliver type systems possessing either syntax or semantics inclusive of certain practical applications, but has struggled to combine these all in one and the same system. Toward resolving this difficulty, I propose a novel categorical interpretation of substructural dependent types, analogous to the use of monoidal categories as models of linear and ordered logic, that encompasses a wide class of mathematical and computational examples. On this basis, I develop a general framework for substructural dependent type theories, and proceed to prove some essential metatheoretic properties thereof. As an application of this framework, I show how it can be used to construct a type theory that satisfactorily addresses the problem of effectively representing cut admissibility for linear sequent calculus in a logical framework.