Foundations of Substructural Dependent Type Theory
C. B. Aberlé
TL;DR
This work addresses the challenge of marrying dependent type theory with substructural logics by proposing left-fibred double categories (LFDCs) as a unifying semantic framework. It develops a detailed LFDC-based type theory, including a comprehensive set of syntax and inference rules (contexts, unit, sums/products, and both term- and type-level function types) and establishes conditions for syntactic completeness and decidability via substructuralization. A key contribution is the application to linear logic metatheory, where LLFDC_sigma encodes cut-admissibility in a way that avoids known pitfalls of prior approaches. The framework sets the stage for further work on universes, inductive types, and the integration of modal and monoidal-topos techniques to extend substructural dependent types to richer semantic settings with practical implications for formalizing substructural reasoning and linear sequent calculi.
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.