Type Theory offers a tantalising promise: that we can program and reason within a single unified system. However, this promise slips away when we try to produce efficient programs. Type Theory offers little control over the intensional aspect of programs: how are computational resources used, and when can they be reused. Tracking resource usage via types has a long history, starting with Girard's Linear Logic and culminating with recent work in contextual effects, coeffects, and quantitative type theories. However, there is conflict with full dependent Type Theory when accounting for the difference between usages in types and terms. Recently, McBride has proposed a system that resolves this conflict by treating usage in types as a zero usage, so that it doesn't affect the usage in terms. This leads to a simple expressive system, which we have named Quantitative Type Theory (QTT).
McBride presented a syntax and typing rules for the system, as well as an erasure property that exploits the difference between “not used” and “used”, but does not do anything with the finer usage information. In this paper, we present present a semantic interpretation of a variant of McBride's system, where we fully exploit the usage information. We interpret terms simultaneously as having extensional (compile-time) content and intensional (runtime) content. In our example models, extensional content is set-theoretic functions, representing the compile-time or type-level content of a type-theoretic construction. Intensional content is given by realisers for the extensional content. We use Abramsky et al.'s Linear Combinatory Algebras as realisers, yield a large range of potential models from Geometry of Interaction, graph models, and syntactic models. Read constructively, our models provide a resource sensitive compilation method for QTT.
To rigorously define the structure required for models of QTT, we introduce the concept of a Quantitative Category with Families, a generalisation of the standard Category with Families class of models of Type Theory, and show that this class of models soundly interprets Quantitative Type Theory.