Dependent Multiplicities in Dependent Linear Type Theory

Abstract: We present a novel dependent linear type theory in which the multiplicity of some variable - i.e., the number of times the variable can be used in a program - can depend on other variables. This allows us to give precise resource annotations to many higher-order functions that cannot be adequately typed in any other system. Inspired by the Dialectica translation, our typing discipline is obtained by embedding linear logic into dependent type theory and specifying how the embedded logic interacts with the host theory. We can then use a standard natural numbers type to obtain a quantitative typing system with dependent multiplicities. We characterise the semantics for our theory as a combination of standard models of dependent type theory and linear logic. Our system can be added to any dependently typed language, which we demonstrate with an implementation in Agda.