A Core Quantitative Coeffect Calculus

Linear logic is well known for its resource-awareness, which has inspired the design of several resource management mechanisms in programming language design. Its resource-awareness arises from the distinction between linear, single-use data and non-linear, reusable data. The latter is marked by the so-called exponential modality, which, from the categorical viewpoint, is a monoidal comonad. Monadic notions of computation are well-established mechanisms used to express effects in pure functional languages. Less well-established is the notion of comonadic computation. However, recent works have shown the usefulness of comonads to structure context dependent computations. In this work, we present a language $\ell \mathcal{R}$ PCF inspired by a generalized interpretation of the exponential modality. In $\ell \mathcal{R}$ PCF the exponential modality carries a label--an element of a semiring $\mathcal{R}$ --that provides additional information on how a program uses its context. This additional structure is used to express comonadic type analysis.