Categorical Models of Amortized Cost: An Adjoint Relationship between Cost and Potential

Abstract

Various type systems have been developed to track the cost $\kappa$ of a computation using a cost-tracking monad $M\ \kappa \tau$. On its own, this only tracks the worst-case cost of a computation. If we also want to track amortized cost, then we can add a type $[\kappa]\tau$ which stores potential $\kappa$ with a type $\tau$, together with operations for storing and releasing potential. In this work, we build on one such system, $\lambda$-amor: $\lambda$-amor allows to track cost and potential in the type system and subsumes effect and coeffect-based systems, call-by-value and call-by-name based languages. In this paper, we identify the abstract properties that denotational models of type theories for cost and potential have to satisfy: Cost and potential must be modelled by an adjoint pair of graded functors, where the functor modelling cost forms both a graded monad and a compatible graded comonad. We present three concrete instances of this general abstract scheme: (1) A simple set-theoretic model that ignores the cost tracked by the type system, (2) the Kripke logical relations model in the original $\lambda$-amor paper (which we show can be turned into an instance of the adjoint model), and (3) a novel model based on copresheaves on a monoidal category of costs, where we model pairs and functions by Day convolution and its right-adjoint.