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

2026-08-10

AI Summary

This work addresses the challenge of uniformly formalizing worst-case and amortized costs within a type system. To this end, it proposes a categorical semantic framework in which cost and potential are modeled by an adjoint pair of graded functors. The cost functor is equipped with both a graded monad and a compatible graded comonad structure, enabling a unified treatment of effects and coeffects across different evaluation strategies. Building upon graded monads, adjunctions, Kripke logical relations, and copresheaf models under Day convolution, the framework yields three concrete semantic models: a cost-erasing set-theoretic model, the original Kripke logical relation model for λ-amor, and a novel copresheaf model over a cost-enriched monoidal category.

Abstract

Various type systems have been developed to track the cost $κ$ of a computation using a cost-tracking monad $M\ κτ$. 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 $[κ]τ$ which stores potential $κ$ with a type $τ$, together with operations for storing and releasing potential. In this work, we build on one such system, $λ$-amor: $λ$-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 $λ$-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.

Problem

• amortized cost
• type systems
• denotational semantics
• graded functors
• adjoint models

Innovation

• graded adjunction
• amortized cost
• potential semantics
• Day convolution
• denotational models