Plotkin and Pretnar’s effect handlers offer a versatile abstraction for modular programming with user-defined effects. This paper focuses on foundations for implementing effect handlers, for the three different kinds of effect handlers that have been proposed in the literature: deep, shallow, and parameterised. Traditional deep handlers are defined by folds over computation trees and are the original construct proposed by Plotkin and Pretnar. Shallow handlers are defined by case splits (rather than folds) over computation trees. Parameterised handlers are deep handlers extended with a state value that is threaded through the folds over computation trees. We formulate the extensions both directly and via encodings in terms of deep handlers and illustrate how the direct implementations avoid the generation of unnecessary closures. We give two distinct foundational implementations of all the kinds of handlers we consider: a continuation-passing style (CPS) transformation and a CEK-style abstract machine. In both cases, the key ingredient is a generalisation of the notion of continuation to accommodate stacks of effect handlers. We obtain our CPS translation through a series of refinements as follows. We begin with a first-order CPS translation into untyped lambda calculus which manages a stack of continuations and handlers as a curried sequence of arguments. We then refine the initial CPS translation by uncurrying it to yield a properly tail-recursive translation and then moving towards more and more intensional representations of continuations in order to support different kinds of effect handlers. Finally, we make the translation higher order in order to contract administrative redexes at translation time. Our abstract machine design then uses the same generalised continuation representation as the CPS translation. We have implemented both the abstract machine and the CPS transformation (plus extensions) as backends for the Links web programming language.

We present Continuation Passing Style (CPS) translations for Plotkin and Pretnar’s effect handlers with Hillerström and Lindley’s row-typed fine-grain call-by-value calculus of effect handlers as the source language. CPS translations of handlers are interesting theoretically, to explain the semantics of handlers, and also offer a practical implementation technique that does not require special support in the target language’s runtime.

We begin with a first-order CPS translation into untyped lambda calculus which manages a stack of continuations and handlers as a curried sequence of arguments. We then refine the initial CPS translation first by uncurrying it to yield a properly tail-recursive translation and second by making it higher-order in order to contract administrative redexes at translation time. We prove that the higher-order CPS translation simulates effect handler reduction. We have implemented the higher-order CPS translation as a JavaScript backend for the Links programming language.