On Complete Categorical Semantics for Effect Handlers

TL;DR

This paper establishes soundness and completeness for categorical models of effect handlers, including free monads and CPS semantics, expanding the understanding of semantic models for algebraic effects.

Contribution

It characterizes categorical models of effect handlers that support soundness and completeness, beyond just free monad models, including CPS semantics.

Findings

Identified categorical models supporting effect handlers

Established soundness and completeness results

Captured CPS semantics as valid models

Abstract

Soundness and completeness with respect to equational theories for programming languages are fundamental properties in the study of categorical semantics. However, completeness results have not been established for programming languages with algebraic effects and handlers, which raises a question of whether the commonly used models in the literature, i.e., free model monads generated from algebraic theories, are the only valid semantic models for effect handlers. In this paper, we show that this is not the case. We identify the precise characterizations of categorical models of effect handlers that allow us to establish soundness and completeness results with respect to a certain equational theory for effect handling constructs. Notably, this allows us to capture not only free monad models but also the CPS semantics for effect handlers as models of the calculus.