Undecidability of Asynchronous Session Subtyping

TL;DR

This paper proves that various forms of asynchronous session subtyping are undecidable, highlighting fundamental limits in verifying communication protocols in distributed systems, but also identifies specific restrictions that restore decidability.

Contribution

It establishes the undecidability of multiple asynchronous session subtyping notions and identifies structural restrictions that make subtyping decidable.

Findings

All three considered asynchronous subtyping notions are undecidable.

Restrictions on type structure can make subtyping decidable.

A core undecidable subtyping relation is identified.

Abstract

Session types are used to describe communication protocols in distributed systems and, as usual in type theories, session subtyping characterizes substitutability of the communicating processes. We investigate the (un)decidability of subtyping for session types in asynchronously communicating systems. We first devise a core undecidable subtyping relation that is obtained by imposing limitations on the structure of types. Then, as a consequence of this initial undecidability result, we show that (differently from what stated or conjectured in the literature) the three notions of asynchronous subtyping defined so far for session types are all undecidable. Namely, we consider the asynchronous session subtyping by Mostrous and Yoshida for binary sessions, the relation by Chen et al. for binary sessions under the assumption that every message emitted is eventually consumed, and the one by Mostrous et al. for multiparty session types. Finally, by showing that two fragments of the core subtyping relation are decidable, we evince that further restrictions on the structure of types make our core subtyping relation decidable.