Undecidability of Asynchronous Session Subtyping

The most prominent proposals of subtyping for asynchronous session types are by Mostrous and Yoshida for binary sessions, by Chen et al. for binary sessions under the assumption that every message emitted is eventually consumed, and by Mostrous et al. for multiparty session types. We prove that, differently from what stated or conjectured in above proposals, all of these three subtyping relations are undecidable. Additionally, we identify two subclasses of types where the relation becomes decidable.