Undecidability of asynchronous session subtyping

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 [1] for binary sessions, the relation by Chen et al. [2] for binary sessions under the assumption that every message emitted is eventually consumed, and the one by Mostrous et al. [3] 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.Formally, subtyping relations have been defined for session types to precisely capture this safe replacement notion.Gay and Hole [6] are the first ones who studied subtyping for session types in a context where protocols involve only two peers (i.e. are binary) and communication is synchronous. Later, Mostrous et al. [3] extended this notion to multiparty session types with asynchronous communication. Both articles propose an algorithm for checking subtying, but the one proposed by Mostrous et al. [3], differently from what stated therein, is not always terminating in the sense that there are cases in which it diverges and never gives an answer. An example of divergent execution is discussed in the Remark paragraph of §4.4.Later work by Mostrous and Yoshida [1], Mostrous [7] and Chen et al. [2] addresses subtyping in variants of an asynchronous setting for binary sessions. In particular Chen et al. [2] focus on binary sessions in which messages sent by a partner are guaranteed to be eventually received. Such articles conjecture that an algorithm for checking asynchronous session subtyping exists, although, in his PhD thesis, Mostrous [7] expresses a few doubts about the decidability of asynchronous subtyping (pp. 178-180), because of the need for infinite simulations. In this work, we prove that the subtyping relations defined by Mostrous and Yoshida [1], Chen et al. [2], and Mostrous et al. [3] are undecidable. We proceed by identifying a core asynchronous subtyping relation and show it is undecidable: all other undecidability results are obtained by reduction from this initial relation.The core relation, denoted by < <, is named asynchronous single-choice relation. Such a relation is obtained by first defining (following the approach by Mostrous and Yoshida [1]) a standard asynchronous subtyping ≤ and then reduce it by imposing additional constraints: T and S are in single-choice relation...

On the boundary between decidability and undecidability of asynchronous session subtyping

“…Recently, Bravetti et al [10] and Lange and Yoshida [11] have independently shown that asynchronous subtyping (the subtyping relation with output anticipation) is undecidable. In particular, in Bravetti…”

“…The aim of this paper is to detect significant decidable fragments of asynchronous session subtyping and to establish a more precise boundary between decidability and undecidability. In particular, concerning decidability, as discussed above, the few decidable fragments of asynchronous subtyping known so far are extremely restrictive: our relations < < sin , < < sout [10] and the decidable relations considered by Lange and Yoshida [11]. Here, for the first time, we show decidability of a fragment that does not impose any limitation on communication buffers and allows both the subtype and the supertype to include multiple choices (either for input branchings or for output selections), thus opening the possibility for some practical applicability in restricted specific scenarios (e.g.…”

“…The combination of non restricted buffers and presence of multiple choices on both related types requires a totally new approach for guaranteeing the termination of the subtyping algorithm (both for the termination condition itself and for the related decidability proof). For instance, if multiple choices are admitted for input branchings, the termination condition has to deal with complex recurrent patterns to be checked on the leafs of trees representing input branchings (with multiple choices), instead of detecting simple repetitions on strings representing sequences of single-choice inputs (as in previous work [10,11]).…”

“…Concerning undecidability, all previous results [10,11] exploit the capability of asynchronous subtyping of matching input branchings/output selections by means of covariance/contravariance. We here show that asynchronous subtyping remains undecidable even if we restrict it by disallowing this feature.…”

Open Multiparty Sessions

“…We expect the compatibility could be extended, since the subtyping for asynchronous multiparty sessions is more permissive than the subtyping for the synchronous ones [25]. Of course this extension requires care, being the subtyping of [25] undecidable, as shown in [2,24].…”