Two Notions of Sub-behaviour for Session-based Client/ServerSystems

We propose a refinement and a simplification of the behavioural semantics of session types, based on the concepts of compliance and sub-behaviour from the theory of web contracts. We introduce two relations, representing the idea of sub-behaviour from the point of view of the client and the server, respectively, and characterize the sub-behaviour relation (from the literature) as the intersection of the other two. We show that a proper subclass of behaviours, called "session behaviors'', and the sub-behaviour relations model session types and subtyping, clarifying the otherwise problematic extension of session type subtyping with concepts from the theory of contracts.