Sub-behaviour relations for session-based client/server systems

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 three relations on a suitable class of behaviours with higher-order input/output, called 'session behaviours'. Such relations, depending on each other, represent the idea of sub-behaviour from the point of view of a client, a server or a peer, respectively. A restriction of the intersection of the first two relations characterizes the 'usual' sub-behaviour relation from the literature. We then device an algorithmic formal system for three subtyping relations (dubbed CSP-subtyping) for session types that takes into account the role played by a user of a channel during an interaction, so extending Gay and Hole subtyping theory. We show that our session behaviours and sub-behaviour relations provide sound and complete semantics for CSP-subtyping, and for Gay and Hole subtyping as a by-product.