We generalise a theory of multiparty session types for the π-calculus through asynchronous communication subtyping, which allows partial commutativity of actions with maximal flexibility and safe optimisation in message choreography. A sound and complete algorithm for the subtyping relation, which can calculate conformance of optimised end-point processes to an agreed global specification, is presented. As a complementing result, we show a type inference algorithm for deriving the principal global specification from end-point processes which is minimal with respect to subtyping. The resulting theory allows a programmer to choose between a top-down and a bottom-up style of communication programming, ensuring the same desirable properties of typable processes.

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Behavioural type systems, usually associated to concurrent or distributed computations, encompass concepts such as interfaces, communication protocols, and contracts, in addition to the traditional input/output operations. The behavioural type of a software component specifies its expected patterns of interaction using expressive type languages, so types can be used to determine automatically whether the component interacts correctly with other components. Two related important notions of behavioural types are those of session types and behavioural contracts. This article surveys the main accomplishments of the last 20 years within these two approaches.

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Abstract. This paper proposes two typing systems for session interactions in higherorder mobile processes. Session types for the HOπ-calculus capture high-level structures of communication protocols and code mobility as type abstraction, and can be used to statically check the safe and consistent process composition in communication-centric distributed software. Integration of arbitrary higher-order code mobility and sessions leads to technical difficulties in type soundness, because linear usage of session channels and completion of sessions are required in executed code. By using techniques from the linear λ-calculus, we develop a coherent and tractable session typing system for the HOπ-calculus. We also present an alternative system based on fine-grained process types. The formal comparison between the two systems offers insight on the interplay between higher-order code mobility and session types.