This is the second part of a two-part special issue on Behavioural Types, which has its origin in a workshop we organized in April 2011, in Lisbon. The aim of the workshop was to bring together the active and expanding community of researchers using type-theoretic approaches to describe and analyse behavioural aspects of software. A particular concern of this field is the identification and description of structured communication in concurrent and distributed systems, but behavioural typing also addresses issues of liveness, fairness, deadlock-freedom, security, observable equivalence, and typestate.

Throughout the years, several typing disciplines for the π-calculus have been proposed. Arguably, the most widespread of these typing disciplines consists of session types. Session types describe the input/output behaviour of processes and traditionally provide strong guarantees about this behaviour (i.e. deadlock-freedom and fidelity). While these systems exploit a fundamental notion of linearity, the precise connection between linear logic and session types has not been well understood.

This paper proposes a type system for the π-calculus that corresponds to a standard sequent calculus presentation of intuitionistic linear logic, interpreting linear propositions as session types and thus providing a purely logical account of all key features and properties of session types. We show the deep correspondence between linear logic and session types by exhibiting a tight operational correspondence between cut-elimination steps and process reductions. We also discuss an alternative presentation of linear session types based on classical linear logic, and compare our development with other more traditional session type systems.

The subtyping relation defined for dyadic session type theories may compromise the liveness of multi-party sessions. In this paper, we define a fair subtyping relation for multi-party session types that preserves liveness, we relate it with the subtyping relation for dyadic session types and provide coinductive, axiomatic and algorithmic characterizations for it.

Session types describe and constrain the input/output behaviour of systems. Existing session typing systems have limited support for polymorphism. For example, existing systems cannot provide the most general type for a generic proxy process that forwards messages between two channels. We provide a polymorphic session typing system for the π calculus, and demonstrate the utility of session-type-level functions in combination with polymorphic session typing. The type system guarantees subject reduction and safety properties, but not deadlock freedom. We describe a formalization of the type system in Coq. The proofs of subject reduction and safety properties, as well as typing of example processes, have been mechanically verified.

Session types and contracts are two formalisms used to study client–server protocols. In this paper, we study the relationship between them. The main result is the existence of a fully abstract model of session types; this model is based on a natural interpretation of these types into a subset of contracts.

This paper formalizes the observable interface behaviour of open systems for a strongly-typed, concurrent object-oriented language with single-class inheritance. We formally characterize the observable behaviour in terms of interactions at the program-environment interface. The behaviour is given by transitions between contextual judgments, where the absent environment is represented abstractly as assumption context. A particular challenge is the fact that, when the system is considered as open, code from the environment can be inherited to the component and vice versa. This requires to incorporate an abstract version of the heap into the environment assumptions when characterizing the interface behaviour. We prove the soundness of the abstract interface description.