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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Baeten, Bergstra, and Klop (and later Caucal) have proved the remarkable result that bisimulation equivalence is decidable for irredundant context-free grammars. In this paper we provide a much simpler and much more direct proof of this result using a tableau decision method involving goal-directed rules. The decision procedure also provides the essential part of the bisimulation relation between two processes which underlies their equivalence. We also show how to obtain a sound and complete sequent-based equational theory for such processes from the tableau system and how one can extract what Caucal calls a fundamental relation from a successful tableau.

A recent theorem 3, 7, 19] shows that strong bisimilarity is decidable for the class of normed BPA processes, which correspond to a class of context-free grammars generating the-free context-free languages. In 21] Huynh and Tian have shown that readiness and failure equivalence are undecidable for BPA processes. In this paper we examine all other equivalences in the linear/branching time hierarchy 13] and show that none of them are decidable for normed BPA processes.

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