Psi-calculi extend the pi-calculus with nominal datatypes to represent data, communication channels, and logics for facts and conditions. This general framework admits highly expressive formalisms such as concurrent higher-order constraints and advanced cryptographic primitives. We here establish the theory of weak bisimulation, where the τ actions are unobservable. In comparison to other calculi the presence of assertions poses a significant challenge in the definition of weak bisimulation, and although there appears to be a spectrum of possibilities we show that only a few are reasonable. We demonstrate that the complications mainly stem from psi-calculi where the associated logic does not satisfy weakening. We prove that weak bisimulation equivalence has the expected algebraic properties and that the corresponding observation congruence is preserved by all operators. These proofs have been machine checked in Isabelle. The notion of weak barb is defined as the output label of a communication action, and weak barbed equivalence is bisimilarity for τ actions and preservation of barbs in all static contexts. We prove that weak barbed equivalence coincides with weak bisimulation equivalence.

A psi-calculus is an extension of the pi-calculus with nominal data types for data structures and for logical assertions representing facts about data. These can be transmitted between processes and their names can be statically scoped using the standard pi-calculus mechanism to allow for scope migrations. Other proposed extensions of the pi-calculus can be formulated as psi-calculi; examples include the applied pi-calculus, the spi-calculus, the fusion calculus, the concurrent constraint pi-calculus, and calculi with polyadic communication channels or pattern matching. Psi-calculi can be even more general, for example by allowing structured channels, higher-order formalisms such as the lambda calculus for data structures, and a predicate logic for assertions. Our labelled operational semantics and definition of bisimulation is straightforward, without a structural congruence. We establish minimal requirements on the nominal data and logic in order to prove general algebraic properties of psi-calculi. The proofs have been checked in the interactive proof checker Isabelle. We are the first to formulate a truly compositional labelled operational semantics for calculi of this calibre. Expressiveness and therefore modelling convenience significantly exceeds that of other formalisms, while the purity of the semantics is on par with the original pi-calculus.