On the bisimulation congruence in χ-calculus

In this paper, we study weak bisimulation congruences for the χ-calculus, a symmetric variant of the π-calculus. We distinguish two styles of such bisimulation definitions, i.e. “open” and “closed” bisimulation, the difference between which lies in that in open style the equivalence is closed under context in every bisimulation step whereas in closed style the equivalence is closed under context only at the very beginning. As a result, we show that both in labelled and barbed congruence, the open and closed style definitions coincide. Thus all bisimulation congruences collapse into two equivalences, that is, the well-known open congruence and open barbed congruence, which are the same in the strong case, while in the weak case their difference can be reflected by one axiom. The results of this paper close some conjectures in the literatures and shed light on the algebraic theory of a large class of mobile process calculi