In this chapter we apply the theory developed for aPi in Chapter 4 to aDpi. For technical reasons it is best to work relative to the extended typing system developed in Section 5.5, although to avoid unnecessary clutter we will abbreviate ⊢rc to simply ⊢.

The motivation for this theory, the relativisation of behavioural equivalences to the partial knowledge of observers as explained in Chapter 4, applies equally well, if not more so, in a distributed setting. Consequently it is not repeated here. Moreover the technical framework, developed in detail in Chapter 4, is easily adapted to aDpi. So we simply apply it, modifying the various concepts as necessary to the distributed setting, in what we hope is a natural manner.

In the first section we discuss distributed actions-in-context and the associated typed bisimulation equivalence, ≈bis, for aDpi systems. Most proofs of the expected properties of this equivalence are left to the reader as they can be easily constructed from the corresponding ones in Sections 4.1 and 4.2.

This is followed by a section on examples, which demonstrates that at least in principle standard bisimulation-based proof methodologies can be adapted to our setting. Necessarily there are more syntactic details to contend with; but in time appropriate software tools, based for example on theorem provers, could be developed to partially alleviate the tedium.

In the final section we revisit the subject first broached in the untyped setting of aDpi, in Section 2.5, namely the justification of our typed bisimulation equivalence for aDpi.