Specifying and Refining Internal Operations in Z

Abstract.

An important aspect in the specification of distributed systems is the role of the internal (or unobservable) operation. Such operations are not part of the interface to the environment (i.e. the user cannot invoke them), however, they are essential to our understanding and correct modelling of the system. In this paper we are interested in the use of the formal specification notation Z for the description of distributed systems. Various conventions have been employed to model internal operations when specifying such systems in Z. If internal operations are distinguished in the specification notation, then refinement needs to deal with internal operations in appropriate ways.

Using an example of a telecommunications protocol we show that standard Z refinement is inappropriate for refining a system when internal operations are specified explicitly. We present a generalisation of Z refinement, called weak refinement, which treats internal operations differently from observable operations when refining a system. We discuss the role of internal operations in a Z specification, and in particular whether an equivalent specification not containing internal operations can be found. The nature of divergence through livelock is also discussed.