Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

20 CHAPTER 2. THE LANGUAGE CCS
Since τ actions are supposed to be unobservable, the following process seems to
be an appropriate high level specification of the behaviour exhibited by process
SmUni:
Spec def
= pub.Spec .
Indeed, we expect that SmUni and Spec describe the same observable behaviour,
albeit at different levels of abstraction. We shall see in the remainder of this book
that one of the big questions in process theory is to come up with notions of ‘behavioural
equivalence’ between processes that will allow us to establish formally
that, for instance, SmUni and Spec do offer the same behaviour. But this is getting
ahead of our story.
2.2 CCS, formally
Having introduced CCS by example, we now proceed to present formal definitions
for its syntax and semantics.
2.2.1 The model of labelled transition systems
We have already indicated in our examples how the operational semantics for CCS
can be given in terms of automata—which we have called labelled transition systems,
as customary in concurrency theory. These we now proceed to define, for the
sake of clarity. We first introduce the ingredients in the model of labelled transition
systems informally, and then provide its formal definition.
In the model of labelled transition systems, processes are represented by vertices
of certain edge-labelled directed graphs (the labelled transition systems themselves)
and a change of process state caused by performing an action is understood
as moving along an edge, labelled by the action name, that goes out of that state.
A labelled transition system consists therefore of a set of states (also referred to
as processes or configurations), a set of labels (or actions), and a transition relation
→ describing changes in process states: if a process p can perform an action a and
become a process p ′ , we write p a → p ′ . Sometimes a state is singled out as the start
state in the labelled transition system under consideration. In that case, we say that
the labelled transition system is rooted.
Example 2.1 Let us start with a variation on the classic example of a tea/coffee
vending machine. The very simplified behaviour of the process which determines
the interaction of the machine with a customer can be described as follows. From
the initial state—say, p—representing the situation ‘waiting for a request’, two possible
actions are enabled. Either the tea button or the coffee button can be pressed