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

64 CHAPTER 3. BEHAVIOURAL EQUIVALENCES
erased from the behaviour of processes because, in light of their pre-emptive power
in the presence of nondeterministic choices, they may affect what we may observe.
Note that the pre-emptive power of internal transitions is unimportant in the
standard theory of automata as there we are only concerned with the possibility
of processing our input strings correctly. Indeed, as you may recall from your
courses in the theory of automata, the so-called ε-transitions do not increase the
expressive power of nondeterministic finite automata—see, for instance, the textbook
(Sipser, 2005, Chapter 1). In a reactive environment, on the other hand, this
power of internal transitions must be taken into account in a reasonable definition of
process behaviour because it may lead to undesirable consequences, e.g., the deadlock
situation in the above example. We therefore expect that the behaviour of the
process SmUni is not equivalent to that of the process (CMb | CS) \ {coin, coffee}
since the latter may deadlock after outputting a publication, whereas the former
cannot.
In order to define a notion of bisimulation that allows us to abstract from internal
transitions in process behaviours, and to differentiate the process SmUni from
(CMb | CS) \ {coin, coffee}, we begin by introducing a new notion of transition
relation between processes.
Definition 3.3 Let P and Q be CCS processes, or, more generally, states in an
LTS. For each action α, we shall write P α ⇒ Q iff either
• α = τ and there are processes P ′ and Q ′ such that
• or α = τ and P ( τ →) ∗ Q,
P ( τ →) ∗ P ′ α → Q ′ ( τ →) ∗ Q
where we write ( τ →) ∗ for the reflexive and transitive closure of the relation τ →.
Thus P α ⇒ Q holds if P can reach Q by performing an α-labelled transition, possibly
preceded and followed by sequences of τ-labelled transitions. For example,
a.τ.0 a ⇒ 0 and a.τ.0 a ⇒ τ.0 both hold, as well as a.τ.0 τ ⇒ a.τ.0. In fact, we have
P τ ⇒ P for each process P .
In the LTS depicted in Table 3.1, apart from the obvious one step pub-labelled
transition, we have that
Start
Start
Start
pub
⇒ Good ,
pub
⇒ Bad , and
pub
⇒ Start .