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

66 CHAPTER 3. BEHAVIOURAL EQUIVALENCES
We can readily argue that a.0 ≈ a.τ.0 by establishing a weak bisimulation that
relates these two processes. (Do so by renaming the states in the labelled transition
system and in the bisimulation above!) On the other hand, there is no weak bisimulation
that relates the process SmUni and the process Start in Table 3.1. In fact,
the process SmUni is observationally equivalent to the process
but the process Start is not.
Spec def
= pub.Spec ,
Exercise 3.20 Prove the claims that we have just made.
Exercise 3.21 Prove that the behavioural equivalences claimed in Exercise 2.11
hold with respect to observational equivalence (weak bisimilarity).
The definition of weak bisimilarity is so natural, at least to our mind, that it is easy
to miss some of its crucial consequences. To highlight some of these, consider the
process
A?
B?
def
= a.0 + τ.B?
def
= b.0 + τ.A? .
Intuitively, this process describes a ‘polling loop’ that may be seen as an implementation
of a process that is willing to receive on port a and port b, and then
terminate. Indeed, it is not hard to show that
A? ≈ B? ≈ a.0 + b.0 .
(Prove this!) This seems to be non-controversial until we note that A? and B? have
a livelock (that is, a possibility of divergence) due to the τ-loop
A? τ → B? τ → A? ,
but a.0 + b.0 does not. The above equivalences capture one of the main features of
observational equivalence, namely the fact that it supports what is called ‘fair abstraction
from divergence’. (See (Baeten, Bergstra and Klop, 1987), where Baeten,
Bergstra and Klop show that a proof rule embodying this idea, namely Koomen’s
fair abstraction rule, is valid with respect to observational equivalence.) This means
that observational equivalence assumes that if a process can escape from a loop
consisting of internal transitions, then it will eventually do so. This property of observational
equivalence, that is by no means obvious from its definition, is crucial
in using it as a correctness criterion in the verification of communication protocols,