Lecture Notes in Computer Science 3472

182 Machiel van der Bijl and Fabien Peureux
Example. Another similar difference is illustrated in Figure 7.4. We have the IOA
p1 and p2, both can perform an arbitrary number of a input actions followed
by one y output action, followed again by an arbitrary amount of a actions. p1
and p2 are equivalent according to the quiescent preorder (both p1 ⊑Q p2 and
p2 ⊑Q p1), because they have the same external traces and their quiescent traces
contain at least a y action. However, p1 and p2 are not equivalent according to
the fair preorder, when considering the same partitioning as before: {y,τ}. This
is because a ω isafairtraceofp1 but not of p2. This might not be easy to see
at first glance, but remember that the partition of locally controlled actions is
{y,τ}. Inp1 we can do the fair execution (q0·a·q1·τq0) ω (and thus the fair trace
a ω ).
q1
q0
!x !x
τ
q2
q3
!y
Fig. 7.5. Quiescent versus fair preorder example 3
Divergence (i.e., the possibility for a system of to do an infinite number of
internal transitions) shows another difference between the quiescent and fair
preorder. Because of divergence, a fair trace is not necessarily a quiescent trace,
as is illustrated in the next example.
Example. In Figure 7.5, we have two IOA p1 and p2 with output actions x and
y. According to the quiescent preorder both automata are equivalent. But they
are not equivalent according to the fair preorder, since x isafairtraceofp1, but
not of p2.
As we can see from these examples, the fair preorder is a stronger relation
than the quiescent preorder. Because of the definition of the quiescent preorder,
this is not very surprising. As a side step, Segala shows a way to make the
quiescent preorder and the fair preorder equivalent by making some restrictions
on the IOA’s that we allow [Seg97]. Basically, the property we are looking for is
that we can approximate an infinite fair trace by a finite trace and can extend a
finite fair trace to an infinite fair trace. This is expressed by the properties fair
continuity and fair approximability. AnIOAp is fair continuous if the limit of
any chain of fair traces of p is also a fair trace. Fair continuity is nicely illustrated
in Figure 7.3. The sequence a n is a fair trace of p2, but if we take n to infinity
this is not the case. In other words the trace a n is not fair continuous. An IOA
p is fair approximable if each infinite trace of p is the limit of a chain of fair
traces of p. This is illustrated in Figure 7.4. The trace a ω is a fair trace of p1
but it is not fair approximable because the finite trace a n is not fair.
t0
t1
t2
!x
!y