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

144 CHAPTER 6. HML WITH RECURSION
Furthermore, if p a → p ′ then p ′ ∈ Der(a, p). Therefore p has the property
[a]
Xp ′
,
p ′ .p a → p ′
for each action a. The above property states that, by performing action a, process p
(and any other process that is bisimilar to it) must become a process satisfying the
characteristic property of a state in Der(a, p). (Note that if p a , then Der(a, p)
is empty. In that case, since an empty disjunction is just the formula ff, the above
formula becomes simply [a]ff—which is what we would expect.)
Since action a is arbitrary, we have that
p |=
[a] Xp ′
.
a
p ′ .p a → p ′
If we summarize the above requirements, we have that
p |=
〈a〉Xp ′
∧ [a]
a,p ′ .p a → p ′
a
p ′ .p a → p ′
As this property is apparently a complete description of the behaviour of process
p, this is our candidate for its characteristic property. Xp is therefore defined as
a solution to the equational system obtained by giving the following equation for
each q ∈ Proc:
Xq =
〈a〉Xq ′ ∧ [a] Xq ′
. (6.14)
a,q ′ .q a → q ′
a
q ′ .q a → q ′
The solution can either be the least or the largest one (or, in fact, any other fixed
point for what we know at this stage).
The following example shows that the least solution to (6.14) in general does
not yield the characteristic property for a process.
Example 6.10 Let p be the process given in Figure 6.5. In this case, assuming for
the sake of simplicity that a is the only action, the equational system obtained by
using (6.14) will have the form
Xp = 〈a〉Xp ∧ [a]Xp .
Since 〈·a·〉∅ = ∅, you should be able to convince yourselves that [Xp ] = ∅ is
the least solution to this equation. This corresponds to taking Xp = ff as the
characteristic formula for p. However, p does not have the property ff, which
therefore cannot be the characteristic property for p.
Xp ′
.