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

6.7. MIXING LARGEST AND LEAST FIXED POINTS 151
In light of the above discussion, a specification of the property mentioned at the
beginning of this section using HML with recursive definitions can be given using
the following system of equations:
P os(LivelockNow) min
= LivelockNow ∨ 〈Act〉P os(LivelockNow)
LivelockNow max
= 〈τ〉LivelockNow .
This looks natural and innocuous. However, first appearances can be deceiving!
Indeed, the equational systems we have considered so far have only allowed us to
express formulae purely in terms of largest or least solutions to systems of recursion
equations. (See Section 6.5.) For instance, in defining the characteristic formulae
for bisimulation equivalence, we only used systems of equations in which
the largest solution was sought for all of the equations in the system.
Our next question is whether we can extend our framework in such a way that
it can treat systems of equations with mixed solutions like the one describing the
formula P os(LivelockNow) above. How can we, for instance, compute the set of
processes in the labelled transition system
s
a
p
τ
τ
τ q
r
that satisfy the formula P os(LivelockNow)? In this case, the answer is not overly
difficult. In fact, you might have already noted that we can compute the set of
processes satisfying the formula P os(LivelockNow) once we have in our hands
the collection of processes satisfying the formula LivelockNow. As you saw in
Exercise 6.16, the only state in the above labelled transition system satisfying the
formula LivelockNow is p. Therefore, we may obtain the collection of states satisfying
the formula P os(LivelockNow) as the least solution of the set equation
S = {p} ∪ 〈·Act·〉S , (6.16)
where S ranges over subsets of {s, p, q, r}. We can calculate the least solution of
this equation using the iterative methods we introduced in Section 6.2.
Since we are looking for the least solution of the above equation, we begin by
obtaining our first approximation S (1) to the solution by computing the value of the
expression on the right-hand side of the equation when S = ∅, which is the least
element in the complete lattice consisting of the subsets of {s, p, q, r} ordered by
inclusion. We have that
S (1) = {p} ∪ 〈·Act·〉∅ = {p} .