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

126 CHAPTER 6. HML WITH RECURSION
transition system. We therefore have that our first approximation to the largest
fixed point is the set
OFX ({s, s1, s2, t, t1}) = (〈·b·〉{s, s1, s2, t, t1}) ∩ [·b·]{s, s1, s2, t, t1}
= {s1, s2, t1} ∩ {s, s1, s2, t, t1}
= {s1, s2, t1} .
Note that our candidate solution to the equation has shrunk in size, since an application
of OFX to the set of all processes has removed the states s and t from our
candidate solution. Intuitively, this is because, by applying OFX to the set of all
states, we have found a reason why s and t do not afford the property specified by
X max
= 〈b〉tt ∧ [b]X ,
namely that s and t do not have a b-labelled outgoing transition, and therefore that
neither of them is in the set 〈·b·〉{s, s1, s2, t, t1}.
Following our iterative algorithm for the computation of the largest fixed point,
we now apply the function OFX to the new candidate largest solution, namely
{s1, s2, t1}. We now have that
OFX ({s1, s2, t1}) = (〈·b·〉{s, s1, s2, t, t1}) ∩ [·b·]{s1, s2, t1}
= {s1, s2, t1} ∩ {s, s1, s2, t, t1}
= {s1, s2, t1} .
(You should convince yourselves that the above calculations are correct!) We have
now found that {s1, s2, t1} is a fixed point of the function OFX . By Theorem 6.1,
this is the largest fixed point and therefore states s1, s2 and t1 are the only states in
our labelled transition system that satisfy the property
X max
= 〈b〉tt ∧ [b]X .
This is in complete agreement with our intuition because those are the only states
that can perform a b-action in all states that they can reach by performing sequences
of b-labelled transitions.
Exercise 6.6 Consider the property
Y min
= 〈b〉tt ∨ 〈{a, b}〉Y .
Use Theorem 6.1 to compute the set of processes in the labelled transition system
above that satisfy this property.