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

92 CHAPTER 4. THEORY OF FIXED POINTS
For instance, the largest fixed point of the function f : 2 N → 2 N defined by
f(X) = X ∪ {1, 2} is
{X ⊆ N | X ⊆ X ∪ {1, 2}} = N .
On the other hand, the least fixed point of f is
{X ⊆ N | X ∪ {1, 2} ⊆ X} = {1, 2} .
This follows because X ∪ {1, 2} ⊆ X means that X already contains 1 and 2, and
the smallest set with this property is {1, 2}.
The following important theorem gives a characterization of the largest and
least fixed points for monotonic functions over finite complete lattices. We shall
see in due course how this result gives an algorithm for computing the fixed points
which will find application in equivalence checking and in the developments in
Chapter 6.
Definition 4.5 Let D be a set, d ∈ D, and f : D → D. For each natural number
n, we define f n (d) as follows:
f 0 (d) = d and
f n+1 (d) = f(f n (d)) .
Theorem 4.2 Let (D, ⊑) be a finite complete lattice and let f : D → D be monotonic.
Then the least fixed point for f is obtained as
zmin = f m (⊥) ,
for some natural number m. Furthermore the largest fixed point for f is obtained
as
for some natural number M.
zmax = f M (⊤) ,
Proof: We only prove the first statement as the proof for the second one is similar.
As f is monotonic we have the following non-decreasing sequence
⊥ ⊑ f(⊥) ⊑ f 2 (⊥) ⊑ . . . ⊑ f i (⊥) ⊑ f i+1 (⊥) ⊑ . . .