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

4.2. TARSKI’S FIXED POINT THEOREM 89
4.2 Tarski’s fixed point theorem
Now that we have some familiarity with posets and complete lattices, we are in a
position to state and prove Tarski’s fixed point theorem—Theorem 4.1. As you will
see in due course, this apparently very abstract result plays a key role in computer
science because it is a general tool that allows us to make sense of recursively
defined objects. If you are interested in the uses of the theorem rather than in the
reason why it holds, you can safely skip the proof of Theorem 4.1 on first reading.
None of the future applications of that result in this textbook depend on its proof,
and you should feel free to use it as a ‘black box’.
In the statement of Tarski’s fixed point theorem, and in the applications to
follow, the collection of monotonic functions will play an important role. We now
proceed to define this type of function for the sake of completeness.
Definition 4.4 [Monotonic functions and fixed points] Let (D, ⊑) be a poset. A
function f : D → D is monotonic iff d ⊑ d ′ implies that f(d) ⊑ f(d ′ ), for all
d, d ′ ∈ D.
An element d ∈ D is called a fixed point of f iff d = f(d).
For example, the function f : 2 N → 2 N defined, for each X ⊆ N, by
f(X) = X ∪ {1, 2}
is monotonic. The set {1, 2} is a fixed point of f because
f({1, 2}) = {1, 2} ∪ {1, 2} = {1, 2} .
Exercise 4.6 Can you give another example of a fixed point of f? Can you characterize
all of the fixed points of that function? Argue for your answers.
Exercise 4.7 Consider the function that is like f above, but maps the set {2} to
{1, 2, 3}. Is such a function monotonic?
As another example, consider the poset
⊤
0
1
⊥