Chapter 3 Context-Free Grammars, Context-Free Languages, Parse ...

52 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
Remark: Note that we are not requiring that an ω-continuous function f: A1 → A2
preserve least elements, i.e., it is possible that f(⊥1) =⊥2.
We now define the crucial concept of a least fixed-point.
Definition 3.7.3 Let 〈A, ≤〉 be a partially ordered set, and let f: A → A be a function. A
fixed-point of f is an element a ∈ A such that f(a) =a. The least fixed-point of f is an
element a ∈ A such that f(a) =a, and for every b ∈ A such that f(b) =b, thena ≤ b.
The following lemma gives sufficient conditions for the existence of least fixed-points. It
is one of the key lemmas in denotational semantics.
Lemma 3.7.4 Let 〈A, ≤〉 be an ω-chain complete poset with least element ⊥. Every ωcontinuous
function f: A → A has a unique least fixed-point x0 given by
x0 = f n (⊥).
Furthermore, for any b ∈ A such that f(b) ≤ b, thenx0 ≤ b.
Proof . First, we prove that the sequence
⊥ ,f(⊥) ,f 2 (⊥), ..., f n (⊥), ...
is an ω-chain. This is shown by induction on n. Since⊥ is the least element of A, wehave
⊥≤ f(⊥). Assuming by induction that f n (⊥) ≤ f n+1 (⊥), since f is ω-continuous, it is
monotonic, and thus we get f n+1 (⊥) ≤ f n+2 (⊥), as desired.
Since A is an ω-chain complete poset, the ω-chain (f n (⊥)) has a least upper bound
Since f is ω-continuous, we have
and x0 is indeed a fixed-point of f.
x0 = f n (⊥).
f(x0) =f( f n (⊥)) = f(f n (⊥)) = f n+1 (⊥) =x0,
Clearly, if f(b) ≤ b implies that x0 ≤ b, thenf(b) =b implies that x0 ≤ b. Thus, assume
that f(b) ≤ b for some b ∈ A. We prove by induction of n that f n (⊥) ≤ b. Indeed, ⊥≤ b,
since ⊥ is the least element of A. Assuming by induction that f n (⊥) ≤ b, by monotonicity
of f, weget
f(f n (⊥)) ≤ f(b),
and since f(b) ≤ b, this yields
f n+1 (⊥) ≤ b.