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

54 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
Then, writing the grammar G as
we define the map
such that
A1 → α1,1 + ···+ α1,n1,
···→···
Ai → αi,1 + ···+ αi,ni ,
···→···
Am → αm,1 + ···+ αm,nn,
ΦG:2 Σ∗
×···×2 Σ∗
→ 2
m
Σ∗
×···×2 Σ∗
m
ΦG(L1,...Lm) = (Φ[Λ]({α1,1,...,α1,n1}),...,Φ[Λ]({αm,1,...,αm,nm}))
for all Λ = (L1,...,Lm) ∈ 2 Σ∗
×···×2 Σ∗
.
m
One should verify that the map Φ[Λ] is well defined, but this is easy. The following
lemma is easily shown:
Lemma 3.8.2 Given a context-free grammar G =(V,Σ,P,S) with m nonterminals A1, ...,
Am, themap
ΦG:2 Σ∗
×···×2 Σ∗
→ 2
m
Σ∗
×···×2 Σ∗
m
is ω-continuous.
Now, 2 Σ∗
×···×2 Σ∗
is an ω-chain complete poset, and the map ΦG is ω-continous. Thus,
m
by lemma 3.7.4, the map ΦG has a least-fixed point. It turns out that the components of
this least fixed-point are precisely the languages generated by the grammars (V,Σ,P,Ai).
Before proving this fact, let us give an example illustrating it.
Example. Consider the grammar G =({A, B, a, b}, {a, b},P,A) defined by the rules
where
A → BB + ab,
B → aBb + ab.
The least fixed-point of ΦG is the least upper bound of the chain
(Φ n G(∅, ∅)) = ((Φ n G,A(∅, ∅), Φ n G,B(∅, ∅)),
Φ 0 G,A(∅, ∅) =Φ 0 G,B(∅, ∅) =∅,