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

3.8. CONTEXT-FREE LANGUAGES AS LEAST FIXED-POINTS 55
and
It is easy to verify that
Φ n+1
G,A (∅, ∅) =Φn G,B(∅, ∅)Φ n G,B(∅, ∅) ∪{ab},
Φ n+1
G,B (∅, ∅) =aΦn G,B(∅, ∅)b ∪{ab}.
Φ 1 G,A(∅, ∅) ={ab},
Φ 1 G,B(∅, ∅) ={ab},
Φ 2 G,A(∅, ∅) ={ab, abab},
Φ 2 G,B(∅, ∅) ={ab, aabb},
Φ 3 G,A(∅, ∅) ={ab, abab, abaabb, aabbab, aabbaabb},
Φ 3 G,B(∅, ∅) ={ab, aabb, aaabbb}.
By induction, we can easily prove that the two components of the least fixed-point are
the languages
LA = {a m b m a n b n | m, n ≥ 1}∪{ab} and LB = {a n b n | n ≥ 1}.
Letting GA =({A, B, a, b}, {a, b},P,A)andGB =({A, B, a, b}, {a, b},P,B), it is indeed
true that LA = L(GA) andLB = L(GB) .
We have the following theorem due to Ginsburg and Rice:
Theorem 3.8.3 Given a context-free grammar G =(V,Σ,P,S) with m nonterminals A1,
..., Am, the least fixed-point of the map ΦG is the m-tuple of languages
where GAi =(V,Σ,P,Ai).
Proof . Writing G as
(L(GA1),...,L(GAm)),
A1 → α1,1 + ···+ α1,n1,
···→···
Ai → αi,1 + ···+ αi,ni ,
···→···
Am → αm,1 + ···+ αm,nn,
let M =max{|αi,j|} be the maximum length of right-hand sides of rules in P .Let
Φ n G(∅,...,∅) =(Φ n G,1(∅,...,∅),...,Φ n G,m(∅,...,∅)).