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

44 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
We claim that G ′ =(V1, Σ,P2 ∪ P3,S1) satisfies the conditions of the lemma. For example,
S1 does not appear on the right-hand side of any production, since the productions in P3
have right-hand sides from P1, andS1does not appear on the right-hand side in P1. It is
also easily shown that L(G ′ )=L(G1) =L(G).
Let us apply the method of lemma 3.3.2 to the grammar
where P is the set of rules
G3 =({E,T,F,+, ∗, (, ),a}, {+, ∗, (, ),a},P,E),
E −→ E + T,
E −→ T,
T −→ T ∗ F,
T −→ F,
F −→ (E),
F −→ a.
We get IE = {T,F}, IT = {F }, andIF = ∅. The new grammar G ′ 3 has the set of rules
E −→ E + T,
E −→ T ∗ F,
E −→ (E),
E −→ a,
T −→ T ∗ F,
T −→ (E),
T −→ a,
F −→ (E),
F −→ a.
At this stage, the grammar obtained in lemma 3.3.2 no longer has ɛ-rules (except perhaps
S ′ → ɛ iff ɛ ∈ L(G)) or chain rules. However, it may contain rules A → α with |α| ≥3, or
with |α| ≥2andwhereα contains terminals(s). To obtain the Chomsky Normal Form. we
need to eliminate such rules. This is not difficult, but notationally a bit messy.
Lemma 3.3.3 Given a context-free grammar G =(V,Σ,P,S), one can construct a contextfree
grammar G ′ =(V ′ , Σ,P ′ ,S ′ ) such that L(G ′ )=L(G) and G ′ is in Chomsky Normal
Form, that is, a grammar whose productions are of the form
A → BC,
A → a, or
S ′ → ɛ,