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

48 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
Lemma 3.4.2 A language L is regular if and only if it is generated by some right-linear
grammar.
Proof . Let L = L(D) for some DFA D =(Q, Σ,δ,q0,F). We construct a right-linear
grammar G as follows. Let V = Q ∪ Σ, S = q0, andletP be defined as follows:
P = {p → aq | q = δ(p, a), p,q∈ Q, a ∈ Σ}∪{p → ɛ | p ∈ F }.
It is easily shown by induction on the length of w that
and thus, L(D) =L(G).
p ∗
=⇒ wq iff q = δ ∗ (p, w),
Conversely, let G =(V,Σ,P,S) be a right-linear grammar. First, let G =(V ′ , Σ,P ′ ,S)be
the right-linear grammar obtained from G by adding the new nonterminal E to N, replacing
every rule in P of the form A → a where a ∈ ΣbytheruleA → aE, and adding the
rule E → ɛ. It is immediately verified that L(G ′ )=L(G). Next, we construct the NFA
M =(Q, Σ,δ,q0,F) as follows: Q = N ′ = N ∪{E}, q0 = S, F = {A ∈ N ′ | A → ɛ}, and
δ(A, a) ={B ∈ N ′ | A → aB ∈ P ′ },
for all A ∈ N and all a ∈ Σ. It is easily shown by induction on the length of w that
and thus, L(M) =L(G ′ )=L(G).
A ∗
=⇒ wB iff B ∈ δ ∗ (A, w),
A similar lemma holds for left-linear grammars. It is also easily shown that the regular
languages are exactly the languages generated by context-free grammars whose rules are of
the form
where A, B ∈ N, andu ∈ Σ ∗ .
A → Bu,
A → u,
3.5 Useless Productions in Context-Free Grammars
Given a context-free grammar G =(V,Σ,P,S), it may contain rules that are useless for
a number of reasons. For example, consider the grammar G3 =({E, A, a, b}, {a, b},P,E),
where P is the set of rules
E −→ aEb,
E −→ ab,
E −→ A,
A −→ bAa.