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

38 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
Lemma 3.2.4 Let G = (V,Σ,P,S) be a context-free grammar. For every w ∈ Σ ∗ , for
every derivation S +
=⇒ w, there is a leftmost derivation S +
=⇒ w, and there is a rightmost
lm
derivation S +
=⇒ w.
rm
Proof . Of course, we have to somehow use induction on derivations, but this is a little
tricky, and it is necessary to prove a stronger fact. We treat leftmost derivations, rightmost
derivations being handled in a similar way.
=⇒ w, then there is a
leftmost derivation α n
=⇒ w.
lm
The claim is proved by induction on n.
For n =1,thereexistsomeλ, ρ ∈ V ∗ and some production A → γ, such that α = λAρ
and w = λγρ. Sincewis a terminal string, λ, ρ, andγ, are terminal strings. Thus, A is the
only nonterminal in α, and the derivation step α 1
=⇒ w is a leftmost step (and a rightmost
step!).
If n>1, then the derivation α n
=⇒ w is of the form
Claim: For every w ∈ Σ ∗ , for every α ∈ V + , for every n ≥ 1, if α n
α =⇒ α1
n−1
=⇒ w.
There are two subcases.
Case 1. If the derivation step α =⇒ α1 is a leftmost step α =⇒ α1, by the induction
lm
hypothesis, there is a leftmost derivation α1
α =⇒
lm α1
n−1
=⇒
lm w, and we get the leftmost derivation
n−1
=⇒
lm w.
Case 2. The derivation step α =⇒ α1 is a not a leftmost step. In this case, there must
be some u ∈ Σ ∗ , µ, ρ ∈ V ∗ , some nonterminals A and B, and some production B → δ, such
that
α = uAµBρ and α1 = uAµδρ,
where A is the leftmost nonterminal in α. Since we have a derivation α1
n − 1, by the induction hypothesis, there is a leftmost derivation
α1
n−1
=⇒
lm w.
n−1
=⇒ w of length
Since α1 = uAµδρ where A is the leftmost terminal in α1, the first step in the leftmost
n−1
derivation α1 =⇒ w is of the form
lm
uAµδρ =⇒
lm uγµδρ,