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

56 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
Then, for any w ∈ Σ ∗ ,observethat
w ∈ Φ 1 G,i(∅,...,∅)
iffthereissomeruleAi → αi,j with w = αi,j, andthat
w ∈ Φ n G,i(∅,...,∅)
for some n ≥ 2 iff there is some rule Ai → αi,j with αi,j of the form
αi,j = u1Aj1u2 ···ukAjk uk+1,
where u1,...,uk+1 ∈ Σ ∗ , k ≥ 1, and some w1,...,wk ∈ Σ ∗ such that
and
We prove the following two claims.
Claim 1: For every w ∈ Σ∗ ,ifAi
Claim 2: For every w ∈ Σ ∗ ,ifw ∈ Φ n G,i
p ≤ (M +1) n−1 .
wh ∈ Φ n−1
G,jh (∅,...,∅),
w = u1w1u2 ···ukwkuk+1.
n
=⇒ w, thenw∈Φ p
G,i (∅,...,∅), for some p ≥ 1.
(∅,...,∅), with n ≥ 1, then Ai
ProofofClaim1. We proceed by induction on n. If Ai
rule A → αi,j, and by the remark just before the claim, w ∈ Φ 1 G,i (∅,...,∅).
If Ai
n+1
=⇒ w with n ≥ 1, then
for some rule Ai → αi,j. If
Ai
where u1,...,uk+1 ∈ Σ ∗ , k ≥ 1, then Ajh
n
=⇒ αi,j =⇒ w
αi,j = u1Aj1u2 ···ukAjk uk+1,
nh
=⇒ wh, wherenh≤n, and
w = u1w1u2 ···ukwkuk+1
for some w1,...,wk ∈ Σ ∗ . By the induction hypothesis,
wh ∈ Φ ph
G,jh (∅,...,∅),
p
=⇒ w for some
1
=⇒ w, thenw = αi,j for some
for some ph ≥ 1, for every h, 1≤ h ≤ k. Letting p =max{p1,...,pk}, since each sequence
(∅,...,∅) for every h, 1≤ h ≤ k, andby
(Φ q
G,i (∅,...,∅)) is an ω-chain, we have wh ∈ Φ p
G,jh
the remark just before the claim, w ∈ Φ p+1
G,i (∅,...,∅).
ProofofClaim2. We proceed by induction on n. If w ∈ Φ1 G,i (∅,...,∅), by the remark
1
just before the claim, then w = αi,j for some rule A → αi,j, andAi=⇒
w.