Introduction to Computational Linguistics

20. Greibach Normal Form 79
Definition 27 A tree domain is a set T of strings of natural numbers such that (i)
if ⃗x if a prefix of ⃗y ∈ T then also ⃗x ∈ T, (b) if ⃗x j ∈ T and i < j then also ⃗xi ∈ T.
We define ⃗x > ⃗y if ⃗x is a proper prefix of ⃗y and ⃗x ⊏ ⃗y iff ⃗x = ⃗ui⃗v and ⃗y = ⃗u j⃗w for
some sequences ⃗u, ⃗v, ⃗w and numbers i < j.
The depth–first search traverses the tree domain in the lexicographical order, the
breadth–first search in the numerical order. Let the following tree domain be
given.
ε
❅
❅
❅
❅
0 1 2
✂ ❇
✂ ❇
00
✂
10
❇
11 20
The depth–first linearization is
(202) ε, 0, 00, 1, 10, 11, 2, 20
The breadth–first linearization, however, is
(203) ε, 0, 1, 2, 00, 10, 11, 20
Notice that with these linearizations the tree domain ω ∗ cannot be enumerated.
Namely, the depth–first linearization begins as follows.
(204) ε, 0, 00, 000, 0000, . . .
So we never reach 1. The breadth–first linearization goes like this.
(205) ε, 0, 1, 2, 3, . . .
So, we never reach 00. On the other hand, ω ∗ is countable, so we do have a
linearization, but it is more complicated than the given ones.