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

62 CHAPTER 3. CONTEXT-FREE GRAMMARS AND LANGUAGES
we use the system X = KY + K to express every nonterminal Ai in terms of expressions
containing strings αi,j involving a terminal as the leftmost symbol (αi,j ∈ ΣV ∗ ), and we
replace all leftmost occurrences of nonterminals in H (occurrences Ai in strings of the form
Aiβ, whereβ ∈ V ∗ ) using the above expressions. In this fashion, we obtain a matrix L, and
it is immediately shown that the system
X = KY + K,
Y = LY + L,
generates the same tuple of languages. Furthermore, this last system corresponds to a weak
Greibach Normal Form.
It we start with a grammar in Chomsky Normal Form (with no production S → ɛ) such
that every nonterminal belongs to T (G), we actually get a Greibach Normal Form (the entries
in K are terminals, and the entries in H are nonterminals). Thus, we have justified lemma
3.6.1. The method is also quite economical, since it introduces only m2 new nonterminals.
However, the resulting grammar may contain some useless nonterminals.
3.10 Tree Domains and Gorn Trees
Derivation trees play a very important role in parsing theory and in the proof of a strong
version of the pumping lemma for the context-free languages known as Ogden’s lemma.
Thus, it is important to define derivation trees rigorously. We do so using Gorn trees.
Let N+ = {1, 2, 3,...}.
Definition 3.10.1 A tree domain D is a nonempty subset of strings in N ∗ + satisfying the
conditions:
(1) For all u, v ∈ N ∗ +,ifuv ∈ D, thenu ∈ D.
(2) For all u ∈ N ∗ +, for every i ∈ N+, ifui ∈ D then uj ∈ D for every j, 1≤ j ≤ i.
The tree domain
is represented as follows:
D = {ɛ, 1, 2, 11, 21, 22, 221, 222, 2211}
ɛ
↙ ↘
1 2
↙ ↙ ↘
11 21 22
↙ ↘
221 222
↓
2211