Introduction to Computational Linguistics

18. Context Free Grammars 71
which is of category B. In the last step we get that the entire string is of category
A under this derivation. Thus, each step in a derivation adds a constituent to the
analysis of the final string. The structure that we get is as follows. There is a
string ⃗α and a set Γ of occurrences of substrings of ⃗α together with a map f that
assigns to each member of Γ a nonterminal (that is, an element of N). We call the
members of Γ constituents and f (C) the category of C, C ∈ Γ.
By comparison, the derivation (??) imposes a different constituent structure
on the string. The different is that it is not (??) that is a constituent but rather
(187) bcbbca
It is however not true that the constituents identify the derivation uniquely. For
example, the following is a derivation that gives the same constituents as (??).
(188) A, BA, BBA, BBBA, BBBa, BBbca, bBbca, bcbbca
The difference between (??) and (??) are regarded inessential, however. Basically,
only the constituent structure is really important, because it may give rise to different
meanings, while different derivations that yield the same structure will give
the same meaning.
The constituent structure is displayed by means of trees. Recall the definition
of a tree. It is a pair 〈T, z. We say that x dominates
y if x > y; and that it immediately dominates y if it dominates y but there is no z
such that x dominates z and z dominates y.
Now, let us return to the constituent structure. Let C = 〈⃗γ 1 , ⃗γ 2 〉 and D =
〈⃗η 1 , ⃗η 2 〉 be occurrences of substrings. We say that C is dominated by D, in symbols
C ≺ D, if C D and (1) ⃗η 1 is a prefix of ⃗γ 1 and (2) ⃗η 2 is a suffix of ⃗γ2. (It
may happen that ⃗η 1 = ⃗γ 1 or that ⃗η 2 = ⃗γ 2 , but not both.) Visually, what the definition
amounts to is that if one underlines the substring of C and the substring of D
then the latter line includes everything that the former underlines. For example,
let C = 〈abc, dx〉 and 〈a, x〉. Then C ≺ D, as they are different and (1) and (2) are
satisfied. Visually this is what we get.
(189) abccddx
Now, let the tree be defined as follows. The set of nodes is the set Γ. The relation
< is defined by ≺.