Introduction to Computational Linguistics

18. Context Free Grammars 69
symbols is very useful. There are several basic types of sentences (assertions,
questions, commands, and so on). These represent different sets of strings. It real
grammars (GPSG, HPSG, for example), one does not start from a single symbol
but rather from different symbols, one for each class of saturated utterance. This
is useful also for other reasons. It is noted that in answers to questions almost
any constituent can function as a complete utterance. To account for that, one
would need to enter a rule of the kind above for each constituent. But there is
a sense in which these constituents are not proper sentences, they are just parts
of constituents. Rather than having to write new rules to encompass the use of
these constituents, we can just place the burden on the start set. So, in certain
circumstances what shifts is the set of start symbols, the rule set however remains
constant.
A derivation is a full specification of the way in which a string has been generated.
This may be given as follows. It is a sequence of strings, where each
subsequent string is obtained by applying a rule to some occurrence of a nonterminal
symbol. We need to specify in each step which occurrence of a nonterminal
is targeted. We can do so by underlining it. For example, here is a grammar.
(180) A → BA | AA, B → AB | BB, A → a, B → b | bc | cb
Here is a derivation:
(181) A, BA, BBA, BBBA, BBBa, BcbBa, bcbBa, bcbbca
Notice that without underlining the symbol which is being replaced some information
concerning the derivation is lost. In the second step we could have applied
the rule A → BA instead, yielding the same result. Thus, the following also is a
derivation:
(182) A, BA, BBA, BBBA, BBBa, BcbBa, bcbBa, bcbbca
(I leave it to you to figure out how one can identify the rule that has been applied.)
This type of information is essential, however. To see this, let us talk about the
second approach to CFGs, the analytic one. Inductively, on the basis of a derivation,
we assign constituents to the strings. First, however, we need to fix some
notation.
Definition 23 Let ⃗x and ⃗y be strings. An occurrence of ⃗y in ⃗x is a pair 〈⃗u,⃗v〉 of
strings such that ⃗x = ⃗u⃗y⃗v. We call ⃗u the left context of ⃗y in ⃗x and ⃗v the right
context.