Introduction to Computational Linguistics

23. Some Metatheorems 96
we take ⃗x = a k b b c k . We shall have to find appropriate ⃗v and ⃗y. First, it is easy
to see that ⃗v⃗y must contain the same number of a, b and c (can you see why?).
The product is nonempty, so at least one a, b and c must occur. We show that
⃗u⃗v 2 ⃗w⃗y 2 ⃗z L. Suppose, ⃗v contains a and b. Then ⃗v 2 contains a b before an a.
Contradiction. Likewise, ⃗v cannot contain a and c. So, ⃗v contains only a. Now
⃗y contains b and c. But then ⃗y 2 contains a c before a b. Contradiction. Hence,
⃗u⃗v 2 ⃗w⃗y 2 ⃗z L.
Now, the Pumping Lemma is not an exact characterization. Here is a language
that satisfies the test but is not context free:
(249) C := {⃗x⃗x : ⃗x ∈ A ∗ }
This is known as the copy language. We shall leave it as an assignment to show
that C fulfills the conclusion of the pumping lemma. We concentrate on showing
that it is not context free. The argument is a bit involved. Basically, we take a
string as in the pumping lemma and fix a decomposition so that ⃗u⃗v j ⃗w⃗y j ⃗z ∈ C for
every j. The idea is now that no matter what string y 0 y 1 · · · y p−1 we are given, if it
contains a constituent A:
(250) y 0 g 1 · · · y p−1 • y p · · · y q−1 • y j · · · y q−1
then we can insert the pair ⃗v, ⃗y like this:
(251) y 0 y 1 · · · y p−1 • x p · · · x j−1 ◦ y i · · · y j−1 • x l · · · x m−1 y p · · · y q−1
Now one can show that since there are a limited number of nonterminals we are
bound to find a string which contains such a constituent where inserting the strings
is inappropriate.
Natural languages do contain a certain amount of copying. For example,
Malay (or Indonesian) forms the plural of a noun by reduplicating it. Chinese
has been argued to employ copying in yes-no questions.