COT 4210
Section E
Spring 2001
Pumping Lemma for Regular Languages
We have completed our study of closure properties of Regular Languages and various formal
systems for defining or describing them.
The question we wish to consider next is, how can
non-Regular languages be identified?
The first such tool is the
pumping lemma for Regular
languages
.
It defines a key property possessed by all Regular languages, specifically all infinite
Regular languages (for it is only the infinite languages that can be non-Regular).
As a logical
assertion, it states that,
“L is regular ] L has property P”.
Negating this implication we have,
“L does not have property P
L is not regular.”
It is the contra-positive form that is useful in
proving languages to be non-Regular.
That is, typical proofs using the Pumping Lemma assert
that a given language, L, is Regular, and then proceed to obtain a contradiction by showing L
does not have property, P.
We’ll explain what property “P” is after the statement of our next
theorem.
Theorem 9 (Pumping Lemma for Regular Languages(PLR)).
Let L be a Regular language
over
.
There exits
a positive integer
α,
depending only on L, such that
for every
x
L with
|x|
α
, then
there exists a decomposition
of x in the form
uvw
, where |uv|
α
and |v|
1,
such that
for every
value of k
0, uv
k
w T L.
The “property P” of Regular languages described in the PLR is: there is a fixed number
α
associated with each regular language, L, such that all strings, x, belonging to L and having
length at least
α
, define an infinite subset of L, denoted L
x
, defined by
L
x
= { uv
k
w | k
0, where x = uvw for some strings u,v,w satisfying |uv|
α
and |v|
1 }
The figure below illustrates this concept.
Feb. 22, 2001
Page 48