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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
nonRegular 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 nonRegular).
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 contrapositive form that is useful in
proving languages to be nonRegular.
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
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View Full DocumentCOT 4210
Section E
Spring 2001
NOTE:
You may be wondering about finite languages.
Since the PLR applies to all Regular
languages, how does the result hold for finite Regular languages?
The answer is that the value of
α
for finite languages, L, is always greater than the length of the longest string in L; all
members of a finite language are therefore considered to be “short” strings.
Before proving the PLR we apply it to an example.
We had stated earlier that certain languages
were known to be nonRegular without giving any proof.
One of these is L = {a
n
b
n
 n
0 }.
Using the PLR we now supply the proof.
Corollary PLR1
:
L = { a
n
b
n

n
0 } is nonRegular.
Proof.
The proof will proceed by contradiction.
Suppose L is Regular, then the conclusion of
the PLR must hold for L.
In particular, consider x = a
α
b
α
h L (a
long
member of L), where
α
is
the parameter defined for L by the PLR.
To obtain a contradiction we must show that for each
decomposition
of the selected long string x in the form uvw, there exists a
k
[ 0, such that
uv
k
w T L.
It is really easier to think of applying the contrapositive of the PLR,
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