1. COSC 3340: Introduction to Theory of Computation
University of Houston
Dr. Verma
Lecture 8
Lecture 8
UofH - COSC Dr. Verma

2. Pumping lemma applications.
Proving L = {anbn | n  0 } is not regular.
Proof:
Assume L is regular. Certainly L is infinite and therefore the pumping lemma applies to L.
Let p be the constant for L (of the pumping lemma).
Lecture 8
UofH - COSC Dr. Verma

3. Pumping lemma applications (contd.)
To show there exist a string w  L of length at least p such that Q where Q is the rest of the statement of pumping lemma.
Let w = apbp such that |w|  p
write
apbp = xyz
But according to pumping lemma,
Lecture 8
UofH - COSC Dr. Verma

4. Pumping lemma applications (contd.)
PL statement (i)  |xy|  p
Therefore,
a…aa…ab…b p p x y z
y = am
m > 0
xyz = apbp
Lecture 8
UofH - COSC Dr. Verma

5. Pumping lemma applications (contd.)
PL statement (ii)  xyiz  L i = 0,1,2,3,…
Therefore,
xy2z  L
xy2z = xyyz = ak+mbk  L
But,
L = {anbn | n  0 }
which means ap+mbp  L since m > 0
CONTRADICTION !!
Lecture 8
UofH - COSC Dr. Verma

6. Pumping lemma applications (contd.)
Therefore our assumption that
L = {anbn | n  0 }
is a regular language cannot be true.
Lecture 8
UofH - COSC Dr. Verma

7. Using Pumping Lemma -- Very Important points
Above example is the typical application of pumping lemma, to show that a language is not regular.
You must choose string w so that w in L and |w| is at least the pumping length.
Example: choosing w = aaabbb is wrong since we do not know the exact value of p.
You must consider all possibilities for x, y and z such that w = xyz and |xy|  p.
The pumping lemma CANNOT be used to show that a language is regular, since it assumes that L is regular.
Lecture 8
UofH - COSC Dr. Verma