Transcript 7

COSC 3340: Introduction to Theory of
Computation
University of Houston
Dr. Verma
Lecture 8
1
Lecture 8
UofH - COSC 3340 - Dr. Verma
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).
2
Lecture 8
UofH - COSC 3340 - Dr. Verma
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
write
such that |w|  p
apbp = xyz
But according to pumping lemma,
3
Lecture 8
UofH - COSC 3340 - Dr. Verma
Pumping lemma applications (contd.)
PL statement (i)  |xy|  p
p
Therefore,
p
a…aa…ab…b
x
y
y = am
z
m>0
xyz = apbp
4
Lecture 8
UofH - COSC 3340 - Dr. Verma
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 !!
5
Lecture 8
UofH - COSC 3340 - Dr. Verma
Pumping lemma applications (contd.)
Therefore our assumption that
L = {anbn | n  0 }
is a regular language cannot be true.
6
Lecture 8
UofH - COSC 3340 - Dr. Verma
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.
–


7
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 3340 - Dr. Verma