CSCI 2670
Fall 2005
HW 4
September 20, 2005
Use the pumping lemma to prove A={ww
R
w
∈
{a,b}
*
} is not regular.
Proof:
Assume A is regular.
Then A has an associated pumping length p such that any
string s in A with s
≥
p can be written as s = xyz such that
(1) xy ≤ p
(2) y > 0
(3) xy
i
z
∈
for every i = 0, 1, 2, …
Consider the string s = a
p
bba
p
.
Since s = ww
R
, where w = a
p
b, s is in A.
Therefore the
pumping lemma holds and s = xyz for some x, y, and z with the above properties.
Since
xy is a prefix of s with at most p symbols, it must be the case that xy = a
k
for some k ≤ p.
Also, since y is a substring of xy, y must be a
j
for some j = 1, 2, …, k (j cannot be 0 by
property 2 above).
Now consider the string xz.
This string is the result of “pumping” y 0
times.
xz = a
pj
bba
p
.
Since j > 0, xz is not in A.
Thus, the pumping lemma fails, which
contradicts our assumption that A is a regular language.
1.54 a. F cannot be regular because if a string in F starts with exactly 1 a, it must be
followed by b
n
c
n
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 Spring '11
 sm
 Formal language, Formal languages, Regular expression, Regular language, Pumping lemma for regular languages

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