Solution to CS243 Assignment2
1. Text (Sipser, second edition) Chapter 1 (p.88) 1.29b [14%]
A
2
= {
ωωω

ω
is in {a,b}
*
}
Assume to the contrary that A
2
is regular, and let
p
be the pumping length given by pumping
lemma. Choose s = a
p
ba
p
ba
p
b, which can be divided into three pieces s =
xyz
, where 
xy

≤
p
. This
means
xy
contains only a’s. Since 
y
 > 0, let y = a
k
, k>0. However,
xy
2
z
= a
p+k
ba
p
ba
p
b, where
p+k>p, is not in A
2
. That is s cannot be pumped. This is a contradiction. Thus, A
2
is not
regular.
2. Text (Sipser, second edition) Chapter 1 (p.89) 1.31 [14%]
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View Full Document3. Text (Sipser, second edition) Chapter 1 (p.89) 1.34 [14%]
The following DFA recognizes D. Thus, D is regular.
4. Text (Sipser, second edition) Chapter 1 (p.89) 1.37 [14%]
We only need to construct a DFA to keep track of the remainder of the input seen so far (from
left to right) divided by n. If it ends up with remainder zero, accept; otherwise reject. Notice the
following relations:
If (
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 Spring '11
 sm
 Division, Remainder, Sipser, following DFA, xy contains

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