hw3-sol - ECS 120 Theory of Computation Homework 3 Due Oct...

ECS 120: Theory of Computation
Homework 3
Due: Oct. 19, beginning of class or 1pm in 2131 Kemper
Problem 1. 1.19 in Sipser.
0,1
0
1
0
0
1
1
q0 q1
q2
q3
1
0
1
0
1
0
0
1
q0 q1
q2
q3
q4
H
0,1
q0
q1
Problem 2. 1.48 in Sipser.
Soln: A DFA cannot “count” the number of occurences of 01’a and 10’s, so we need a
trick. Notice that any string s ∈ { 0 , 1 }
∗
is one and only one of the following, where
n ≥ 1 and i
j
≥ 1:
(a) ε
(b) 0 i
1
1 i
2
0 i
3
. . . 1 i
n - 1
0 i
n
(begins and ends with a 0)
(c) 1 i
1
0 i
2
1 i
3
. . . 0 i
n - 1
1 i
n
(begins and ends with a 1)
(d) 0 i
1
1 i
2
0 i
3
. . . 0 i
n - 1
1 i
n
(begins with a 0 and ends with a 1)
(e) 1 i
1
0 i
2
1 i
3
. . . 1 i
n - 1
0 i
n
(begins with a 1 and ends with a 0)
Now, strings of the form (a), (b) and (c) are in D , and strings of the form (d) and (e)
are not in D (just count the number of occurences of 01 and 10 in each). Therefore,
a string x ∈ { 0 , 1 }
∗
is in D if and only if x starts and ends with the same character
or x = ε . So D is described by the regular expression ε ∪ 0 ∪ 1 ∪ 1(0 ∪ 1) ∗ 1 ∪ 0(0 ∪ 1) ∗ 0,
hence it is regular.