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

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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.
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