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Theory of Computation TOC | Sample Assignment | www.expertsmind.com

Answer 2a. DFA M=(Q,S,d, q0,F).for any two states q,q’ЄQ,

q≈q’ w єΣ*: δ*(q, w) ó δ*(q’, w) where δ*(q,e)=q and δ*(q,ax)= δ*( δ(q,a),x) A relation is ≈ is equivalence if (i) It is reflexive i.e. x≡x (ii) It is symmetric ie x≡y => y≡x (iii) It is transitive ie x≡yΛy≡z =>x≡z wєΣ δ(q, w) ≡ δ(q’, w) ó aєΣ xєΣ* : δ*(δ(q, a),x)єF≡ δ*(δ(q’, a),x)єF ó aєΣ xєΣ* : δ*(q,ax)єF ≡ δ*(q’,ax)єF ó w’єΣ* : δ*(q,w’)єF ≡ δ*(q’,w’)єF óq≡q’


2b. state diagram for M=(Q,S,d, q0,F) 0,1

0

0

q1

q0

0

1

1

0

q2 1

q0

q1

q2

q3

q4

0

-

F

-

F

F

1

-

-

-

-

F

00

F

F

F

F

F

01

-

F

-

F

F

10

-

-

-

-

F

11

-

-

-

-

F

000 F

F

F

F

F

001 F

F

F

F

F

010 -

F

-

F

F

011 -

F

-

F

F

100 F

F

F

F

F

101 -

-

-

-

F

q4 q3

1

[q0] = {q0, q2} [q1] = {q1, q3} [q4] = {q4}


state diagram for M’

0,1

0

0

q4

qq00

0

1

1

0

q2 1 1

4(a) set of all strings that have abba as substring. b

a a

a b

b

a,b

a

b

4(b) set of all strings that do not have abba as substring. a

a

b

a,b b

a b b

b

a


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