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ASSIGNMENT #3 Due Wednesday, September 24, 2014 1. Consider the deterministic nite automaton M = ({q1 , q2 , q3 }, {0, 1}, , q1 , {q2 }) where is...

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ASSIGNMENT #3 Due Wednesday, September 24, 2014 1. Consider the deterministic finite automaton M = ( { q 1 , q 2 , q 3 } , { 0, 1 } , δ , q 1 , { q 2 } ) where δ is defined as follows: δ ( q 1 , 0) = q 1 δ ( q 1 , 1) = q 2 δ ( q 2 , 0) = q 3 δ ( q 2 , 1) = q 2 δ ( q 3 , 0) = q 2 δ ( q 3 , 1) = q 2 Write an equivalent regular expression. 2. Prove that the following languages are not regular sets: (a) L = { a i b j c k | i = 0 j = k , i,j,k 0 } . Example strings include bccc, abbcc, aaa, etc. (b) L = { ww | w ∈ { 0 , 1 } + } . Example strings include 00, 11, 0101, 010010, etc. (c) L = { a 2 n | n 0 } . Example strings include aaaa, a 16 , a 64 , etc. (d) L = { w | w ∈ { 0 , 1 } * , w is of the form (0 i 1) n , for i = 1, 2, . .., n , n 0 } . The strings of this language are ε , 01, 01001, 010010001, . ..., each successive string of 0’s being one larger than the previous. 3. Find the minimum state finite automaton for the language specified by the finite automaton M = ( { q 0 , q 1 , q 2 , q 3 } , { 0, 1 } , δ , q 0 , { q 0 } ) where δ is defined as follows: δ ( q 0 , 0) = q 3 δ ( q 0 , 1) = q 0 δ ( q 1 , 0) = q 0 δ ( q 1 , 1) = q 3 δ ( q 2 , 0) = q 2 δ ( q 2 , 1) = q 1 δ ( q 3 , 0) = q 1 δ ( q 3 , 1) = q 2
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