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# Homework 3 - CSCI 2400 Models of Computation Homework 3...

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CSCI 2400 - Models of Computation Homework 3 Solutions Problem 1 (25 points) Give a regular expression that describes the following language: L = { a n b m : n + m = 3 k , k 0 } . Solution: ( aaa ) * ( aab + abb + λ ) ( bbb ) * Problem 2 (75 = 5x15 points) Show that the following languages are not regular: L 1 = { a n 3 : n 0 } Solution L 1 : This language can be shown to be not regular by using the pump- ing lemma, and the Proof by Contradiction proof technique. Assume the language is regular, and attempt to find a contradiction. For the pumping lemma to hold, there must m 1 such that w L 1 with | w | ≥ m , there a partition xyz of w subject to the constraints | y | ≥ 1 and | xy | ≤ m , such that i 0, xy i z L 1 . For any arbitrary value of m , let us look at the string a m 3 . This string is clearly of length m for all m 1, and it is clearly L 1 . Further, for any arbitrary partition xyz of this string, let us look at the string xy 2 z which must be L 1 for the pumping lemma to hold. | xy 2 z | = m 3 + k , where 1 k m . The length of this string | xy 2 z | cannot equal | a n 3 | for any n : m 3 < m 3 + k < ( m +1) 3 = m 3 + 3 m 2 + 3 m + 1 Thus xy 2 z negationslash∈ L 1 , which is a contradiction. L 1 is not regular. L 2 = { a 3 n : n 0 } Solution L 2 : This language can be shown to be not regular by using the pump- ing lemma, and the Proof by Contradiction proof technique.

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Homework 3 - CSCI 2400 Models of Computation Homework 3...

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