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Unformatted text preview: 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 } . Solution: ( aaa ) * ( aab + abb + ) ( bbb ) * Problem 2 (75 = 5x15 points) Show that the following languages are not regular: L 1 = { a n 3 : n } 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....
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This homework help was uploaded on 04/07/2008 for the course CSCI 2400 taught by Professor Carothers during the Spring '08 term at Rensselaer Polytechnic Institute.
 Spring '08
 CAROTHERS

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