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HW4Soln

# HW4Soln - CSCI 2670 Fall 2005 HW 4 Use the pumping lemma to...

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CSCI 2670 Fall 2005 HW 4 September 20, 2005 Use the pumping lemma to prove A={ww R |w {a,b} * } is not regular. Proof: Assume A is regular. Then A has an associated pumping length p such that any string s in A with |s| p can be written as s = xyz such that (1) |xy| ≤ p (2) |y| > 0 (3) xy i z for every i = 0, 1, 2, … Consider the string s = a p bba p . Since s = ww R , where w = a p b, s is in A. Therefore the pumping lemma holds and s = xyz for some x, y, and z with the above properties. Since xy is a prefix of s with at most p symbols, it must be the case that xy = a k for some k ≤ p. Also, since y is a substring of xy, y must be a j for some j = 1, 2, …, k (j cannot be 0 by property 2 above). Now consider the string xz. This string is the result of “pumping” y 0 times. xz = a p-j bba p . Since j > 0, xz is not in A. Thus, the pumping lemma fails, which contradicts our assumption that A is a regular language. 1.54 a. F cannot be regular because if a string in F starts with exactly 1 a, it must be followed by b n c n

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HW4Soln - CSCI 2670 Fall 2005 HW 4 Use the pumping lemma to...

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