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Unformatted text preview: CS103 HO#33 SlidesContextFree Languages 4/29/11 1 Show that L = { w  0 n where n is prime } is not regular. Suppose L is regular, and that p is the pumping length guaranteed to exist by the lemma. We will find a string s such that s L and s > p but s does not satisfy the conditions of the lemma. Show that L = { w  0 n where n is prime } is not regular. Suppose L is regular, and that p is the pumping length guaranteed to exist by the lemma. We will find a string s such that s L and s > p but s does not satisfy the conditions of the lemma. Let s = 0 k , where k is the smallest prime > p + 1. s L, and s > p. According to the lemma, there must exist strings x, y, and z such that (s = xyz y > 0 xy p) and xy i z L for all i 0. That is, for all i 0, x + z + iy must be prime. We will show that for any x, y, z satisfying (.), there is an i where the last condition fails. Suppose (s = xyz y > 0 xy p) and let i = x + z. Then x + z + iy = x + z + (x + z)y = (x + z) (1 + y) x + z > 1 since s = xyz, s > p + 1, and y p. 1 + y > 1 since y > 0. But that means (x + z)(1 + y) is not prime. So for at least one value of i, xy i z L, and thus L is not regular. Closure Properties of Regular Languages Regular languages are closed under: Union Intersection Concatenation Difference Complement Reversal Closure (star) (You may use these facts in your proofs.) Using Closure Properties to Show Nonregularity Consider D = { w {0, 1}*  the number of 0's in w is the same as the number of 1's } Suppose D is regular. Then D 0*1* is regular since regular languages are closed under intersection. But D 0*1* = { 0 n 1 n  n 0 } , which we have shown to be nonregular. So D is not regular. Regular Grammars A 1 1 B Instead of accepting its language, we can think of the machine as generating it. That is, every transition on a path to the accepting state generates a string containing the symbols on the arcs of the path....
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This note was uploaded on 06/01/2011 for the course EE 103 taught by Professor Plummer during the Spring '11 term at Stanford.
 Spring '11
 PLUMMER

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