# lect10 - Lecture 10 Pumping Lemma A property of regular...

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Lecture 10 Pumping Lemma A property of regular sets

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Weak version Strong version Applications
Pumping Lemma (weak) If a language L is accepted by a DFA M with m states, then any string x in L with | x| > m can be written as x = uvw such that (1) v ≠ε, and (2) uv*w is a subset of L (i.e., for any n> 0, uv w in L ). n

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Proof Consider the path associated with x (|x| > m). x Since |x| > m, # of nodes on the path is At least m+1. Therefore, there is a state Appearing twice.
u v w v ≠ ε because M is DFA uw in L because there is a path associated with uw from initial state to a final state. uv w in L n due to the same reason as above

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L={0 | n is a prime} is not regular . Proof. For contradiction, suppose L is regular. So, L=L(M) for some DFA M. Let m be the number of states of M. Consider a prime p > m . By Pumping Lemma, 0 = uvw such that v≠ε and uv*w is a subset of L . Thus, p = |u| + |v| + |w| and for any k > 0, |u|+k|v|+|w| is a prime. n p
For k =0, |u|+|w| is a prime. For k=|u|+|w|, |u|+k|v|+|w| = (|u|+|w|)(1+|v|) is a prime. (-><-)

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L={0 1 | i > 0 } is not regular. Proof.
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lect10 - Lecture 10 Pumping Lemma A property of regular...

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