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# class14 - More Applications of The Pumping Lemma 1 The...

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1 More Applications of The Pumping Lemma

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2 The Pumping Lemma: there exists an integer such that m for any string m w L w | | , we can write For infinite context-free language L uvxyz w = with lengths 1 | | and | | vy m vxy and it must be: 0 all for , i L z xy uv i i
3 Context-free languages } 0 : { n b a n n Non-context free languages } 0 : { n c b a n n n }*} , { : { b a w ww R }} , { : { b a v vv

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4 Theorem: The language }*} , { : { b a v vv L = is not context free Proof: Use the Pumping Lemma for context-free languages
5 Assume for contradiction that is context-free Since is context-free and infinite we can apply the pumping lemma L L }*} , { : { b a v vv L =

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6 Pumping Lemma gives a magic number such that: m Pick any string of with length at least m we pick: L b a b a m m m m L }*} , { : { b a v vv L =
7 We can write: with lengths and m vxy | | 1 | | vy uvxyz b a b a m m m m = Pumping Lemma says: L z xy uv i i for all 0 i }*} , { : { b a v vv L =

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8 We examine all the possible locations of string in vxy m vxy | | 1 | | vy uvxyz b a b a m m m m = m m m m b a b a }*} , { : { b a v vv L =
9 Case 1: vxy is within the first m a b b a a b b a a ...... ...... ...... ...... v m m m u z m vxy | | 1 | | vy uvxyz b a b a m m m m = x y m 1 k a v = 2 k a y = 1 2 1 + k k }*} , { : { b a v vv L =

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10 b b a a b b a a ...... ...... ...... ...... .......... 2 v 2 1 k k m + + m m u z m vxy | | 1 | | vy uvxyz b a b a m m m m = x 2 y m Case 1: vxy is within the first m a 1 k a v = 2 k a y = 1 2 1 + k k }*} , { : { b a v vv L =
11 m vxy | | 1 | | vy uvxyz b a b a m m m m = Case 1: vxy is within the first m a 1 2 1 + k k L z xy uv b a b a m m m k k m = + + 2 2 2 1 }*} , { : { b a v vv L =

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12 m vxy | | 1 | | vy uvxyz b a b a m m m m = Case 1: vxy is within the first m a L z xy uv 2 2 Contradiction!!! L z xy uv b a b a m m m k k m = + + 2 2 2 1 However, from Pumping Lemma: }*} , { : { b a v vv L =
13 is in the first is in the first Case 2: b b a a b b a a ...... ...... ...... ......

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