class14 - 1 More Applications of The Pumping Lemma 2 The...

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Unformatted text preview: 1 More Applications of The Pumping Lemma 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: all for , ≥ ∈ i L z xy uv i i 3 Context-free languages } : { ≥ n b a n n Non-context free languages } : { ≥ n c b a n n n }*} , { : { b a w ww R ∈ }} , { : { b a v vv ∈ 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 ∈ = 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 ≥ i }*} , { : { b a v vv L ∈ = 8 We examine all the possible locations of string in vxy m vxy ≤ | | | | ≥ 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 ≤ | | | | ≥ 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 ∈ = 10 b b a a b b a a ...... ...... ...... ...... .......... 2 v 2 1 k k m + + m m u z m vxy ≤ | | | | ≥ 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 ≤ | | | | ≥ 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 ∈ = 12 m vxy ≤ | | | | ≥ 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!!!...
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This note was uploaded on 04/07/2008 for the course CSCI 2400 taught by Professor Carothers during the Spring '08 term at Rensselaer Polytechnic Institute.

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class14 - 1 More Applications of The Pumping Lemma 2 The...

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