automata-8 - Automata Chapter 8. Properties of Context-Free...

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Automata Chapter 8. Properties of Context-Free Languages
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Chapter 8. Properties of Context-Free Languages Closure property Membership Classifying languages
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8.1 Two Pumping Lemmas A pumping lemma for context-free languages Theorem 8.1 Let L be an infinite context-free language. Then 5 some positive m s.t. 2200 w L with |w| m can be decomposed as w=uvxyz, with |vxy| m, |vy| 1, s.t. uv i xy i z L, for all i=0, 1, 2, …. Proof: Consider a derivation tree for a long string. # of variables is finite 5 variable repeated. S * uAz * uvAyz *uvxyz and A *vAy and A *x, so uv i xy i z can be generated by G for all i and |vxy| m. no unit-prod and no λ -prod |vy| 1. S A A u v x y z
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8.1 Two Pumping Lemmas Ex) A BC|a, B BA|b, C BA A B C B A B A b B C b B A b a b a w= b bb a λ ba u v x y z What is m? If G is in CNF and # of variables is n, then m 2 n .
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Ex 8.1 Show L ={a n b n c n | n 0} is not context-free. Ex 8.2 L={ww| w
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automata-8 - Automata Chapter 8. Properties of Context-Free...

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