14 - 14: Pumping Theorem for CFLs The idea behind the...

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Unformatted text preview: 14: Pumping Theorem for CFLs The idea behind the pumping theorem for CF languages is similar to the pumping theorem for regular languages. If the parse tree of a string is deep enough, then at least one nonterminal must be repeated along a certain branch. Some relevant substrings can then either be trimmed away or be pasted repeatedly. A A z y x v u S A A S z y v u A x v y w = uvxyz , then uxz ∈ L , and uv 2 xy 2 z ∈ L , and so on. 1 Height of tree = length of the longest path from the root to some leaf. Let b be the branching factor of a tree, i.e., the maximum number of branches from any node. A tree of height h with a branching factor of b can have at most b h leaves. Therefore, a tree with b h leaves must have height ≥ h . <= b <= b 2 h=2 Let b be the fanout of a grammar G , i.e., the maximum number of symbols on the RHS of any rule of G . A string of length b h must have its parse tree of height ≥ h . 2 Properties of CFLs The pumping theorem for CFLs Theorem 1 Let L be a context free language. Then there is an integer N ≥ 1 such that for every w ∈ L and | w | ≥ N , w can be split into five parts w = uvxyz such that 1. vy = e (i.e. v and y cannot both be e ) 2. | vxy | ≤ N 3. uv i xy i z ∈ L for i = 0 , 1 , 2 , .... 3 Properties of CFLs Proof : • Let G = ( V, Σ , R, S ) be a CFG that generates the CFL L . • Let b = max {| α | : A → α in G } • Let N = b | NT | +1 = b | V − Σ | +1 • Let w ∈ L and | w | ≥ N . • Let T be a parse tree for w that has the smallest num- ber of leaves....
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14 - 14: Pumping Theorem for CFLs The idea behind the...

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