Definiton: Derivation tree Let G = ( V, T, S, P ) be a cfg. An - - PowerPoint PPT Presentation

Definiton: Derivation tree Let G = (V, T, S, P) be a cfg. An ordered tree is called a derivation tree for G if

• the root is labelled S
• every interior vertex has a label from V
• Every leaf has a label from T ∪ {λ}
• If a vertex has a label A and its children are labelled

X1, X2, X3, . . . Xn from left to right, then A → X1X2 . . . Xn is a production in P.

• A leaf labelled λ has no siblings.

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Chomsky Normal Form (CNF) A cfg G is in CNF if all productions are of the form A → BC

• r A → a

where A, B, C ∈ V and a ∈ T.

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Given a cfg G, to put it into CNF, we perform the following steps:

• 1. Remove all λ-productions.
• 2. Remove all unit productions.
• 3. Remove all useless productions.
• 4. Remove productions not of the form A → BC or A → a.

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Griebach Normal Form A cfg G is in GNF if all productions are of the form A → ax where a ∈ T and x ∈ V ∗

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HW Q2 (revised): Let G be a context-free grammar in CNF, and let w ∈ L(G) be the yield of a parse tree for w according to the grammar G. Prove using induction that if the length of the longest path in the tree is n, then |w| ≤ 2n−1.

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Pumping Lemma for CFLs Let L be a CFL. Then ∃m > 0 such that ∀w ∈ L with |w| ≥ m ∃ a way to break up w such that w = uvxyz with |vxy| ≤ m and |vy| ≥ 1 ∀i ≥ 0 uvixyiz ∈ L

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To use the pumping lemma to show that a given L is not context- free, we play an adversary game as follows:

• 1. Assume adversary picks m, the constant of the pumping

lemma/

• 2. We get to pick w, a string in L that is longer than m.
• 3. Adversary gets to break up w into uvxyz, while obeying the

rules that |vxy| ≤ m and |vy| ≥ 1.

• 4. We get to pick i (the number of times to pump v and y. We

win if we show that uvixyiz / ∈ L.

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Closure properties Operation Regular CFLs DCFLs Union Yes Yes No Concatenation Yes Yes No Kleene closure Yes Yes No Intersection Yes No No

• Int. with regular language

Yes Yes Yes Complementation Yes No Yes Reversal Yes Yes No

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Decision algorithms

Given a cfg G, there are algorithms to determine the following questions:

• 1. Is L(G) empty?
• 2. Is L(G) finite?
• 3. Given a string x, is x ∈ L(G)?

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Decision algorithms

But for many questions about CFLs that no algorithm can an- swer (we will formalize what we mean by this later). Here are some examples: Let G be a cfg.

• 1. Is L(G) = Σ∗?
• 2. Is L(G) a regular set?
• 3. Is L(G) a DCFL?
• 4. Is L(G) inherently ambiguous?
• 5. Is the complement of L(G) a cfl?

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Let G1, G2 be cfgs and let R be a regular set.

• 6. Is L(G1) = L(G2)?
• 7. Is L(G1) ∩ L(G2) = ∅?
• 8. Is L(G1) ∩ L(G2) a cfl?
• 9. Is L(G1) ⊆ L(G2)?
• 10. Is L(G) = R?