Properties of Context-Free Languages Wen-Guey Tzeng Computer - - PowerPoint PPT Presentation

Properties of Context-Free Languages

Wen-Guey Tzeng Computer Science Department National Chiao Tung University

Pumping Lemma for CFL

• Let L be a CFL.

– There is a constant integer m. – For any wL, |w|m, we can decompose w into u, v, x, y, z with

• w=uvxyz
• |vxy|m
• |vy| 1

– such that uvixyiz L for all i=0, 1, 2, …

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Example: G=({S, A, B, C,}, T, S, P) in CNF, w=bbaabbab

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• For every path longer than

5 = |V|+1, there are repetitive internal nodes.

• S-A-B-A-C-a

S C A A S B C B A B C B B b a b b a b B C b a

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Abstract:

• S uAz  uvAyz  uvxyz=w
• S  uAz  uvAyz

 uvvAyyz  …  uviAyiz  uvixyiz for any i  0

• For w to have a high derivation

tree with h >= |V|+1, |w| >= 2|V|=m

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Example

• Show that L={: {a, b}*} is not context-free.

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Example

• Show that L={anbncn : n0} is not context-free.

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Example

• Show that L={anbj : n=j2 } is not context-free.

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Closure Properties

• The family of CFL’s are closed under

– Union – Concatenation – star closure.

• Assume that L1 and L2 are two CFL’s with CFG

G1={V1, T, S1, P1} and G1={V2, T, S2, P2}. Assume that V1V2=.

– Construct G3 for L3=L1L2 – Construct G4 for L4=L1L2 – Construct G5 for L5=L1

*

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• G3
• G4
• G5

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Non-closed operations

• The family of CFL’s are NOT closed under

– intersection – complementation

• Why?

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• Not closed under intersection

– Why? – Example,

• L1={anbncm : n, m0}, L2={anbmcm : n, m0} are both

context-free

• But, L3=L1L2={anbncn : n0} is not context-free

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• Not closed under complementation.

– Otherwise, L3= L1L2 = comp(comp(L1)comp(L2)) is context-free. – Example

• L2=Comp(L1) is context-free, where L1={anbncn : n0}
• But, Comp(L2) = L1 is not context-free

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Closed: regular intersection

• If L1 is context-free and L2 is regular, then

L1L2 is context-free.

• Why?

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• Example

– Show that L={w{a, b, c}* : na(w)=nb(w)=nc(w)} is not context free. – Consider regular language L2=L(a*b*c*). – L3 = LL2 = {anbncn : n0} is not context-free – Thus, L is not context-free.

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