DR. Torng

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DR. Torng

• Closure Properties for CFLs
• Kleene Closure
• Union
• Concatenation
• CFLs versus regular languages
• regular languages subset of CFL

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Closure Properties for CFLs

• Kleene Closure

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CFL closed under Kleene Closure

• Let L be an arbitrary CFL
• Let G1 be a CFG s.t. L(G1) L
• G1 exists by definition of L1 in CFL
• Construct CFG G2 from CFG G1
• Argue L(G2) L
• There exists CFG G2 s.t. L(G2) L
• L is a CFL

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Visualization

• Let L be an arbitrary CFL
• Let G1 be a CFG s.t. L(G1) L
• G1 exists by definition of L1 in CFL
• Construct CFG G2 from CFG G1
• Argue L(G2) L
• There exists CFG G2 s.t. L(G2) L
• L is a CFL

CFL
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Algorithm Specification

• Input
• CFG G1
• Output
• CFG G2 such that L(G2)

CFG G1
CFG G2
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Construction

• Input
• CFG G1 (V1, S, S1, P1)
• Output
• CFG G2 (V2, S, S2, P2)
• V2 V1 union T
• T is a new symbol not in V1 or S
• S2 T
• P2 P1 union ??

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Closure Properties for CFLs

• Kleene Closure Examples

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Example 1
V2 V1 union T T is a new symbol not in
V1 or SS2 TP2 P1 union T ? ST l

• Input grammar
• V S
• S a,b
• S S
• P
• S ? aa ab ba bb

• Output grammar
• V
• S a,b
• Start symbol is
• P

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Example 2
V2 V1 union T T is a new symbol not in
V1 or SS2 TP2 P1 union T ? ST l

• Input grammar
• V S, T
• S a,b
• Start symbol is T
• P
• T ? ST l
• S ? aa ab ba bb

• Output grammar
• V
• S a,b
• Start symbol is
• P

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Construction for Set Union

• Input
• CFG G1 (V1, S, S1, P1)
• CFG G2 (V2, S, S2, P2)
• Output
• CFG G3 (V3, S, S3, P3)
• V3 V1 union V2 union T
• Variable renaming to insure no names shared
  between V1 and V2
• T is a new symbol not in V1 or V2 or S
• S3 T
• P3

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Construction for Set Concatenation

• Input
• CFG G1 (V1, S, S1, P1)
• CFG G2 (V2, S, S2, P2)
• Output
• CFG G3 (V3, S, S3, P3)
• V3 V1 union V2 union T
• Variable renaming to insure no names shared
  between V1 and V2
• T is a new symbol not in V1 or V2 or S
• S3 T
• P3

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CFLs and regular languages
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CFL Closure Properties

• What have we just proven
• CFLs are closed under Kleene closure
• CFLs are closed under set union
• CFLs are closed under set concatenation
• What can we conclude from these 3 results?
• It follows that regular languages are a subset of
  CFLs

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Regular languages subset of CFL

• Similar to reg-exp to NFA-? construction from
  module 22
• Base Case
• , l, a, b are regular languages over
  a,b
• P, PS ? l, PS ? a, PS ? b
• Inductive Case
• If L1 and L2 are are regular languages, then L1,
  L1L2, L1 union L2 are regular languages
• Use previous constructions to see that these
  resulting languages are also context-free

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Other CFL Closure Properties

• We will show that CFLs are NOT closed under many
  other set operations
• Examples include
• set complement
• set intersection
• set difference

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Language class hierarchy
REG
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Nested language examples

• Prove the following languages are CFLs
• anbncmdm m,n 0
• anbmcmdn m,n 0
• amnbmcn m,n 0
• What happens if we change bounds above to 1
  instead of 0?