# 13 - 13: Closure Properties of CFLs Theorem 1 CFLs are...

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13: Closure Properties of CFLs Theorem 1 CFLs are closed under 1. union, 2. concatenation, 3. Kleene Star. Proof Let G 1 =( V 1 , Σ 1 ,R 1 ,S 1 ), and G 2 =( V 2 , Σ 2 ,R 2 ,S 2 )beth e two CFG generating the CFLs L 1 and L 2 . Rename the nonterminals, if necessary, so that V 1 Σ 1 and V 2 Σ 2 are disjoint sets. 1. Let G =( V 1 V 2 ∪{ S } , Σ 1 Σ 2 , R 1 R 2 ∪{ S S 1 ,S S 2 } ,S ). Prove that L ( G )= L ( G 1 ) L ( G 2 ): w L ( G )i± S G w . S G S 1 G w or S G S 2 G w . S 1 G w or S 2 G w . S 1 G 1 w or S 2 G 2 w ,s in c e S ±∈ V 1 V 2 and ( V 1 Σ 1 ) ( V 2 Σ 2 )= . w L ( G 1 )or w L ( G 2 ). w L ( G 1 ) L (

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2. Let G =( V 1 V 2 ∪{ S } , Σ 1 Σ 2 , R 1 R 2 ∪{ S S 1 S 2 } ,S ). Prove that
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## This note was uploaded on 09/22/2011 for the course COMP 272 taught by Professor Prof.tai during the Spring '10 term at HKUST.

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13 - 13: Closure Properties of CFLs Theorem 1 CFLs are...

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