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Unformatted text preview: 1 Module 31 • Closure Properties for CFL’s – Kleene Closure – Union – Concatenation • CFL’s versus regular languages – regular languages subset of CFL 2 Closure Properties for CFL’s Kleene Closure 3 CFL closed under Kleene Closure • Let L be an arbitrary CFL • Let G 1 be a CFG s.t. L(G 1 ) = L – G 1 exists by definition of L 1 in CFL • Construct CFG G 2 from CFG G 1 • Argue L(G 2 ) = L * • There exists CFG G 2 s.t. L(G 2 ) = L * • L * is a CFL 4 Visualization L * L CFL CFG’s • Let L be an arbitrary CFL • Let G 1 be a CFG s.t. L(G 1 ) = L – G 1 exists by definition of L 1 in CFL • Construct CFG G 2 from CFG G 1 • Argue L(G 2 ) = L * • There exists CFG G 2 s.t. L(G 2 ) = L * • L * is a CFL G 1 G 2 5 Algorithm Specification • Input – CFG G 1 • Output – CFG G 2 such that L(G 2 ) = CFG G 1 CFG G 2 A 6 Construction • Input – CFG G 1 = (V 1 , Σ , S 1 , P 1 ) • Output – CFG G 2 = (V 2 , Σ , S 2 , P 2 ) • V 2 = V 1 union {T} – T is a new symbol not in V 1 or Σ • S 2 = T • P 2 = P 1 union ?? 7 Closure Properties for CFL’s Kleene Closure Examples 8...
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 Fall '07
 TORNG
 Naive set theory, Empty set, Formal language, CFL, CFG G2

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