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# 17b - Closure properties of recursive and r.e languages...

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Closure properties of recursive and r.e. languages Theorem 1 The class of recursive languages is closed un- der 1. complementation 2. union 3. concatenation 4. Kleene star (proof similar to (3)) 5. intersection, (proof similar to (2)) Note: the set of context-free languages is not closed under com- plementation, but the set of recursive languages is closed under complementation. L is CF L is recursive L 0 is recursive, but L 0 may or may not be CF. 1

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If L is recursive, then L 0 is recursive. Proof : Let M = ( K, Σ , δ, s, { y, n } ) be the TM that decides L . We can construct a TM M 0 that decides L 0 by reversing the roles of states y and n in M . That is: M 0 = ( K, Σ , δ 0 , s, { y, n } ) δ 0 ( q, σ ) = ( n, γ ) if δ ( q, σ ) = ( y, γ ) ( y, γ ) if δ ( q, σ ) = ( n, γ ) ( p, γ ) if δ ( q, σ ) = ( p, γ ) and p 6 = y, p 6 = n Question : Is the complement of a recursively enumerable language always recursively enumerable? No (example later). 2
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17b - Closure properties of recursive and r.e languages...

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