automata-4 - Automata Chapter 4. Properties of Regular...

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Automata Chapter 4. Properties of Regular Languages
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4. Properties of Regular Languages How general are regular languages? Closure question? Finite or not? Membership? Nature or characteristics of regular language family
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4.1 Closure Properties of Regular Languages Closure Question: Given regular languages L 1 and L 2 , we perform an operation. Is the resulting language still regular? If yes, the family of regular languages is closed under the operation.
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4.1 Closure Properties of Regular Languages Closure under Simple Set Operations Theorem 4.1 If L 1 and L 2 are regular languages, then so are L 1 L 2 , L 1 L 2 , L 1 L 2 , L 1 and L 1 *. The family of regular languages is closed under union, intersection, concatenation, complementation, and star-closure. Pf) L 1 L 2 , L 1 L 2 , and L 1 *. by definition of regular expression
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5 4.1 Closure Properties of Regular Languages Let be a DFA that accepts L 1 Then =(Q, Σ , δ ,q 0 ,Q-F) accepts . Proof by construction L 1 = L(M 1 ) M 1 =(Q, Σ , δ 1 ,q 0 ,F 1 ) L 2 = L(M 2 ) M 2 =(P, Σ , δ 2 ,p 0 ,F 2 ) M=(QxP, Σ , δ ,(q 0 ,p 0 ), F 1 xF 2 ) δ ((q i ,p j ),a) = ( δ 1 (q i ,a), δ 2 (p j ,a)) M =( Q, Σ , δ ,q 0 ,F) 1 L M 2 1 2 1 L L L L =
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4.1 Closure Properties of Regular Languages Ex 4.1 Is the family of regular languages closed under difference?
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4.1 Closure Properties of Regular Languages Theorem 4.2 The family of regular languages is closed under reversal. Pf) Suppose L is regular and L=L(M) for some NFA M with a single final state. Construct M R such that L R = L(M R ) (1) initial state final state (2) reverse the direction on all the edges
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4.1 Closure Properties of Regular Languages Definition 4.1 Let and Γ alphabets. Then a function
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This note was uploaded on 11/03/2009 for the course CS automata taught by Professor Prof.jung during the Fall '09 term at 홍익대학교.

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automata-4 - Automata Chapter 4. Properties of Regular...

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