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# 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
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
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?
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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