Session-19.pdf - Closure Properties of Regular Languages...

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Closure Properties of Regular Languages What is Closure property ? If a set of regular languages are combined using an operator, then the resulting language is also regular. Regular languages are closed under: 1. Union 2. Intersection 3. Complement 4. Difference 5. Reversal 6. Kleene closure 7. Concatenation 8. Homomorphism 9. Inverse homomorphism Let L and M be regular languages. Then the following languages are all regular: 1. Union: L M 2. Intersection: L M 3. Complement: N bar or N C 4. Difference: L\M 5. Reversal: LR=wR:w L 6. Closure: L 7. Concatenation: L.M 8. Homomorphism: h(L) ={h(w)|w L, h is a homomorphism } 9. Inverse homomorphism: h 1(L) ={w Σ|h(w)L, h: Σ→∆ is a homomorphism } Proofs: 1. Closure under Union Let us take two regular expressions RE 1 = a(aa)* and RE 2 = (aa)* So, L 1 = {a, aaa, aaaaa, ..... } (Strings of odd length excluding Null)
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and L 2 ={ ε, aa, aaaa, aaaaaa, ....... } (Strings of even length including Null) L 1 L 2 = { ε, a, aa, aaa, aaaa, aaaaa, aaaaaa, ....... } (Strings of all possible lengths including Null) RE (L 1 L 2
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  • Fall '14
  • rani
  • Formal language, Regular expression, Closure

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