Regular Languages

• Regular expressions
• Regular languages
• Equivalence between languages accepted by FA and regular languages
• Closure Properties

Regular Expressions

Regular expression over  alphabet Σ

•   -- regular expression.
• ε -- regular expression.
• For any aΣ∈, a is a ordinary expression.
• If r1 and r2 are regular expressions, then

-    (r1 + r2)  is an ordinary expression.

-    (r1 × r2) is an ordinary expression.

-    (r1* ) is an ordinary expression.

• Anything else is not a regular expression.

•   is a regular language equivalent to the regular expression .
• {ε} is a regular language equivalent to the regular expression e.
• For any symbol  aΣ∈, {a} is a regular language equivalent to the regular expression a.
• If L1 and L2 are regular languages equivalent to the regular expression r1 and r2, then

-     are regular languages equivalent to (r1 + r2) , (r1 × r2), and (r1*).

Simple examples

Let Σ= {0,1}.

• { Σ∈a*|a does not contain 1's}

-    (0*)

• { Σ∈a *|a contains 1's only}

-    (1×(1*)) (which can can be denoted by (1+))

• Σ*

-    ((0+1)*)

• { Σ∈a*|acontains only 0's or only 1's}

-    ((00*)+(11*))

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