Closure Properties of Regular Languages
San Skulrattanakulchai
March 4, 2019
Closure
- Consider a set \(S\) of elements under some operation, say \(\cdot\). We say that \(S\) is closed under \(\cdot\) if applying \(\cdot\) to the element(s) of \(S\) always results in an element of \(S\).
- For example, the set of natural numbers is closed under addition but not closed under subtraction.
Closure of the Regular Languages
- The class of Regular Languages is closed under
- complement
- union
- intersection
- set difference
- symmetric difference
- concatenation
- Kleene closure
- positive closure
- reversal
- homomorphism
- inverse homomorphism
- etc.
- [Proofs to be done in class.]