## What is Closure Properties

Closure property refers to some operation on language which returns the language with the same type in the result. If a language satisfies the condition for some operation, we can say that the language is closed under that operation.

We will discuss the closure properties of the following formal languages:

• Regular Language (REG)
• Deterministic Context Free Language (DCFL)
• Context Free Language (CFL)
• Context Sensitive Language (CSL)
• Recursive Language (REC)
• Recursive Enumerable Language (REL)

## Closure Properties Table

 Operation REG DCFL CFL CSL REC REL Union YES NO YES YES YES YES Concatenation YES NO YES YES YES YES Kleen Closure YES NO YES YES YES YES Reversal YES NO YES YES YES YES Intersection YES NO NO YES YES YES Complement YES YES NO YES YES NO Inverse Homomorphism YES YES YES YES YES YES Homomorphism YES NO YES YES YES YES Substitution YES NO YES YES YES YES Subset NO NO NO NO NO NO Infinite - Union NO NO NO NO NO NO Infinite - Intersection NO NO NO NO NO NO Infinite - Difference NO NO NO NO NO NO Prefix YES YES YES YES YES YES Quotient YES NO YES YES YES YES Cycle YES NO YES YES YES YES Min YES NO NO Max YES NO NO Half YES NO NO

## Closure Properties with Regular Languages

 Operation L = REG L = DCFL L = CFL L = CSL L = REC L = REL L ∪ REG YES YES YES YES YES YES L ∩ REG YES YES YES YES YES YES L - REG YES YES YES YES YES YES REG - L YES YES NO YES YES NO L / REG YES YES YES YES YES YES