View source for Closure Property of Language Families ← Closure Property of Language Families

Closure property is a helping to know what will be the resulting language when we do an operation on two languages of the same class. That is, suppose <math>L_1</math> and <math>L_2</math> belong to CFL and if CFL is closed under operation <math>\cup</math>, then <math>L_1 \cup L_2</math> will be CFL. But if CFL is not closed under <math>\cap</math>, that doesn't mean <math>L_1 \cap L_2</math> won't be CFL. For a class to be closed under an operation, it should hold true for all languages in that class. So, if a class is not closed under an operation, we cannot say anything about the result of the operation, it may or may not belong to that class. In short, closure property is useful, only when a language is closed under an operation.

Now, while applying closure property do remember the language hierarchy.

Regular <math>\subset</math> DCFL <math>\subset</math>CFL <math>\subset</math> REC <math>\subset</math> RE. 

So, if CFL is closed under Union, and <math>L_1</math> and <math>L_2</math> belong to CFL, then <math>L_1\cup L_2</math> will be CFL. But <math>L_1 \cupL_2</math> may also be regular, which closure property can't tell.

Closure properties of language families (<math>L_1</math> Op <math>L_2</math> where both <math>L_1</math> and <math>L_2</math> are in the language family given by the column).

Operation	Regular	DCFL	CFL	CSL	Recursive	RE
Union	Yes	No	Yes	Yes	Yes	Yes
Intersection	Yes	No	No	Yes	Yes	Yes
Complement	Yes	Yes	No	Yes	Yes	No
Concatenation	Yes	No	Yes	Yes	Yes	Yes
Kleene star	Yes	No	Yes	Yes	Yes	Yes
Homomorphism	Yes	No	Yes	No	No	Yes
e-free Homomorphism	Yes	No	Yes	Yes	Yes	Yes
Substitution	Yes	No	Yes	Yes	No	Yes
Inverse Homomorphism	Yes	Yes	Yes	Yes	Yes	Yes
Reverse	Yes	No	Yes	Yes	Yes	Yes
Intersection with a regular language	Yes	Yes	Yes	Yes	Yes	Yes