Arjun Suresh (talk | contribs) |
Arjun Suresh (talk | contribs) |
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Now, while applying closure property do remember the language hierarchy. | 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. | 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 \ | + | 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 a CFL. But <math>L_1 \cup L_2</math> may also be regular, which closure property can't tell. For this we need to see <math>L_1</math> and <math>L_2</math>. |
:{| class="wikitable" | :{| class="wikitable" |
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 a CFL. But <math>L_1 \cup L_2</math> may also be regular, which closure property can't tell. For this we need to see <math>L_1</math> and <math>L_2</math>.
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 |
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.
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 |