Difference between revisions of "Closure Property of Language Families"

 
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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>\union</math>, then <math>L_1 \union L_2</math> will be CFL. But if CFL is not closed under \intersection, that doesn't mean <math>L_1 \intersection 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 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.
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Closure property is a helping technique to know the class of 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 a 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 a 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 class of the resulting language of the operation - it may or may not belong to the class of the operand languages. In short, '''closure property is applicable, only when a language is closed under an operation'''.
  
 
Now, while applying closure property do remember the language hierarchy.  
 
Now, while applying closure property do remember the language hierarchy.  
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  Regular <math>\subset</math> DCFL <math>\subset</math>CFL <math>\subset</math> REC <math>\subset</math> RE.  
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So, if CFL is closed under Union, and <math>L_1</math> and <math>L_2</math> belong to CFL,  then <math>L_1 U L_2</math> will be CFL. But  <math>L_1 U L_2</math> may also be regular, which closure property can't tell.
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Regular <math>\subset</math> DCFL <math>\subset</math>CFL <math>\subset</math> REC <math>\subset</math> RE.  
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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> can also be a regular language, which closure property can't tell'''. For this we need to see <math>L_1</math> and <math>L_2</math>.  
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{{alert|Please don't by heart this table. This is just to check your understanding and some of them are not required for GATE|alert-danger}}
  
 
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|+ align="top"|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).
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|+ align="top"|Closure properties of language families
 
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Latest revision as of 20:51, 7 December 2015

Closure property is a helping technique to know the class of 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 a 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 a 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 class of the resulting language of the operation - it may or may not belong to the class of the operand languages. In short, closure property is applicable, 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> can also be a regular language, which closure property can't tell. For this we need to see <math>L_1</math> and <math>L_2</math>.

Heads Up! Please don't by heart this table. This is just to check your understanding and some of them are not required for GATE
Closure properties of language families
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
<math>\epsilon</math>-free Homomorphism Yes No Yes Yes Yes Yes
Substitution (<math>\epsilon</math>-free) 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







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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>\union</math>, then <math>L_1 \union L_2</math> will be CFL. But if CFL is not closed under \intersection, that doesn't mean <math>L_1 \intersection 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 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 U L_2</math> will be CFL. But <math>L_1 U L_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







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