Difference between revisions of "Closure Property of Language Families"

(Created page with "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 L_1 and L_2 belong to CF...")
 
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! Operation
 
! Operation
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! Regular
 
! Regular
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! [[Deterministic context-free language|DCFL]]
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! DCFL
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! [[Context free language|CFL]]
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! CFL
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! [[Indexed language|IND]]
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! CSL
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! [[Context sensitive language|CSL]]
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! Recursive
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! [[Recursive language|recursive]]
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! RE
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! [[Recursively enumerable language|RE]]
 
 
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|[[Union (set theory)|Union]]
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|Union
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| <math>\{w | w \in L_1 \lor w \in L_2\} </math>
 
 
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|[[Intersection (set theory)|Intersection]]
 
|[[Intersection (set theory)|Intersection]]
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| <math>\{w | w \in L_1 \land w \in L_2\}</math>
 
 
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| {{No}}
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|[[Complement (set theory)|Complement]]
 
|[[Complement (set theory)|Complement]]
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| <math>\{w | w \not\in L_1\}</math>
 
 
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| {{Yes}}
 
| {{Yes}}
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|[[Concatenation]]
 
|[[Concatenation]]
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| <math>L_1\cdot L_2 = \{w\cdot z | w \in L_1 \land z \in L_2\}</math>
 
 
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|Kleene star
 
|Kleene star
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| <math>L_1^{*} = \{\epsilon\} \cup \{w \cdot z | w \in L_1 \land z \in L_1^{*}\}</math>
 
 
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|Homomorphism
 
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|e-free Homomorphism
 
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|Substitution
 
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|Inverse Homomorphism
 
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|Reverse
 
|Reverse
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| <math>\{w^R | w \in L\} </math>
 
 
| {{Yes}}
 
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| {{No}}
 
| {{No}}
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|Intersection with a [[regular language]]
 
|Intersection with a [[regular language]]
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| <math>\{w | w \in L_1 \land w \in R\}, R \text{ regular}</math>
 
 
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|Min
 
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|Init
 
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|Cycle
 
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|Shuffle
 
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|Perfect Shuffle
 
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| {{Yes}}
 
| {{?}}
 
| {{?}}

Revision as of 11:44, 26 February 2014

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 L_1 and L_2 belong to CFL and if CFL is closed under operation U, then L_1 U L_2 will be CFL. But if CFL is not closed under \intersection, that doesn't mean L_1 \intersect L_2 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 \subset \subset \DCFL \subset \CFL \subset REC \subset \RE. So, if CFL is closed under Union, and L_1 and L_2 belong to CFL, then L_1 U L_2 will be CFL. But L_1 U L_2 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). After Hopcroft and Ullman.
Operation Regular DCFL CFL CSL Recursive RE
Union Yes No Yes Yes Yes Yes Yes
Intersection Yes No No No Yes Yes Yes
Complement Yes Yes No No Yes Yes No
Concatenation Yes No Yes Yes Yes Yes Yes
Kleene star Yes No Yes Yes Yes Yes Yes
Homomorphism Yes No Yes Yes No No Yes
e-free Homomorphism Yes No Yes Yes Yes Yes Yes
Substitution Yes No Yes Yes Yes No Yes
Inverse Homomorphism Yes Yes Yes Yes Yes Yes Yes
Reverse Yes No Yes Yes Yes Yes Yes
Intersection with a regular language Yes 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 L_1 and L_2 belong to CFL and if CFL is closed under operation U, then L_1 U L_2 will be CFL. But if CFL is not closed under \intersection, that doesn't mean L_1 \intersect L_2 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 \subset \subset \DCFL \subset \CFL \subset REC \subset \RE. So, if CFL is closed under Union, and L_1 and L_2 belong to CFL, then L_1 U L_2 will be CFL. But L_1 U L_2 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). After Hopcroft and Ullman.
Operation Regular DCFL CFL CSL Recursive RE
Union Yes No Yes Yes Yes Yes Yes
Intersection Yes No No No Yes Yes Yes
Complement Yes Yes No No Yes Yes No
Concatenation Yes No Yes Yes Yes Yes Yes
Kleene star Yes No Yes Yes Yes Yes Yes
Homomorphism Yes No Yes Yes No No Yes
e-free Homomorphism Yes No Yes Yes Yes Yes Yes
Substitution Yes No Yes Yes Yes No Yes
Inverse Homomorphism Yes Yes Yes Yes Yes Yes Yes
Reverse Yes No Yes Yes Yes Yes Yes
Intersection with a regular language Yes Yes Yes Yes Yes Yes Yes







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