Closure properties Table in TOC
The Below Table shows the Closure Properties of Formal Languages :
REG = Regular Language
DCFL = deterministic context-free languages,
CFL = context-free languages,
CSL = context-sensitive languages,
RC = Recursive.
RE = Recursive Enumerable
Consider L and M are regular languages :
- The Kleene star –
∑*, is a unary operator on a set of symbols or strings, ∑, that gives the infinite set of all possible strings of all possible lengths over ∑ including λ. - Kleen Plus –
The set ∑+ is the infinite set of all possible strings of all possible lengths over ∑ excluding λ. - Complement –
The complement of a language L (with respect to an alphabet E such that E* contains L) is E*–L. Since E* is surely regular, the complement of a regular language is always regular. - Reverse Operator –
Given language L, LR is the set of strings whose reversal is in L. - Complement –
The complement of a language L (with respect to an alphabet E such that E* contains L) is E*–L. Since E* is surely regular, the complement of a regular language is always regular. - Union –
Let L and M be the languages of regular expressions R and S, respectively. Then R+S is a regular expression whose language is(L U M). - Intersection –
Let L and M be the languages of regular expressions R and S, respectively, then it is a regular expression whose language is L intersection M. - Set Difference operator –
If L and M are regular languages, then so is L – M = strings in L but not M. - Homomorphism –
A homomorphism on an alphabet is a function that gives a string for each symbol in that alphabet. - Inverse Homomorphism –
Let h be a homomorphism and L a language whose alphabet is the output language of h. h-1 (L) = {w | h(w) is in L}. - Substitution –
substitution is a letter-to-language mapping, which is likewise extended to a string-to-language mapping. By identifying a singleton language {x} to the tx, morphisms are seen as special cases of substitutions. - Left quotient – Left quotient , or quotient of a language L by a word w is The language Lw = {x ∈ Σ* | wx ∈ L}
The right quotient of L1 with L2 is the set of all strings x where you can pick some y from L2 and append it to x to get something from L1. That is, x is in the quotient if there is y in L2 for which xy is in L1.)
Operations | REG | DCFL | CFL | CSL | RC | RE |
Union | Y | N | Y | Y | Y | Y |
Intersection | Y | N | N | Y | Y | Y |
Set Difference | Y | N | N | Y | Y | N |
Complement | Y | Y | N | Y | Y | N |
Intersection with a Regular Language | Y | Y | Y | Y | Y | Y |
Union with a Regular Language | Y | Y | Y | Y | Y | Y |
Concatenation | Y | N | Y | Y | Y | Y |
Kleene Star | Y | N | Y | Y | Y | Y |
Kleene Plus | Y | N | Y | Y | Y | Y |
Reversal | Y | N | Y | Y | Y | Y |
Epsilon-free Homomorphism | Y | N | Y | Y | Y | Y |
Homomorphism | Y | N | Y | N | N | Y |
Inverse Homomorphism | Y | Y | Y | Y | Y | Y |
Epsilon-free Substitution | Y | N | Y | Y | Y | Y |
Substitution | Y | N | Y | N | N | Y |
Subset | N | N | N | N | N | N |
Left Difference with a Regular Language (L-Regular) | Y | Y | Y | Y | Y | Y |
Right Difference with a Regular Language (Regular-R) | Y | Y | N | Y | Y | N |
Left Quotient with a Regular Language | Y | Y | Y | N | Y | Y |
Right Quotient with a Regular Language | Y | Y | Y | N | Y | Y |
NOTE : If we do union, intersection or set difference of any language with regular language, language doesn’t change .
Example
- CFL ∩ Regular is CFL.
- CFL ∪ Regular is CFL.
It’s always a good idea to convert the secondary operations into primary operations.
Let L1 and L2 be two languages.
NOTE : For ⊆ , ⊇ , infinite union, infinite intersection, infinite set difference, No language is closed.
Under these operations, language may or may not be regular.
Let us consider some cases with concatenation operation :
- Regular . Regular ⇒ Regular
- Regular . Non-Regular ⇒ May or may not be regular
- Non-Regular . Non-Regular ⇒ May or may not be regular
- If L1 . L2 is regular ⇒ L1 may or may not be regular
- If L1 . L2 is regular ⇒ L2 may or may not be regular
- If L1 . L2 is Non-Regular ⇒ Atleast one of them should be Non-regular.
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