Closure properties of Regular languages

Closure properties on regular languages are defined as certain operations on regular language which are guaranteed to produce regular language. Closure refers to some operation on a language, resulting in a new language that is of same “type” as originally operated on i.e., regular.

Regular languages are closed under following operations.

Consider L and M are regular languages:



  1. Kleen Closure:
    RS is a regular expression whose language is L, M. R* is a regular expression whose language is L*.

  2. Positive closure:
    RS is a regular expression whose language is L, M. R^+ is a regular expression whose language is L^+.

  3. 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.

  4. Reverse Operator:
    Given language L, L^R is the set of strings whose reversal is in L.
    Example: L = {0, 01, 100};
    L^R ={0, 10, 001}.
    Proof: Let E be a regular expression for L. We show how to reverse E, to provide a regular expression E^R for L^R.

  5. 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.

  6. 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).

  7. Intersection:
    Let L and M be the languages of regular expressions R and S, respectively then it a regular expression whose language is L intersection M.
    proof: Let A and B be DFA’s whose languages are L and M, respectively. Construct C, the product automaton of A and B make the final states of C be the pairs consisting of final states of both A and B.

  8. Set Difference operator:
    If L and M are regular languages, then so is L – M = strings in L but not M.

    Proof: Let A and B be DFA’s whose languages are L and M, respectively. Construct C, the product automaton of A and B make the final states of C be the pairs, where A-state is final but B-state is not.

  9. Homomorphism:
    A homomorphism on an alphabet is a function that gives a string for each symbol in that alphabet. Example: h(0) = ab; h(1) = E. Extend to strings by h(a1…an) =h(a1)…h(an). Example: h(01010) = ababab.


    If L is a regular language, and h is a homomorphism on its alphabet, then h(L)= {h(w) | w is in L} is also a regular language.
    Proof: Let E be a regular expression for L. Apply h to each symbol in E. Language of resulting R, E is h(L).

  10. 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}.

Note: There are few more properties like symmetric difference operator, prefix operator, substitution which are closed under closure properties of regular language.

Decision Properties:
Approximately all the properties are decidable in case of finite automaton.

(i) Emptiness 
(ii) Non-emptiness 
(iii) Finiteness 
(iv) Infiniteness 
(v) Membership 
(vi) Equality  

These are explained as following below.

(i) Emptiness and Non-emptiness: