Closure properties of Regular languages

A closure property is a characteristic of a class of languages (such as regular, context-free, etc.) where applying a specific operation (like union, intersection, concatenation, etc.) to languages within that class results in a language that is also within the same class.

Closure refers to some operation on a language, resulting in a new language that is of the same "type" as originally operated on i.e., regular. Regular languages are closed under the following operations:

Consider that 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. 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).
6. 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.
7. 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.
8. 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).
9. 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:

• Step-1: select the state that cannot be reached from the initial states & delete them (remove unreachable states).
• Step 2: if the resulting machine contains at least one final states, so then the finite automata accepts the non-empty language.
• Step 3: if the resulting machine is free from final state, then finite automata accepts empty language.
  • Step-1: select the state that cannot be reached from the initial state & delete them (remove unreachable states).
  • Step-2: select the state from which we cannot reach the final state & delete them (remove dead states).
  • Step-3: if the resulting machine contains loops or cycles then the finite automata accepts infinite language.
  • Step-4: if the resulting machine do not contain loops or cycles then the finite automata accepts finite language.