The concept of closure is fundamental in both mathematics and computer science.
A set is considered closed under an operation if performing that operation on elements of the set results in an element that is also within the set.
Regular languages exhibit closure properties with respect to several operations, which are crucial for understanding their behavior in computation.
Complement
Definition: If ( L ) is a language over a finite alphabet, then the complement of ( L ) contains all strings over the alphabet not in ( L ).
Theorem 2: Regular languages are closed under the complement operation.
Proof: For a regular language ( L ) recognized by a deterministic finite automata (DFA) ( M ):
Construct a DFA ( M' ) with the same states, alphabet, and transitions as ( M ).
( M' ) retains the same start state.
The final states of ( M' ) are all non-final states of ( M ).
This construction ensures ( M' ) recognizes the complement of ( L ).
Example: A DFA for a unary language accepting strings whose lengths are multiples of 3. Its complement accepts strings whose lengths are not multiples of 3.
Union
Definition: The union of two languages ( L_1 ) and ( L_2 ) over finite alphabets ( \Sigma_1 ) and ( \Sigma_2 ) contains all strings that are in either ( L_1 ) or ( L_2 ).
Theorem 3: Regular languages are closed under finite union.
Proof: Given DFAs recognizing ( L_1 ) and ( L_2 ):
Construct DFA ( M ) with state set as the Cartesian product of the sets of ( L_1 ) and ( L_2 ).
Use the union of the two alphabets.
Define transitions that track states through the input string.
Accept strings if any final state of ( L_1 ) or ( L_2 ) is reached.
Example: DFA states and transitions for the union of two regular languages ( L_1 ) and ( L_2 ).
Intersection
Definition: The intersection of two languages ( L_1 ) and ( L_2 ) contains strings present in both ( L_1 ) and ( L_2 ).
Theorem 4: Regular languages are closed under finite intersection.
Proof: Similar to union, use Cartesian product for the states.
Accept a string if final states from both ( L_1 ) and ( L_2 ) are reached simultaneously.
Example: DFA construction for the intersection of two regular languages.
Concatenation
Definition: The concatenation of two languages ( L_1 ) and ( L_2 ) consists of strings formed by taking a string from ( L_1 ) followed by a string from ( L_2 ).
Theorem 5: Regular languages are closed under concatenation.
Proof: Exists for both DFAs and non-deterministic finite state automata (NFA).
Reference to Theorems 1.26 and 1.47 from Michael Sipser's textbook.
Kleene Star
Definition: The Kleene star of a language ( L ) includes all strings that can be formed by concatenating zero or more strings from ( L ).
Theorem 6: Regular languages are closed under Kleene star.
Proof: Exists for both DFAs and NFAs.
Reference to Theorem 1.49 from Michael Sipser's textbook.
Conclusion
Understanding these closure properties helps in the analysis and design of algorithms and automata for regular languages.