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Math and science::Theory of Computation

Regular language operators

Let \( A \) and \( B \) be languages. We define the regular operations union, concatenation and star as follows:

Union
\( A \cup B = \{x \mid x \in A \lor x \in B\} \)  (i.e. as per standard axiom of union for sets)
Concatenation
\( A \circ B = \{ xy \mid x \in A \land y \in B \} \)
Star
\( A^\star = \{ x_1x_2...x_k \mid k \ge 0 \text{ and each } x_i \in A \} \)

The class of regular languages is closed under these three operations!


Visual Justification

It is quite easy to grasp how regular languages are closed under the above three operations. To do so, consider the state diagrams representing two nondeterministic finite automata \( N_1 \) and \( N_2 \) which accepts languages \( A_1 \) and \( A_2 \) respectively. We can combine them in different ways to achieve union, concatenation and star operations.

Union, \( A_1 \cup A_2 \)

The set of start states is the union of the start states of \( N_1 \) and \( N_2 \).

Concatenation, \( A_1 \circ A_2 \)

Each accept state of \( N_1 \) has an ε-transition to the start state of \( N_2 \).

Star operator, \( A_1^{\star} \)

Each accept state of \( N_1 \) has an ε-transition to the start state of \( N_1 \).

Source

Introduction to the Theory of Computation, Sipser
p44