If a language \( L \) is regular, then its reverse \( L^R \), defined as \( L^R = \{ w^R \mid w \in L \} \), is also regular. This can be demonstrated in two main ways. First, by utilizing a regular expression \( r \) that generates \( L \), one can construct a new expression for \( L^R \) through recursive transformation based on the structure of \( r \). Second, by taking a deterministic finite automaton (DFA) that accepts \( L \), one can reverse the direction of all transitions, introduce a new initial state, and create \( \varepsilon \)-transitions from this new state to all original accepting states. Both methods effectively show that the reverse of a regular language remains regular.
