What happens when apply reversal to the empty language?

Question

Reversal changes start states to final states and final states to start states, and changes the direction of arrows. The empty language has no final states, so does this reversal creates something that does not exist, and if so, is the reversal of the empty language the only way to achieve this? To clarify I'm talking about {} and NOT the language containing the empty string {ε}.

Answer 1

There is some confusion in your question, as your definition of reversal applies to nondeterministic automata and not to languages.

In a more formal way, if $\mathcal A = (Q, A, E, I, F)$ is a nondeterministic automaton, where $Q$ is the set of states, $A$ the alphabet, $I$ the set of initial states, $F$ the set of final states and $E \subseteq Q \times A \times Q$ is the set of transitions, then the reversal of $\mathcal A$ is $\mathcal A^r = (Q, A, E^r, F, I)$, where $E^r = \{(q, a, p) \in Q \times A \times Q \mid (p,a,q) \in E\}$.

If you start with a language, I suppose you want to consider the reversal of its minimal deterministic automaton. In your case, the minimal DFA of the empty language has exactly one state, which is initial but not final, and no transitions. Formally, $\mathcal A = (\{q\}, A, \emptyset, \{q\}, \emptyset)$. Thus its reversal is $\mathcal A^r = (\{q\}, A, \emptyset, \emptyset, \{q\})$. This NFA has no initial state (but it is well defined anyway) and it accepts the empty language.