evinda said:
We suppose that $L$ is regular. Then there is a dfa that recognizes $L$.
Assume that $q_0$ is the starting state and $q_n$ is an accepting state, where $n \in \mathbb{N}$.
A DFA can have several accepting states.
evinda said:
Taking as $q_n$ the starting state and as $q_0$ the accepting state, we get a dfa that recognizes $L^R=\{ w | w^R \in L \}$.
No need to reverse transitions?
evinda said:
Since $L$ is regular , there is a finite automaton $M=(Q, \Sigma, \delta, q_0, F)$ that recognizes it.
I assume it's a DFA, not an NFA.
evinda said:
Could you explain to me why it has to hold that $Q'=\mathcal{P}(Q)$?
It does not. The author of the proof chose to give this definition. Another person can give a different constructions for the automaton that accepts $L^R$ and come up with a different proof. Strictly speaking, questions about definitions are illegitimate. Only questions like "Why does something hold?" or "Why does this follow from that?" are legitimate.
That said, the idea behind the suggested proof is to reverse the transitions of the original automaton $M$. However, it may happen that $\delta(q,a)=q'$ for several $q$ and given $a$, $q'$. Therefore, we can't define $\delta'(q',a)=q$ because there are several $q$s to choose from. In other words, the automaton that acceots $L^R$ is nondeterministic in general. Therefore, this proof combines the idea of reversing transitions with converting the resulting NFA into a DFA, where the key idea is to consider sets of states of NFA as individual states of the DFA.
Another legitimate question is
what to prove. Most of claims about finite automata is ultimately proved by induction, and induction often requires strengthening the inductive hypothesis. Coming up with such hypothesis may be nontrivial. Suppose that $\delta$ is extended from $Q\times\Sigma\to Q$ to $Q\times\Sigma^*\to Q$, and similarly for $\delta'$, in the natural way. Then one proves by induction on word length that
\[
\{q\in Q\mid\delta(q,w)\in F\}=\delta'(q_0',w^R)\text{ for all }w\in\Sigma^*.\qquad(*)
\]
I would also change the definitions of $F'$ and $\delta'$.
\[
F'=\{P\subseteq Q\mid q_0\in P\}\\
\delta'(P,a)=\{q\in Q\mid\delta(q,a)\in P\}
\]
Then the required statement is a corollary of (*).
\[
w\in L\iff\delta(q_0,w)\in F\overset{(*)}{\iff} q_0\in\delta'(q_0',w^R)\iff\delta'(q_0',w^R)\in F'\iff w^R\text{ is accepted by }M'
\]
But as I said, this proof basically repeats the proof of the theorem about the equivalence of NFAs and DFAs. It may be easier to start with a nondeterministic $M$. One can assume without loss of generality that an NFA has a single accepting state. Then we simply reverse the transitions.
Yet another way to prove that $L^R$ is regular is to construct a regular expression that describes $L^R$ from the one that describes $L$.