I'm a newbee learning DFA minimization. And I found that(strangely) Brzozowki's algorithm cannot give me a minimized DFA on this example:
In this DFA, $S_0$ and $S_1$ are nondistinguishable and should be merged after minimization, but they are not.
The process of minimization is like this:
I tried to find out why by reading @Hendrik Jan's proof here, Proof of Brzozowski's algorithm for DFA minimization?
However, I find that this case bypasses the proof, because that in the reversed DFA, the initial state $q_0$ cannot be reached by $q_0$ using any words (but initial state is always considered reachable and cannot be removed). So did I miss anything, or does Brzozowki's algorithm has any specification for this corner case?
(It bypasses the proof like this: in the second series of reversal-determinization, $S_0 = \{q_1\}$ and $S_1 = \{q_0, q_1\}$, for any $w$, $\delta(q_0, w)$ in $S_0$ iff $\delta(q_0, w)$ in $S1$, so we say $S_0 = S_1$, assuming all states can be reached by $\delta(q_0, w)$, but $\delta(q_0,w)$ will never be $q_0$)