# Brzozowki's algorithm doesn't work for this corner case

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 $$q_0$$)

the initial DFA I was working on:

Rici made things clearer as he commented "$$S_0$$ is not a final state and $$S_1$$ is. So they cannot be merged", since "the merged DFA accepts $$\epsilon$$. The original one does not. So they are not equivalent".