Often I read this sentence about DFA:
The empty string $\epsilon$ distinguishes any accept state from any reject state.
Here the source: http://pages.cs.wisc.edu/~shuchi/courses/520-S08/handouts/Lec7.pdf
I don't understand the above statement, since by definition a DFA can't have $\epsilon$ transitions. Please can you explain me better? Many thanks!
That claim has nothing to do with $\epsilon$-transitions. What is says intuitively is that even before you read any input letter, you can tell an accepting state from a non-accepting state.
In minimizing DFAs, we look for indistinguishable states. Two states are indistinguishable if the same language is accepted starting from both. To compute the indistinguishable pairs, we define $k$-indistinguishable states; that is, those states from which the words of length up to $k$ accepted from the two states are exactly the same. As a special case, two states are $0$-indistinguishable if and only if they are either both accepting or both rejecting. This is what those lecture notes presumably talk about.
