I am looking for an example, which corresponds to what I've learned in my Applied Automata Theory Class:
Given a NFA $\mathcal{A}$,
- a $\approx _\mathcal{A}$ quotient automaton can be bigger then a $\sim_\mathcal{A}$ quotient automaton and this can be bigger than an optimal quotient
- and an optimal quotient does not need to be the smallest equivalent NFA.
$\approx _\mathcal{A}$ denotes the bisimulation relation (block refinement algorithm) and $\sim_\mathcal{A}$ denotes the canonical congruence, where $p \sim_\mathcal{A} q \text{ iff } \delta^*(p,w) \in F \text{ iff } \delta^*(q,w) \in F$, where DFA $\mathcal{A}= (Q, \Sigma, q_0, \delta, F)$.