Indistinguishable states of a deterministic finite automaton

Two states are indistinguishable when every continuation is accepted from both or rejected from both. Distinguishable states admit at least one suffix with different acceptance outcomes.

Indistinguishability is a right-invariant equivalence relation. The quotient has states $[q]$, transition $[q]\mapsto [\delta (q,a)]$, initial state $[q_0]$, and accepting classes represented by accepting states. It accepts the original language and has no two distinct indistinguishable states.