Minimal deterministic finite automaton

An irreducible deterministic automaton has the fewest states among deterministic automata for its language and is unique up to isomorphism. Its states are the Myhill--Nerode classes.

The canonical residual automaton of a regular language has its distinct left quotients of a formal language as states, initial state $L$, accepting states those containing the empty word, and letter transition $R\mapsto a^{-1}R$. It is a complete minimal deterministic finite automaton; every other complete minimal deterministic finite automaton for that language is isomorphic to it.

In a minimal deterministic finite automaton accepting a finite language, any state lying on a nonempty directed cycle is the unique rejecting sink state. It cannot reach an accept state, since traversing the cycle arbitrarily many times would produce infinitely many accepted words. Every state reachable from it also rejects every continuation, so minimality forces all such states to be the same state.

For a positive integer $m$, binary strings can be tested for divisibility by $m$ with residue states $0,\dots ,m-1$ and transitionThe initial and accepting residue is zero. For $m=7$, all seven states are reachable and pairwise distinguishable, so this automaton is minimal.

An accessible automaton over one letter consists of a directed tail entering one directed cycle. Its minimal quotient is obtained by merging positions having the same future binary acceptance sequence.