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Let $\hat{\delta }$ be the extended transition function of a deterministic finite automaton. States $p,q\in Q$ are equivalent, or indistinguishable, whenThe quotient deterministic finite automaton by indistinguishable states $D/\sim$ has state set $Q/\sim$, initial state $[q_0]$, accepting states $\{[q]:q\in F\}$, and transitionRight invariance of $\sim$ makes this well-defined, and the quotient accepts $L(D)$.

To prove minimality, choose for each state $q$ of the accessible automaton $D$ a word $w_q$ with ${\hat{\delta }}_D(q_0,w_q)=q$. Run the same word in any DFA $A$ accepting $L(D)$, and assign the class $[q]$ the reached state of $A$. If two distinct classes $[p]\ne [q]$ reached the same state of $A$, every continuation $u$ would have the same acceptance result after $w_p$ and $w_q$. Since $A$ and $D$ accept the same language, this would make $p$ and $q$ indistinguishable in $D$, a contradiction. Thus $A$ has at least one distinct state for every class in $D/\sim$, proving that the quotient is the minimal deterministic finite automaton.

For divisibility by seven, take states $Q=\{0,1,\dots ,6\}$, initial state $0$, accepting set $F=\{0\}$, and transitionsAfter reading a word, the state is its binary value modulo seven, so this binary divisibility automaton accepts exactly the multiples of seven, with leading zeros allowed. Every state is accessible: the three-bit words representing $0,1,\dots ,6$ reach the corresponding residues.

For distinct residues $p,q$, let $w_p$ be the three-bit representation of $-p(\mathrm{mod}7)$, chosen in $\{0,\dots ,6\}$. Since $2^3\equiv 1(\mathrm{mod}7)$, reading $w_p$ from state $p$ ends atwhereas reading it from $q$ ends at $q-p\ne 0$. Thus $w_p$ distinguishes $p$ from $q$. All seven states are pairwise distinguishable, so the DFA is minimal by the Myhill-Nerode theorem.

A deterministic finite automaton is a tuple $(Q,\Sigma ,\delta ,q_0,F)$ with finite state set $Q$, finite input alphabet $\Sigma$, transition function $\delta :Q\times \Sigma \to Q$, initial state $q_0$, and accepting-state set $F\subseteq Q$. It accepts a word when the extended transition function of a deterministic finite automaton carries $q_0$ to a state in $F$. A regular language is a language accepted by some deterministic finite automaton.

The pumping lemma for regular languages says that if a regular language is accepted by an automaton with $N$ states, every accepted word $w$ of length at least $N$ has a decomposition $w=xyz$ with $|xy|\le N$, $|y|>0$, and $x{y}^{i}z$ in the language for every $i\ge 0$. Indeed, among the $N+1$ states visited before and after the first $N$ symbols, two are equal by the pigeonhole principle. The intervening nonempty word $y$ labels a loop, which may be traversed any number of times without changing the final accepting state.

In base two, the powers of two have representations $10^n$, so their language is the regular expression $10^{\ast }$ and is regular.

Suppose instead that their base-ten representations formed a regular language. Apply the pumping lemma to a sufficiently long decimal power of two $xyz$, and put $d=|y|>0$. Pumping gives decimal integers $a_i$ represented by $x{y}^{i}z$, all powers of two. A direct place-value calculation shows thatfor a constant $C$ independent of $i$. The lengths, and hence the exponents in $a_i=2^{n_i}$, tend to infinity. Dividing by $a_i$ givesFor large $i$, the right side has absolute value less than one while the left side is an integer, so it must vanish. This would make a power of two equal to $10^d=2^d{5}^d$, impossible for $d>0$ by unique prime factorization. Thus the decimal powers of two are not a regular language.

For a deterministic finite automaton the extended transition function of a deterministic finite automaton is recursivelyThe accepted language isFor a nondeterministic finite automaton, $\hat{\delta }(q,w)$ is the set of all states reachable from $q$ while reading $w$; recursively,Thus

The powerset construction has state set $\mathcal{P}(Q)$, initial state $\{q_0\}$, transitionand accepting states $\{S\subseteq Q:S\cap F\ne \varnothing \}$. Induction on $|w|$ givesso the two automata accept exactly the same language. If $|Q|=m$ and $F$ has one state, exactly $2^{m-1}$ of the $2^m$ subset states contain it and are accepting, including states that may be unreachable.

Extend $\delta :Q\times \Sigma \to Q$ tobyThe extended transition function of a deterministic finite automaton processes a whole word from left to right. The language accepted by $D$ is