We show that every regular language defines a unique nondeterministic finite
automaton (NFA), which we call "\'atomaton", whose states are the "atoms" of
the language, that is, non-empty intersections of complemented or
uncomplemented left quotients of the language. We describe methods of
constructing the \'atomaton, and prove that it is isomorphic to the reverse
automaton of the minimal deterministic finite automaton (DFA) of the reverse
language. We study "atomic" NFAs in which the right language of every state is
a union of atoms. We generalize Brzozowski's double-reversal method for
minimizing a deterministic finite automaton (DFA), showing that the result of
applying the subset construction to an NFA is a minimal DFA if and only if the
reverse of the NFA is atomic. We prove that Sengoku's claim that his method
always finds a minimal NFA is false.Comment: 29 pages, 2 figures, 28 table
