We show that every regular language defines a unique nondeterministic finite automaton (NFA), which we call "átomaton", 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 theátomaton, 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.

The definition in [3] does not consider the intersection of all the complemented quotients to be an atom. Our new definition adds symmetry to the theory.

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The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs of a given language, i.e., those NFAs that have no equivalent states. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2 n − 1 if r = 0 or r = n; for 1 r n − 1 the bound isFor each n 2, we exhibit a language with 2 n atoms which meet these bounds.Keywords: atoms, finite automaton, atomic NFA, quotient complexity, regular language, state complexity, syntactic semigroup, witness 2000 Mathematics Subject Classification: 68Q45, 68Q19, 68Q70 Terminology and NotationIn this section we provide some background information, introduce atoms of regular languages, and state our reasons for studying them. For basic properties of regular languages and finite automata see [7,9]. If Σ is a non-empty finite alphabet, then Σ * is the free monoid generated by Σ. A word is any element of Σ * , and the empty word is ε. A language over Σ is any subset of Σ * . The reverse of a language L is denoted by L R and defined as L R = {w R | w ∈ L}, where w R is w spelled backwards. The (left) quotient of a regular language L over an alphabet Σ by a word w ∈ Σ * is the language w −1 L = {x ∈ Σ * | wx ∈ L}. It is well known that the quotient of

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