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We examine the NFA minimization problem in terms of atomic NFA's, that is,NFA's in which the right language of every state is a union of atoms, where theatoms of a regular language are non-empty intersections of complemented anduncomplemented left quotients of the language. We characterize all reducedatomic NFA's of a given language, that is, those NFA's that have no equivalentstates. Using atomic NFA's, we formalize Sengoku's approach to NFA minimizationand prove that his method fails to find all minimal NFA's. We also formulatethe Kameda-Weiner NFA minimization in terms of quotients and atoms.

Minimal Nondeterministic Finite Automata and Atoms of Regular Languages