Minimal Nondeterministic Finite Automata and Atoms of Regular Languages

TL;DR

This paper explores atomic NFA's for regular languages, characterizes their minimal forms, and analyzes existing minimization methods, revealing limitations in Sengoku's approach and reformulating Kameda-Weiner's method.

Contribution

It provides a comprehensive characterization of reduced atomic NFA's and formalizes existing NFA minimization techniques, highlighting their limitations.

Findings

Sengoku's method does not find all minimal NFA's.

Characterization of all reduced atomic NFA's.

Reformulation of Kameda-Weiner minimization in terms of quotients and atoms.

Abstract

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 the atoms of a regular language are non-empty intersections of complemented and uncomplemented left quotients of the language. We characterize all reduced atomic NFA's of a given language, that is, those NFA's that have no equivalent states. Using atomic NFA's, we formalize Sengoku's approach to NFA minimization and prove that his method fails to find all minimal NFA's. We also formulate the Kameda-Weiner NFA minimization in terms of quotients and atoms.