Graph Spectral Properties of Deterministic Finite Automata

TL;DR

This paper explores the spectral properties of minimal automata, establishing a link between automaton structure and matrix rank, and introduces the concept of rank-one languages with specialized automata.

Contribution

It proves that minimal automata have minimal adjacency matrix rank and nullity, and introduces rank-one languages and their expanded canonical automata.

Findings

Minimal automata have minimal adjacency matrix rank and nullity.

Introduction of rank-one languages with automata of rank one.

Definition of expanded canonical automata for rank-one languages.

Abstract

We prove that a minimal automaton has a minimal adjacency matrix rank and a minimal adjacency matrix nullity using equitable partition (from graph spectra theory) and Nerode partition (from automata theory). This result naturally introduces the notion of matrix rank into a regular language L, the minimal adjacency matrix rank of a deterministic automaton that recognises L. We then define and focus on rank-one languages: the class of languages for which the rank of minimal automaton is one. We also define the expanded canonical automaton of a rank-one language.