Automata, Boolean Matrices, and Ultimate Periodicity

This paper presents a Boolean-matrix-based method to automata theory, with an application to the study of regularity-preserving functions. A new characterization of such functions is derived in terms of the property of ultimate periodicity with respect to powers of Boolean matrices. This characterization reveals the intrinsic algebraic nature of regularity-preserving functions. It facilitates a concise proof of known, as well as previously unknown, properties of regularity-preserving functions, leading to the solution of the “subtraction problem,” left open by Kosaraju.