Classification of Finite Dynamical Systems

TL;DR

This paper classifies finite dynamical systems on binary strings based on coordinate dependencies, introduces a notion of linearization, and provides an upper bound on the number of such systems, advancing the mathematical understanding of these systems.

Contribution

It offers a new classification scheme for finite dynamical systems and establishes a sharp upper bound related to coordinate dependencies.

Findings

Classification based on dependency relations among coordinate functions

Introduction of a natural notion of linearization

Sharp upper bound on the number of systems

Abstract

This paper is motivated by the theory of sequential dynamical systems, developed as a basis for a mathematical theory of computer simulation. It contains a classification of finite dynamical systems on binary strings, which are obtained by composing functions defined on the coordinates. The classification is in terms of the dependency relations among the coordinate functions. It suggests a natural notion of the linearization of a system. Furthermore, it contains a sharp upper bound on the number of systems in terms of the dependencies among the coordinate functions. This upper bound generalizes an upper bound for sequential dynamical systems.