Classification of Discrete Dynamical Systems Based on Transients

TL;DR

This paper introduces a new classification method for deterministic discrete dynamical systems based on their transient behavior, identifying phase transitions from order to chaos, and applied to cellular automata, Turing machines, and Boolean networks.

Contribution

A novel, broadly applicable classification technique based on asymptotic computation time behavior, enabling analysis of complex dynamics and phase transitions in various deterministic systems.

Findings

Identified a critical region indicating phase transition from order to chaos.

Successfully classified cellular automata, Turing machines, and Boolean networks.

Enabled automatic detection of complex dynamics in 2D cellular automata.

Abstract

In order to develop systems capable of artificial evolution, we need to identify which systems can produce complex behavior. We present a novel classification method applicable to any class of deterministic discrete space and time dynamical systems. The method is based on classifying the asymptotic behavior of the average computation time in a given system before entering a loop. We were able to identify a critical region of behavior that corresponds to a phase transition from ordered behavior to chaos across various classes of dynamical systems. To show that our approach can be applied to many different computational systems, we demonstrate the results of classifying cellular automata, Turing machines, and random Boolean networks. Further, we use this method to classify 2D cellular automata to automatically find those with interesting, complex dynamics. We believe that our work can be used to design systems in which complex structures emerge. Also, it can be used to compare various versions of existing attempts to model open-ended evolution (Ray (1991), Ofria et al. (2004), Channon (2006)).