Semipredictable dynamical systems

TL;DR

This paper introduces semipredictable dynamical systems, a class of deterministic systems with both predictable and unpredictable traits, and provides a decomposition framework for cellular automata based on their state divisors.

Contribution

It proves a decomposition theorem for nonlinear cellular automata into layers based on divisors of the state space size, and shows how certain traits can be predicted despite overall unpredictability.

Findings

Decomposition of CA dynamics into layers involving divisors of the number of states.

Existence of systems where some traits are exactly predictable despite overall unpredictability.

Explicit examples using Wolfram's elementary CA and complex CA rules.

Abstract

A new class of deterministic dynamical systems, termed semipredictable dynamical systems, is presented. The spatiotemporal evolution of these systems have both predictable and unpredictable traits, as found in natural complex systems. We prove a general result: The dynamics of any deterministic nonlinear cellular automaton (CA) with $p$ possible dynamical states can be decomposed at each instant of time in a superposition of $N$ layers involving $p_{0}$, $p_{1}$,... $p_{N-1}$ dynamical states each, where the $p_{k\in \mathbb{N}}$, $k \in [0, N-1]$ are divisors of $p$. If the divisors coincide with the prime factors of $p$ this decomposition is unique. Conversely, we also prove that $N$ CA working on symbols $p_{0}$, $p_{1}$,... $p_{N-1}$ can be composed to create a graded CA rule with $N$ different layers. We then show that, even when the full spatiotemporal evolution can be unpredictable, certain traits (layers) can exactly be predicted. We present explicit examples of such systems involving compositions of Wolfram's 256 elementary CA and a more complex CA rule acting on a neighborhood of two sites and 12 symbols and whose rule table corresponds to the smallest Moufang loop $M_{12}(S_{3},2)$.