Decidability and Universality in Symbolic Dynamical Systems

TL;DR

This paper introduces a unified definition of universality for symbolic dynamical systems based on undecidability, providing new insights into cellular automata, subshifts, and chaotic systems.

Contribution

It proposes a general, robust definition of universality applicable to various dynamical systems, extending classical notions to cellular automata and subshifts.

Findings

Universal systems must have a sensitive point and a proper subsystem.

Necessary conditions for undecidability include certain structural features.

A universal chaotic system is explicitly constructed.

Abstract

Many different definitions of computational universality for various types of dynamical systems have flourished since Turing's work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as undecidability of a model-checking problem. For Turing machines, counter machines and tag systems, our definition coincides with the classical one. It yields, however, a new definition for cellular automata and subshifts. Our definition is robust with respect to initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for undecidability and universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have infinite number of subsystems. We also discuss the thesis according to which computation should occur at the `edge of chaos' and we exhibit a universal chaotic system.