Computational Dynamical Systems

TL;DR

This paper explores the computational limits of smooth dynamical systems, showing that certain classes cannot simulate universal Turing machines and establishing links between system stability, decidability, and complexity.

Contribution

It provides formal definitions for simulation of Turing machines by dynamical systems and characterizes which systems can or cannot perform universal computation.

Findings

Chaotic and integrable systems cannot robustly simulate universal Turing machines.

Structurally stable 1D systems encode only decidable halting problems.

Highlights the importance of encoding schemes in computational dynamical systems.

Abstract

We study the computational complexity theory of smooth, finite-dimensional dynamical systems. Building off of previous work, we give definitions for what it means for a smooth dynamical system to simulate a Turing machine. We then show that 'chaotic' dynamical systems (more precisely, Axiom A systems) and 'integrable' dynamical systems (more generally, measure-preserving systems) cannot robustly simulate universal Turing machines, although such machines can be robustly simulated by other kinds of dynamical systems. Subsequently, we show that any Turing machine that can be encoded into a structurally stable one-dimensional dynamical system must have a decidable halting problem, and moreover an explicit time complexity bound in instances where it does halt. More broadly, our work elucidates what it means for one 'machine' to simulate another, and emphasizes the necessity of defining low-complexity 'encoders' and 'decoders' to translate between the dynamics of the simulation and the system being simulated. We highlight how the notion of a computational dynamical system leads to questions at the intersection of computational complexity theory, dynamical systems theory, and real algebraic geometry.