Correspondences in computational and dynamical complexity I

TL;DR

This paper introduces a novel approach linking computational complexity to dynamical systems, showing that high complexity in associated decision problems implies positive topological entropy, thus enabling classification of dynamical behaviors.

Contribution

It develops a method connecting computational complexity bounds with dynamical properties, specifically associating telic problems to systems and deriving entropy implications.

Findings

High complexity telic problems imply positive topological entropy.

Methods for classifying systems via complexity bounds on telic problems.

Establishment of reductions between telic problems from different systems.

Abstract

We begin development of a method for studying dynamical systems using concepts from computational complexity theory. We associate families of decision problems, called telic problems, to dynamical systems of a certain class. These decision problems formalize finite-time reachability questions for the dynamics with respect to natural coarse-grainings of state space. Our main result shows that complexity-theoretic lower bounds have dynamical consequences: if a system admits a telic problem for which every decider runs in time $2^{\Omega(n)}$, then it must have positive topological entropy. This result and others lead to methods for classifying dynamical systems through proving bounds on the runtime of algorithms solving their associated telic problems, or by constructing polynomial-time reductions between telic problems coming from distinct dynamical systems.