Correspondences in computational and dynamical complexity II: forcing complex reductions

TL;DR

This paper explores the relationship between algebraic telic problems in dynamical systems and their computational complexity, establishing that reductions imply similar system behavior and identifying barriers for algorithmic solutions.

Contribution

It introduces a framework linking reductions between telic problems to dynamical system behavior and demonstrates explicit computational barriers for complex systems.

Findings

Reductions imply similar dynamical behavior.

Certain telic problems cannot be decided by simple arithmetic circuits.

Explicit complexity barriers for systems with positive topological entropy.

Abstract

An algebraic telic problem is a decision problem in $\textsf{NP}_\mathbb{R}$ formalizing finite-time reachability questions for one-dimensional dynamical systems. We prove that the existence of "natural" mapping reductions between algebraic telic problems coming from distinct dynamical systems implies the two dynamical systems exhibit similar behavior (in a precise sense). As a consequence, we obtain explicit barriers for algorithms solving algebraic telic problems coming from complex dynamical systems, such as those with positive topological entropy. For example, some telic problems cannot be decided by uniform arithmetic circuit families with only $+$ and $\times$ gates.