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Correspondences in computational and dynamical complexity I

Samuel Everett

TL;DR

This work builds a bridge between dynamical systems and computational complexity by associating telic decision problems to efficiently discretizable dynamical systems. It shows that exponential-time lower bounds for telic problems imply positive topological entropy, and that reductions and conjugacies preserve or translate hardness in principled ways, enabling a dynamical classification via algorithmic tractability. The development includes a rigorous framework for discretizing state spaces, structuring telic problems, and proving a dichotomy between pure and impure sets, with a topological-dichotomy result connecting dynamical behavior to complexity. The results provide a new toolkit for diagnosing dynamical properties through computational lower bounds, with potential implications for reductions, invariants, and the speculative link between hard telic problems and cryptographic primitives.

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 $2^{Ω(n)}$, 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.

Correspondences in computational and dynamical complexity I

TL;DR

This work builds a bridge between dynamical systems and computational complexity by associating telic decision problems to efficiently discretizable dynamical systems. It shows that exponential-time lower bounds for telic problems imply positive topological entropy, and that reductions and conjugacies preserve or translate hardness in principled ways, enabling a dynamical classification via algorithmic tractability. The development includes a rigorous framework for discretizing state spaces, structuring telic problems, and proving a dichotomy between pure and impure sets, with a topological-dichotomy result connecting dynamical behavior to complexity. The results provide a new toolkit for diagnosing dynamical properties through computational lower bounds, with potential implications for reductions, invariants, and the speculative link between hard telic problems and cryptographic primitives.

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 , 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.
Paper Structure (23 sections, 16 theorems, 41 equations, 1 figure)

This paper contains 23 sections, 16 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the main results of this paper
  3. Historical context and motivation
  4. Structure of the work
  5. Preliminaries
  6. Dynamical systems background
  7. Computational complexity background and notation
  8. Discretizing dynamical systems
  9. Notation
  10. Discretizing state space
  11. Discretizations of dynamical systems in practice
  12. Finite telic problems
  13. Telic problems and first results
  14. Basic properties of telic problems
  15. Reductions and transformations between telic problems
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $(X, T)$ be an efficiently discretizable topological dynamical system. If $(X, T)$ admits a telic problem such that every decider for the language runs in time $2^{\Omega(n)}$, then the topological entropy of the system is positive.

Figures (1)

  • Figure 1: A representation of the central idea of telic problems. A map $h^{(i)}$ carries points from the unit dyadic rationals (tuples of binary strings of length $i$) into points in state space in an orderly fashion. The problem is to decide whether one of the points --- $\hat{x}$ in this case --- evolves to a region nearby or inside of the target set $B^{(i)}_{v(i)}$ after $i$ of iterations of the discretization of the map.

Theorems & Definitions (47)

  • Theorem 1.1: \ref{['thm:entropyHardness']}
  • Theorem 1.2: \ref{['thmTopDichotomy']}
  • Theorem 1.3: \ref{['thmConjugacyHardness']}
  • Definition 2.1
  • Definition 2.2
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.5
  • ...and 37 more