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Correspondences in computational and dynamical complexity II: forcing complex reductions

Samuel Everett

TL;DR

This work develops a framework linking reductions between algebraic telic problems—finite-time reachability questions derived from computable one-dimensional dynamical systems—to dynamical conjugacy. It introduces four levels of natural reductions and analyzes when such reductions can or cannot exist between telic problems arising from dynamically distinct systems, including chaotic vs. regular ones. The key results show explicit barriers: level-1 and level-2 natural reductions fail for mismatched systems, higher-level reductions (level-4 and level-4a) must exhibit highly irregular structure in many cases, and some telic problems resist even uniform arithmetic-circuit-based reductions. By embedding dynamical properties (like entropy) into the reduction framework, the paper separates the intrinsic complexity of dynamics from the algorithmic techniques available to solve telic problems, yielding lower bounds and barriers that connect dynamical structure to real-number computation. The findings illuminate how dynamical richness imposes fundamental limits on reductions and, consequently, on the efficiency of algorithms solving telic decision and search problems in the real-number setting.

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.

Correspondences in computational and dynamical complexity II: forcing complex reductions

TL;DR

This work develops a framework linking reductions between algebraic telic problems—finite-time reachability questions derived from computable one-dimensional dynamical systems—to dynamical conjugacy. It introduces four levels of natural reductions and analyzes when such reductions can or cannot exist between telic problems arising from dynamically distinct systems, including chaotic vs. regular ones. The key results show explicit barriers: level-1 and level-2 natural reductions fail for mismatched systems, higher-level reductions (level-4 and level-4a) must exhibit highly irregular structure in many cases, and some telic problems resist even uniform arithmetic-circuit-based reductions. By embedding dynamical properties (like entropy) into the reduction framework, the paper separates the intrinsic complexity of dynamics from the algorithmic techniques available to solve telic problems, yielding lower bounds and barriers that connect dynamical structure to real-number computation. The findings illuminate how dynamical richness imposes fundamental limits on reductions and, consequently, on the efficiency of algorithms solving telic decision and search problems in the real-number setting.

Abstract

An algebraic telic problem is a decision problem in 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 gates.
Paper Structure (13 sections, 16 theorems, 31 equations)

This paper contains 13 sections, 16 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Overview of main definitions and results
  3. Organization
  4. Preliminaries
  5. Background on dynamical systems
  6. Background on real complexity theory and the Blum--Shub--Smale model
  7. Algebraic telic problems
  8. Natural reductions between algebraic telic problems
  9. Ruling out natural telic reductions
  10. Ruling out natural search reductions
  11. Placing structure on level 4 natural search reductions
  12. A concluding comment on Level 4a reductions
  13. Ruling out reductions in a general setting

Key Result

Theorem 1.1

There exists a dynamical system $(I, F)$ with positive topological entropy and a regular dynamical system $(I, T)$ with zero topological entropy, both computable by BSS-machines, as well as homeomorphisms $g:I\rightarrow I$ and $\hat{g}:I\rightarrow I$ computable by BSS-machines, for which $\mathbb{

Theorems & Definitions (44)

  • Theorem 1.1: \ref{['thmNoNaturalSearch']}
  • Theorem 1.2: \ref{['thm:fixedPoint']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 3.1: Algebraic telic problems
  • Lemma 3.2
  • proof
  • ...and 34 more