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A Smoothed Analysis of the Space Complexity of Computing a Chaotic Sequence

Naoaki Okada, Shuji Kijima

TL;DR

This paper tackles the problem of computing chaotic sequences generated by the tent map, focusing on the space complexity of recognizing tent codes. It introduces the tent language $\mathcal{L}_n$ and employs an automaton/Markov-extension framework together with smoothed analysis to study the decision problem: given $x$ and $\mathbf{b}_n$, decide if $\mathbf{b}_n$ equals the tent code $\gamma^n(x)$, under $\epsilon$-perturbations. The main results show that a smoothed decision can be performed in $O(\log^2 n)$ space and that a constant-space approximate calculation exists to produce valid tent codes within $\mathcal{L}_n(x,\epsilon)$, with rigorous bounds derived via segment-types and Markov-model analyses. These findings bridge chaotic dynamics with space-efficient symbolic computation and offer robustness insights for applications in pseudo-random generation and cryptography under perturbations.

Abstract

This work is motivated by a question whether it is possible to calculate a chaotic sequence efficiently, e.g., is it possible to get the $n$-th bit of a bit sequence generated by a chaotic map, such as $β$-expansion, tent map and logistic map in $\mathrm{o}(n)$ time/space? This paper gives an affirmative answer to the question about the space complexity of a tent map. We show that the decision problem of whether a given bit sequence is a valid tent code is solved in $\mathrm{O}(\log^{2} n)$ space in a sense of the smoothed complexity.

A Smoothed Analysis of the Space Complexity of Computing a Chaotic Sequence

TL;DR

This paper tackles the problem of computing chaotic sequences generated by the tent map, focusing on the space complexity of recognizing tent codes. It introduces the tent language and employs an automaton/Markov-extension framework together with smoothed analysis to study the decision problem: given and , decide if equals the tent code , under -perturbations. The main results show that a smoothed decision can be performed in space and that a constant-space approximate calculation exists to produce valid tent codes within , with rigorous bounds derived via segment-types and Markov-model analyses. These findings bridge chaotic dynamics with space-efficient symbolic computation and offer robustness insights for applications in pseudo-random generation and cryptography under perturbations.

Abstract

This work is motivated by a question whether it is possible to calculate a chaotic sequence efficiently, e.g., is it possible to get the -th bit of a bit sequence generated by a chaotic map, such as -expansion, tent map and logistic map in time/space? This paper gives an affirmative answer to the question about the space complexity of a tent map. We show that the decision problem of whether a given bit sequence is a valid tent code is solved in space in a sense of the smoothed complexity.
Paper Structure (23 sections, 28 theorems, 54 equations, 2 figures, 5 algorithms)

This paper contains 23 sections, 28 theorems, 54 equations, 2 figures, 5 algorithms.

Table of Contents

  1. Brief Introduction
  2. Background
  3. Chaos.
  4. Tent map: 1-D, piecewise-linear and chaotic.
  5. Smoothed analysis.
  6. Contribution
  7. This work.
  8. Related works.
  9. Organization
  10. Issues and Results
  11. Tent code
  12. What is the issue?
  13. Problems and Results
  14. Proof strategy of the theorems.
  15. Underlying Technology
  16. ...and 8 more sections

Key Result

Proposition 2.1

Suppose $\gamma_{\mu}^{\infty}(x) = b_1b_2\cdots$ for $x \in [0,1)$. Then, $(\mu - 1) \sum_{i=1}^{\infty} b_{i} \mu^{-i} = x$.

Figures (2)

  • Figure 1: A tent map $f(x)$ and its cobweb.
  • Figure 4: Transition diagram over $\mathcal{T}_n$ for $\mu=1.6$.

Theorems & Definitions (40)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7: Approximate calculation
  • Theorem 2.8: Decision for $\epsilon$-perturbed input
  • Lemma 3.1: OK23
  • ...and 30 more