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

TL;DR

This paper investigates the space complexity of computing chaotic sequences, specifically showing that the decision problem for tent map codes can be solved in logarithmic squared space under smoothed analysis.

Contribution

It provides the first known efficient space complexity bound for deciding tent map codes, demonstrating feasibility in smoothed analysis context.

Findings

Decision problem for tent map codes is in O(log^2 n) space

Smoothed analysis framework applied to chaotic sequence computation

Shows potential for efficient computation of chaotic sequences

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 $\beta$-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.