Core Concepts
It is possible to correctly recognize whether a given bit sequence is a valid tent code of a chaotic sequence, using only logarithmic space in the length of the sequence.
Abstract
The paper investigates the computational complexity of computing tent codes, which are bit sequences that represent the iterates of a chaotic tent map. The main contributions are:
It shows that a standard calculation using rounding-off can easily produce invalid tent codes, due to the sensitivity to initial conditions of chaotic sequences.
It presents an algorithm that can calculate an approximate tent code using only O(log^2 ε^-1 log d / log^2 μ + log n) space, where ε is the allowed error, μ is the parameter of the tent map, and n is the length of the sequence.
It then provides a smoothed analysis algorithm that can decide whether a given bit sequence is a valid tent code of an ε-perturbed input, using only O(log^2 n / log^3 d + log ε^-1 / log d) space on average.
The key ideas are the use of a space-efficient automaton representation of the tent language, and a Markov chain analysis of the transitions in this automaton. This allows the algorithms to operate in logarithmic space, despite the inherent complexity of chaotic sequences.