Computational Complexity of Space-Bounded Real Numbers

TL;DR

This paper investigates the space complexity of computable real numbers, revealing that transcendental numbers can be computed in logarithmic space while algebraic irrationals may require linear space, and introduces a new method using tally sets.

Contribution

It introduces a novel technique to analyze the space complexity of real numbers via their tally sets, providing new insights into the complexity of transcendental numbers.

Findings

Transcendental numbers can be logspace computable.

Algebraic irrational numbers may require linear space.

A new approach using tally sets to study real number complexity.

Abstract

In this work we study the space complexity of computable real numbers represented by fast convergent Cauchy sequences. We show the existence of families of trascendental numbers which are logspace computable, as opposed to algebraic irrational numbers which seem to required linear space. We characterized the complexity of space-bounded real numbers by quantifying the space complexities of tally sets. The latter result introduces a technique to prove the space complexity of real numbers by studying its corresponding tally sets, which is arguably a more natural approach. Results of this work present a new approach to study real numbers whose transcendence is unknown.