Kolmogorov Complexity Theory over the Reals

TL;DR

This paper explores the parallels and differences between classical discrete Kolmogorov Complexity and its real-number counterpart within the framework of BSS-machines, highlighting how classical concepts adapt to the continuous domain.

Contribution

It provides a comparative analysis of discrete and real Kolmogorov Complexity, extending the theoretical understanding of complexity over the reals and identifying key distinctions and similarities.

Findings

Real Kolmogorov Complexity exhibits natural counterparts to classical notions.

Differences in proofs highlight unique aspects of the real domain.

The work advances understanding of complexity theory in continuous settings.

Abstract

Kolmogorov Complexity constitutes an integral part of computability theory, information theory, and computational complexity theory -- in the discrete setting of bits and Turing machines. Over real numbers, on the other hand, the BSS-machine (aka real-RAM) has been established as a major model of computation. This real realm has turned out to exhibit natural counterparts to many notions and results in classical complexity and recursion theory; although usually with considerably different proofs. The present work investigates similarities and differences between discrete and real Kolmogorov Complexity as introduced by Montana and Pardo (1998).