On information content in certain objects

TL;DR

This paper explores the difference between total and plain conditional complexities for natural objects, showing that total complexity can be significantly larger even when plain complexity is low, highlighting new insights into information dependence.

Contribution

It demonstrates that natural objects can have large mutual total conditional complexities despite negligible plain conditional complexities, a novel finding in complexity theory.

Findings

Total conditional complexity can be much larger than plain conditional complexity for natural objects.

Natural objects like string counts and lexicographically first strings exhibit large mutual total complexities.

This is the first example showing plain complexity is much less than total complexity for natural objects.

Abstract

The fine approach to measure information dependence is based on the total conditional complexity CT(y|x), which is defined as the minimal length of a total program that outputs y on the input x. It is known that the total conditional complexity can be much larger than than the plain conditional complexity. Such strings x, y are defined by means of a diagonal argument and are not otherwise interesting. In this paper we investigate whether this happens also for some natural objects. More specifically, we consider the following objects: the number of strings of complexity less than n and the lex first string of length n and complexity at least n. It is known that they have negligible mutual conditional complexities. In this paper we prove that their mutual total conditional complexities may be large. This is the first example of natural objects whose plain conditional complexity is much less than the total one.