On the Conditional Complexity of Sets of Strings

TL;DR

This paper investigates the existence of a simple string within a set, conditional on all other set members, using Kolmogorov complexity, especially for sets with low mutual information with the halting sequence.

Contribution

It proves the existence of such a simple string for sets with low mutual information with the halting sequence, based on bounds involving conditional complexities.

Findings

Existence of a member with low conditional complexity in certain sets.

Results depend on maximum conditional complexity between elements.

Results also depend on expected conditional complexity relative to set members.

Abstract

Given a set X of finite strings, one interesting question to ask is whether there exists a member of X which is simple conditional to all other members of X. Conditional simplicity is measured by low conditional Kolmogorov complexity. We prove the affirmative to this question for sets that have low mutual information with the halting sequence. There are two results with respect to this question. One is dependent on the maximum conditional complexity between two elements of X, the other is dependent on the maximum expected value of the conditional complexity of a member of X relative to each member of X.