Conditional Kolmogorov Complexity and Universal Probability

TL;DR

This paper provides the first written proofs in English for the conditional versions of Levin's Coding Theorem, which relate conditional Kolmogorov complexity to universal probability, using a different definition than standard conditional probability.

Contribution

It offers new rigorous proofs for the conditional Coding Theorem, clarifying the relationship between conditional Kolmogorov complexity and universal probability.

Findings

Proofs of conditional Coding Theorem provided in English

Different definition of conditional universal distribution used

No conditional version exists under classic probability definition

Abstract

The Coding Theorem of L.A. Levin connects unconditional prefix Kolmogorov complexity with the discrete universal distribution. There are conditional versions referred to in several publications but as yet there exist no written proofs in English. Here we provide those proofs. They use a different definition than the standard one for the conditional version of the discrete universal distribution. Under the classic definition of conditional probability, there is no conditional version of the Coding Theorem.