## Question:

Did Kolmogorov develop a set of axioms for probability theory motivated by Algorithmic Information Theory in the 1960s?

## Context:

In 1965, Andrey Kolmogorov considered three approaches to information theory(combinatorial, probabilistic, and algorithmic) [1] where in the algorithmic approach he introduces the Invariance theorem for the minimal description of datasets $X$ relative to Universal description languages $U$ and $U'$:

\begin{equation} \lvert K_U(X)-K_{U'}(X) \rvert \leq \text{Cst} \tag{1} \end{equation}

which allows us to formulate the Universal Distribution, an effective formalisation of Occam's razor [2].

In essence, he introduced the foundations for Algorithmic Information Theory and in the conclusion he clarifies that his motivation for doing so is to provide a clear definition of randomness as well as a robust foundation for probability theory where all probabilities have a deterministic and frequentist nature.

In fact, his PhD student Leonid Levin would rigorously define the algorithmic probability of observing a dataset $X$:

\begin{equation} -\log_2 P(X) = K_U(X) - \mathcal{O}(1) \tag{2} \end{equation}

which would later become known as Levin's Coding theorem [3].

## References:

A. N. Kolmogorov Three approaches to the quantitative definition of information. Problems of Information and Transmission, 1(1):1--7, 1965

Peter Grünwald and Paul Vitanyí. Shannon Information and Kolmogorov Complexity. Arxiv. 2004.

L.A. Levin. Laws of information conservation (non-growth) and aspects of the foundation of probability theory. Problems Inform. Transmission, 10:206–210, 1974.