arXiv subjects

Fouad B. Chedid

Publications and source records attributed to Fouad B. Chedid.

3 recordsLinked to original sources

arXiv2019

Kolmogorov's Algorithmic Mutual Information Is Equivalent to Bayes' Law

Given two events $A$ and $B$, Bayes' law is based on the argument that the probability of $A$ given $B$ is proportional to the probability of $B$ given $A$. When probabilities are interpreted in the Bayesian sense, Bayes' law constitutes a learning algorithm which shows how one can learn from a new observation to improve their belief in a theory that is consistent with that observation. Kolmogorov's notion of algorithmic information, which is based on the theory of algorithms, proposes an objective measure of the amount of information in a finite string about itself and concludes that for any two finite strings $x$ and $y$, the amount of information in $x$ about $y$ is almost equal to the amount of information in $y$ about $x$. We view this conclusion of Kolmogorov as the algorithmic information version of Bayes' law. This can be easily demonstrated if one considers the work of Levin on prefix Kolmogorov complexity and then expresses the amount of Kolmogorov mutual information between two finite strings using Solomonoff's a priori probability.

cs.IT↗

arXiv2017

Kolmogorov Complexity and Information Content

In this paper, we revisit a central concept in Kolmogorov complexity in which one would equate program-size complexity with information content. Despite the fact that Kolmogorov complexity has been widely accepted as an objective measure of the information content of a string, it has been the subject of many criticisms including the fundamental one directed by logicians and philosophers towards the statistical and semantical theories of information, which is about confusing an object and its name. In this paper, we clarify a number of subtle issues that are at the center of this debate.

cs.IT↗

arXiv2017

A Variation of Levin Search for All Well-Defined Problems

In 1973, L.A. Levin published an algorithm that solves any inversion problem $π$ as quickly as the fastest algorithm $p^*$ computing a solution for $π$ in time bounded by $2^{l(p^*)}.t^*$, where $l(p^*)$ is the length of the binary encoding of $p^*$, and $t^*$ is the runtime of $p^*$ plus the time to verify its correctness. In 2002, M. Hutter published an algorithm that solves any well-defined problem $π$ as quickly as the fastest algorithm $p^*$ computing a solution for $π$ in time bounded by $5.t_{p}(x)+d_p.time_{t_{p}}(x)+c_p$, where $d_p=40.2^{l(p)+l(t_{p})}$ and $c_p=40.2^{l(f)+1}.O(l(f)^2)$, where $l(f)$ is the length of the binary encoding of a proof $f$ that produces a pair $(p,t_p)$, where $t_p(x)$ is a provable time bound on the runtime of the fastest program $p$ provably equivalent to $p^*$. In this paper, we rewrite Levin Search using the ideas of Hutter so that we have a new simple algorithm that solves any well-defined problem $π$ as quickly as the fastest algorithm $p^*$ computing a solution for $π$ in time bounded by $O(l(f)^2).t_p(x)$.

cs.CC↗