On entropic convergence of algorithms in terms of domain partitions

TL;DR

This paper proposes an entropy-based method to analyze algorithm convergence by examining how input partitions evolve, aiming to understand information transformation and complexity in data processing.

Contribution

It introduces an entropy-like measure based on input partitions and the Principle of Maximal Uncertainty to study algorithm convergence from an information-theoretic perspective.

Findings

The approach quantifies convergence through entropic weight of events.

Illustrated with two example algorithms.

Provides a new perspective on algorithm complexity analysis.

Abstract

The paper describes an approach to measuring convergence of an algorithm to its result in terms of an entropy-like function of partitions of its inputs of a given length. The goal is to look at the algorithmic data processing from the viewpoint of information transformation, with a hope to better understand the work of algorithm, and maybe its complexity. The entropy is a measure of uncertainty, it does not correspond to our intuitive understanding of information. However, it is what we have in this area. In order to realize this approach we introduce a measure on the inputs of a given length based on the Principle of Maximal Uncertainty: all results should be equiprobable to the algorithm at the beginning. An algorithm is viewed as a set of events, each event is an application of a command. The commands are very basic. To measure the convergence we introduce a measure that is called entropic weight of events of the algorithm. The approach is illustrated by two examples.