Properties of calculus in r-Complexity 2025

TL;DR

This paper explores the fundamental properties and conversion rules of r-Complexity calculus, aiming to enhance understanding and application of this complexity measurement system for real-world algorithms.

Contribution

It introduces and analyzes key properties and arithmetic principles of r-Complexity calculus, facilitating its broader adoption and application.

Findings

Established reflexivity, transitivity, symmetry in r-Complexity

Compared r-Complexity addition with traditional notation

Provided conversion rules between complexity classes

Abstract

This paper presents a series of general properties of the r-Complexity calculus, a complexity measurement for assessing the performance and asymptotic behaviour of real-world algorithms. This research describes characteristics such as reflexivity, transitivity, or symmetry and discusses several conversion rules between different classes of r-Complexity, as well as establishing fundamental arithmetic principles. The work also examines the behaviour of the addition property within this system and compares its characteristics with those frequently used in the traditional Bachmann-Landau notation. Through utilizing these properties, this research seeks to promote the exploration and development of novel applications for r-Complexity, as well as accelerating the adoption rate of calculus in this refined complexity model.