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Properties of calculus in r-Complexity 2025

Rares Folea, Emil Slusanschi

TL;DR

The paper addresses the insufficiency of traditional Bachmann-Landau notations to distinguish closely related algorithms, especially at finite input sizes. It introduces the r-Complexity rc framework, defining core sets $\Theta_r$, $\mathcal{O}_r$, and $\Omega_r$ via limit-based criteria and developing corollaries and conversion rules to classical notations. Key contributions include establishing reflexivity, transitivity, symmetry, and projection properties, plus addition rules for r-Complexity classes and their small counterparts, enabling finer-grained algorithm analysis and embeddings. The work demonstrates practical benefits such as differentiating between algorithms with similar asymptotic growth but different finite-time behavior (e.g., Strassen vs. standard matrix multiplication) and supporting applications like code embeddings for performance-informed decision making. Overall, r-Complexity provides a more nuanced toolkit for analyzing and comparing real-world algorithms beyond traditional asymptotics, with potential for accelerated adoption in software engineering workflows.

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.

Properties of calculus in r-Complexity 2025

TL;DR

The paper addresses the insufficiency of traditional Bachmann-Landau notations to distinguish closely related algorithms, especially at finite input sizes. It introduces the r-Complexity rc framework, defining core sets , , and via limit-based criteria and developing corollaries and conversion rules to classical notations. Key contributions include establishing reflexivity, transitivity, symmetry, and projection properties, plus addition rules for r-Complexity classes and their small counterparts, enabling finer-grained algorithm analysis and embeddings. The work demonstrates practical benefits such as differentiating between algorithms with similar asymptotic growth but different finite-time behavior (e.g., Strassen vs. standard matrix multiplication) and supporting applications like code embeddings for performance-informed decision making. Overall, r-Complexity provides a more nuanced toolkit for analyzing and comparing real-world algorithms beyond traditional asymptotics, with potential for accelerated adoption in software engineering workflows.

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.
Paper Structure (6 sections, 34 theorems, 55 equations, 1 figure)

This paper contains 6 sections, 34 theorems, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Definition of the r-Complexity sets
  4. Common properties in r-Complexity
  5. Addition properties
  6. Conclusions

Key Result

corollary 1

Criteria for a function $f$ to be in the Big r-Theta complexity class defined by a function $g$ is a direct result of the following:

Figures (1)

  • Figure 1: An example of the regression fitting process in cbce, with a set of two raw metrics: temporarily and space. The goal of the research is to approximate, using simplified functions, the actual Big r-Theta complexity function that models program behaviour across multiple performance metrics, which later can be used to build complexity-based code embeddings. One part of the embedding that analyses the time complexity, POLY 2 3 6, denotes a second degree polynomial, with the coefficient $3$ for the squared term of the polynomial and the intercept $6$.

Theorems & Definitions (47)

  • definition 1
  • corollary 1
  • corollary 2
  • corollary 3
  • corollary 4
  • corollary 5
  • theorem 1
  • proof
  • theorem 2
  • proof
  • ...and 37 more