Properties of calculus in r-Complexity 2025

2026-01-26

AI Summary

Existing methods for algorithmic complexity analysis exhibit expressive limitations in capturing both practical performance and asymptotic behavior, and lack a systematic study of transformations and operations between complexity classes. This work proposes the first axiomatic framework for r-Complexity calculus, formally integrating complexity theory with algebraic structures. Within this framework, fundamental properties such as reflexivity and transitivity are rigorously derived, and systematic rules for inter-class transformations and arithmetic operations are established. The approach significantly enhances the expressiveness and applicability of complexity analysis, while demonstrating the theoretical advantages of r-Complexity over the traditional Bachmann–Landau notation system, thereby laying a foundation for novel applications in emerging computational contexts.

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Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesSearch and Optimization: Other Foundations of Search & Optimization

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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.

Problem

• r-Complexity
• calculus
• algorithm complexity
• asymptotic behaviour
• complexity measurement

Innovation

• r-Complexity
• complexity calculus
• asymptotic behavior
• Bachmann-Landau notation
• algorithmic complexity