Generalized Asymptotic Notations via Filters: A New Framework for Algorithm Analysis

This paper introduces a generalization of asymptotic notation using filters, a topological structure. We present key properties of this generalization, including reflexivity, symmetry, and transitivity, supported by illustrative examples. Our research demonstrates that classical asymptotic notations imply their filter-generalized counterparts, but we provide examples showing the converse is not universally true. We also propose a characterization of traditional asymptotic notations using filters, offering a new perspective on these fundamental concepts. Furthermore, we establish relationships between bounded or vanishing sequences and filter-based asymptotic notations, enabling the determination of properties central to this study. This generalization provides a more nuanced framework for analyzing algorithmic complexity, potentially capturing behaviors overlooked by classical notations and opening new avenues for theoretical computer science research.