# Granular Generalized Variable Precision Rough Sets and Rational Approximations

arXiv: https://arxiv.org/abs/2205.14365 · HTML: https://mlsanity.com/paper/2205.14365/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 121 references that are in the map.
- First posted on arXiv 28 May 2022.

## Builds on (most cited first, 5 of 5)

- [Dialectical Rough Sets, Parthood and Figures of Opposition-1](https://mlsanity.com/paper/1703.10251/) (2017)
- [General Rough Modeling of Cluster Analysis](https://mlsanity.com/paper/2106.04683/) (2021)
- [Algebraic Semantics of Proto-Transitive Rough Sets](https://mlsanity.com/paper/1410.0572/) (2014)
- [Algebraic Semantics of Similarity-Based Bitten Rough Set Theory](https://mlsanity.com/paper/1301.0647/) (2013)
- [Algebraic Semantics of Generalized RIFs](https://mlsanity.com/paper/2109.12998/) (2021)

## Abstract (arXiv)

Rational approximations are introduced and studied in granular graded rough sets and generalizations thereof by the first author in recent research papers. The concept of rationality is determined by related ontologies and coherence between granularity, mereology and approximations in the context. In addition, a framework for rational approximations is introduced by her in the mentioned paper(s). Granular approximations constructed as per the procedures of variable precision rough sets (VPRS) are likely to be more rational than those constructed from a classical perspective under certain conditions. This may continue to hold for some generalizations of the former. However, a formal characterization of such conditions is not available in the previously published literature. In this research, theoretical aspects of the problem are critically examined, uniform generalizations of granular VPRS are introduced, new connections with granular graded rough sets are proved, appropriate concepts of substantial parthood are introduced, their extent of compatibility with the framework is accessed, and the framework is extended. Basic assumptions are explained in detail, and additional examples are constructed for readability. Furthermore, meta applications to cluster validation, image segmentation and dynamic sorting are invented. Extensions to direct generalizations of VPRS such as probabilistic rough sets are a natural consequence of the work.