[Paper Review] High Granular Operator Spaces, and Less-Contaminated General Rough Mereologies

A. Mani|arXiv (Cornell University)|Nov 15, 2018

Rough Sets and Fuzzy Logic52 references4 citations

TL;DR

This paper introduces high granular operator spaces (HGOS) and granular rough inclusion functions (GRIFs) to reduce contamination in general rough mereologies by grounding approximations in granular, data-driven structures. It proposes a partial algebraic framework for HGOS, establishes representation theorems using matrix forms, and applies GRIFs to decision-making and inverse problems with minimal assumptions, significantly reducing reliance on potentially contaminating functions like standard RIFs.

ABSTRACT

Granular operator spaces and variants had been introduced and used in theoretical investigations on the foundations of general rough sets by the present author over the last few years. In this research, higher order versions of these are presented uniformly as partial algebraic systems. They are also adapted for practical applications when the data is representable by data table-like structures according to a minimalist schema for avoiding contamination. Issues relating to valuations used in information systems or tables are also addressed. The concept of contamination introduced and studied by the present author across a number of her papers, concerns mixing up of information across semantic domains (or domains of discourse). Rough inclusion functions ( extsf{RIF}s), variants, and numeric functions often have a direct or indirect role in contaminating algorithms. Some solutions that seek to replace or avoid them have been proposed and investigated by the present author in some of her earlier papers. Because multiple kinds of solution are of interest to the contamination problem, granular generalizations of RIFs are proposed, and investigated. Interesting representation results are proved and a core algebraic strategy for generalizing Skowron-Polkowski style of rough mereology (though for a very different purpose) is formulated. A number of examples have been added to illustrate key parts of the proposal in higher order variants of granular operator spaces. Further algorithms grounded in mereological nearness, suited for decision-making in human-machine interaction contexts, are proposed by the present author. Applications of granular extsf{RIF}s to partial/soft solutions of the inverse problem are also invented in this paper.

Motivation & Objective

• To address the contamination problem in rough set theory, where information from different semantic domains is inadmissibly mixed via functions like rough inclusion functions (RIFs).
• To develop a granular, high-order algebraic framework—high granular operator spaces (HGOS)—that avoids questionable assumptions and supports non-invasive data analysis.
• To generalize RIFs into granular versions (GRIFs) that are inherently tied to granules and data table structures, reducing algorithmic contamination.
• To provide a partial algebraic system formalism for the inverse problem in general rough sets, enabling cleaner, more interpretable solutions.
• To connect HGOS and GRIFs with rough mereology, proximity measures, and decision-making in human-machine interaction, especially in safety-critical contexts.

Proposed method

• Formalizes high granular operator spaces (HGOS) as partial algebraic systems, enabling higher-order generalizations of granular rough sets.
• Proposes granular rough inclusion functions (GRIFs) as generalizations of standard RIFs, with values derived from granular intersections and cardinality-based ratios.
• Applies t-norms and s-norms to model ideal representations of GRIFs, ensuring consistency and minimizing contamination.
• Uses matrix representation theorems to characterize GRIFs and their aggregation, enabling computational implementation.
• Introduces weighted chain decompositions of attribute sets to handle non-uniform attribute importance, extending GRIFs to multi-level granular structures.
• Develops a new algorithm—Pilot’s Algorithm—for approximate decision-making in human-machine interaction, relying on mereological nearness and GRIFs.

Experimental results

Research questions

• RQ1 How can contamination in rough set algorithms be minimized by redefining rough inclusion functions in terms of granular structures rather than abstract sets?
• RQ2 What algebraic framework supports high-order generalizations of granular operator spaces while preserving foundational rigor and avoiding non-essential assumptions?
• RQ3 Can granular RIFs be formally characterized and represented via matrix forms to enable practical computation and aggregation?
• RQ4 How can GRIFs be applied to solve the inverse problem in general rough sets within a clean, partial algebraic system?
• RQ5 To what extent can GRIFs and HGOS support robust decision-making in human-machine interaction, especially when human users have limited knowledge of system internals?

Key findings

• The paper establishes that high granular operator spaces (HGOS) can be formalized as partial algebraic systems, enabling higher-order generalizations of granular rough sets.
• Granular rough inclusion functions (GRIFs) are formally characterized and shown to reduce contamination by anchoring approximations in granular, data-reflective structures.
• Matrix representation theorems are proven, showing that GRIFs can be represented via structured matrices, enabling algorithmic implementation and aggregation.
• The proposed Pilot’s Algorithm for human-machine interaction uses mereological nearness and GRIFs to support safe, approximate decision-making under uncertainty.
• GRIFs are successfully applied to partial and soft solutions of the inverse problem in general rough sets, demonstrating practical utility beyond theoretical modeling.
• The study reveals that even Pre-GGS (generalized granular systems) are equivalent to certain single-sorted partial algebras, suggesting broad applicability to fuzzy and rough set theories.