Two-sorted Modal Logic for Formal and Rough Concepts

TL;DR

This paper introduces two-sorted modal logics to model and reason about concepts from rough set theory and formal concept analysis, providing a formal framework for their representation and interrelation.

Contribution

It develops two-sorted modal logics, KB and KF, with a correspondence theorem linking them, to represent and analyze rough and formal concepts within a unified logical framework.

Findings

KB captures rough set-based concepts.

KF represents formal concepts.

The logics can model lattice structures of concepts.

Abstract

In this paper, we propose two-sorted modal logics for the representation and reasoning of concepts arising from rough set theory (RST) and formal concept analysis (FCA). These logics are interpreted in two-sorted bidirectional frames, which are essentially formal contexts with converse relations. On one hand, the logic $\textbf{KB}$ contains ordinary necessity and possibility modalities and can represent rough set-based concepts. On the other hand, the logic $\textbf{KF}$ has window modality that can represent formal concepts. We study the relationship between \textbf{KB} and \textbf{KF} by proving a correspondence theorem. It is then shown that, using the formulae with modal operators in \textbf{KB} and \textbf{KF}, we can capture formal concepts based on RST and FCA and their lattice structures.