Kripke Contexts, Double Boolean Algebras with Operators and Corresponding Modal Systems

TL;DR

This paper unifies formal concept analysis and rough set theory through Kripke contexts, introducing complex algebras and modal logics for these structures, with representation theorems and semantics based on protoconcepts.

Contribution

It develops a unified framework for contexts using Kripke structures, introduces double Boolean algebras with operators, and formulates corresponding modal logics with semantics.

Findings

Representation theorems for complex algebras are established.

Modal logics for contextual double Boolean algebras are formulated.

A sequent calculus for these logics is proposed.

Abstract

The notion of a context in formal concept analysis and that of an approximation space in rough set theory are unified in this study to define a Kripke context. For any context (G,M,I), a relation on the set G of objects and a relation on the set M of properties are included, giving a structure of the form ((G,R), (M,S), I). A Kripke context gives rise to complex algebras based on the collections of protoconcepts and semiconcepts of the underlying context. On abstraction, double Boolean algebras (dBas) with operators and topological dBas are defined. Representation results for these algebras are established in terms of the complex algebras of an appropriate Kripke context. As a natural next step, logics corresponding to classes of these algebras are formulated. A sequent calculus is proposed for contextual dBas, modal extensions of which give logics for contextual dBas with operators and topological contextual dBas. The representation theorems for the algebras result in a protoconcept-based semantics for these logics.