A non-distributive logic for semiconcepts of a context and its modal extension with semantics based on Kripke contexts

A non-distributive two-sorted hypersequent calculus \textbf{PDBL} and its modal extension \textbf{MPDBL} are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational semantics for \textbf{PDBL} is next proposed, where any formula is interpreted as a semiconcept of a context. For \textbf{MPDBL}, the relational semantics is based on Kripke contexts, and a formula is interpreted as a semiconcept of the underlying context. The systems are shown to be sound and complete with respect to the relational semantics. Adding appropriate sequents to \textbf{MPDBL} results in logics with semantics based on reflexive, symmetric or transitive Kripke contexts. One of these systems is a logic for topological pure double Boolean algebras. It is demonstrated that, using \textbf{PDBL}, the basic notions and relations of conceptual knowledge can be expressed and inferences involving negations can be obtained. Further, drawing a connection with rough set theory, lower and upper approximations of semiconcepts of a context are defined. It is then shown that, using the formulae and sequents involving modal operators in \textbf{MPDBL}, these approximation operators and their properties can be captured

Two-sorted Modal Logic for Formal and Rough Concepts

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

Conceptual Knowledge Representation and Reasoning

One of the main areas in knowledge representation and logic-based artificial intelligence concerns logical formalisms that can be used for representing and reasoning with concepts. For almost 30 years, since research in this area began, the issue of intensionality has had a special status in that it has been considered to play an important role, yet it has not been precisely established what it means for a logical formalism to be intensional. This thesis attempts to set matters straight. Based on studies of the main contributions to the issue of intensionality from philosophy of language, in particular the works of Gottlob Frege and Rudolf Carnap, we start by defining when a logical formalism is intensional. We then examine whether the current formalizations of concepts are intensional. The result is negative in the sense that none of the prevalent formalizations are intensional. This motivates the development of intensional logics for concepts. Our main contribution is the presentation of such an intensional concept logic

The Word Problem in Semiconcept Algebras (CLA 2011)

International audienceThe aim of this article is to prove that the word problem in semiconcept algebras is PSPACE-complete

The word problem in semiconcept algebras (CiE 2011)

International audienceSemiconcept algebras have attracted interest both for their theoretical merits and for their practical relevance. The aim of this article is to prove that the word problem in semiconcept algebras is PSPACE-complete

The word problem in semiconcept algebras

The conference was dedicated to the memory of Leo Esakia, who passed away in November 2010International audienceThe aim of this article is to prove that the word problem in semiconcept algebras is PSPACE-complete