George Voutsadakis

Package-Based Description Logics

We present the syntax and semantics of a family of modular ontology languages, Package-based Description Logics (P-DL), to support context- specific reuse of knowledge from multiple ontology modules. In particular, we describe a P-DL

Polyadic Concept Analysis

{\mathcal SHOIQP}

Categorical Abstract Algebraic Logic: Equivalent Institutions

that allows the importing of concept, role and nominal names between multiple ontology modules (each of which can be viewed as a

Categorical Abstract Algebraic Logic: Algebraizable Institutions

{\mathcal SHOIQ}

Categorical Abstract Algebraic Logic: Models of π-Institutions

ontology).

Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity

{\mathcal SHOIQP}

Categorical Abstract Algebraic Logic: More on Protoalgebraicity

supports contextualized interpretation, i.e., interpretation from the point of view of a specific package. We establish the necessary and sufficient conditions on domain relations (i.e., the relations between individuals in different local domains) that need to hold in order to preserve the unsatisfiability of concept formulae, monotonicity of inference, transitive reuse of knowledge across modules.

Categorical Abstract Algebraic Logic: Partially Ordered Algebraic Systems

The framework and the basic results of Wille on triadic concept analysis, including his Basic Theorem of Triadic Concept Analysis, are here generalized to n-dimensional formal contexts.

Categorical Abstract Algebraic Logic: Categorical Algebraization of Equational Logic

A category theoretic generalization of the theory of algebraizable deductive systems of Blok and Pigozzi is developed. The theory of institutions of Goguen and Burstall is used to provide the underlying framework which replaces and generalizes the universal algebraic framework based on the notion of a deductive system. The notion of a term π-institution is introduced first. Then the notions of quasi-equivalence, strong quasi-equivalence and deductive equivalence are defined for π-institutions. Necessary and sufficient conditions are given for the quasi-equivalence and the deductive equivalence of two term π-institutions, based on the relationship between their categories of theories. The results carry over without any complications to institutions, via their associated π-institutions. The π-institution associated with a deductive system and the institution of equational logic are examined in some detail and serve to illustrate the general theory.

Categorical Abstract Algebraic Logic: (ℐ,N)-Algebraic Systems

The framework developed by Blok and Pigozzi for the algebraizability of deductive systems is extended to cover the algebraizability of multisignature logics with quantifiers. Institutions are used as the supporting structure in place of deductive systems. In particular, the concept of an algebraic institution and that of an algebraizable institution are made precise using the theory of monads from categorical algebra and the notion of equivalence of institutions introduced by Voutsadakis. Several examples of algebraic and algebraizable institutions are provided.