From Classical to Normal Modal Logics

. Classical modal logics generally neither possess the necessitation rule nor the K-axiom. Their semantics being in terms of minimal models, several specialisations of these have been given such as augmented minimal models, models with queer worlds or models with inaccessible worlds. In this paper we exhibit exact translations from these families into normal multi-modal logics. Such translations allow to reuse the proof systems that have been developed for the latter. 1 Introduction Classical modal logics (Segerberg 1971, Chellas 1980) are weaker than the wellknown normal modal logics: The only rule that is common to all classical modal logics is RE : F $G 2F$ 2G (We nevertheless note that this principle raises problems in systems containing equality (Hughes and Cresswell 1968).) Classical modal logics do not necessarily validate RN : F 2F ? This work has been partially supported by the Esprit projects DRUMS II and MEDLAR II. It has been inspired by Andrew Jones's pr..

Concept Forgetting in ALCOI-Ontologies Using an Ackermann Approach

Abstract. We present a method for forgetting concept symbols in on-tologies specified in the description logic ALCOI. The method is an adaptation and improvement of a second-order quantifier elimination method developed for modal logics and used for computing correspon-dence properties for modal axioms. It follows an approach exploiting a result of Ackermann adapted to description logics. An important feature inherited from the modal approach is that the inference rules are guided by an ordering compatible with the elimination order of the concept sym-bols. This provides more control over the inference process and reduces non-determinism, resulting in a smaller search space. The method is ex-tended with a new case splitting inference rule, and several simplification rules. Compared to related forgetting and uniform interpolation methods for description logics, the method can handle inverse roles, nominals and ABoxes. Compared to the modal approach on which it is based, it is more efficient in time and improves the success rates. The method has been implemented in Java using the OWL API. Experimental results show that the order in which the concept symbols are eliminated significantly affects the success rate and efficiency.

Representation Theorems and the Semantics of Non-Classical Logics , and Applications to Automated Theorem Proving

We give a uniform presentation of representation and decidability results related to the Kripke-style semantics of several nonclassical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, distributive lattices and semilattices) extends in a natural way to several classes of operators and allows to establish a relationship between algebraic and Kripke-style models. We illustrate the ideas on several examples. We conclude by showing how the Kripkestyle models thus obtained can be used (if rst-order axiomatizable) for automated theorem proving by resolution for some non-classical logics