A note on constructive semantics for description logics Loris Bozzato, Mauro Ferrari, and Paola Villa Dipartimento di Informatica e Comunicazione Universit`a degli Studi dell’Insubria Via Mazzini 5, 21100, Varese, Italy Abstract. Following the approaches and motivations given in recent works about constructive interpretation of description logics, we intro- duce the constructive description logic KALC. This logic is based on a Kripke-style semantics inspired by the Kripke semantics for Intuitionis- tic first order logic. In the paper we present the main features of our semantics and we study its relations with other approaches. Moreover, we present a tableau calculus which turns out to be sound and complete for KALC. 1 Introduction In Computer Science it often happens that the introduction of a classically based logical system is followed by an analysis of its constructive or intuitionistic coun- terparts. Indeed, if the applicability of a logical system is often driven from its classical semantics, a constructive analysis allows us to take advantage of the computational properties of its formulas and proofs. In line with this consid- eration, one of the reasons for the success of description logics as a knowledge representation formalism is surely their simple classically-based semantics and only in recent works [3, 5, 6, 9, 11, 12] different proposals of a constructive rein- terpretation of description logics have been motivated. Following this line, we discuss a Kripke-style semantics for the basic descrip- tion logic ALC [2,13] inspired by the Kripke semantics for Intuitionistic first order logic [14]. We call KALC the logic corresponding to our semantics. Basi- cally, we may think of a Kripke model as a set of worlds, representing states of knowledge, partially ordered by their information content. Our semantics differs from the similar semantics described in [5] by the fact that we impose a condi- tion on the partial order. In particular, we require that every world is followed by a classical world, that is a world where concepts are interpreted according to the usual classical semantics for the description logic ALC . As we discuss in Section 3, such a condition seems to be essential to get the finite model property, useful to define a decidable calculus for the logic. An important feature of our semantics is that such a condition can be characterized by the axiom-schema ∀R.¬¬A → ¬¬∀R.A. This schema is obtained by translating in the description logics setting the Kuroda principle for first order logic, a principle which has been deeply studied in the literature of constructive logics [7, 14].