Traditional rst order predicate logic is known to be designed for representing and manipulating static knowledge (e.g. mathematical theories). So are manyof its applications. Knowledge representation systems based on concept description logics are not exceptions.

In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatiotemporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u , which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with muli-dimensional formalisms and discuss possible directions for further research.

In this paper, we construct a new concept description language intended for representing dynamic and intensional knowledge. The most important feature distinguishing this language from its predecessors in the literature is that it allows applications of modal operators to all kinds of syntactic terms: concepts, roles and formulas. Moreover, the language may contain both local (i.e., state-dependent) and global (i.e., state-independent) concepts, roles and objects. All this provides us with the most complete and natural means for re ecting the dynamic and intensional behaviour of application domains. We construct a satis ability checking (mosaic-type) algorithm for this language (based on ALC) in(i) arbitrary multimodal frames, (ii) frames with universal accessibility relations (for knowledge) and (iii) frames with transitive, symmetrical and euclidean relations (for beliefs). On the other hand, it is shown that the satisfaction problem becomes undecidable if the underlying frames are arbitrary strict linear orders, hN; <i, or the language contains the common knowledge operator for n 2 agents. 1

The product of two modal logics L1 and L2 is the modal logic determined by the class of frames of the form F G such that F and G validate L1 and L2, respectively. This paper proves the decidability of the product of converse PDL and polymodal K. Decidability results for products of modal logics of knowledge as well as temporal logics and polymodal K are discussed. All those products form rather expressive but still decidable fragments of modal predicate logics. Based on the equivalence of polymodal K and the description logic ALC we shall discuss the obtained fragments, extend the expressive power a bit, and compare them with other modal description logics. 1

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The aim of this paper is to construct a tableau decision algorithm for the modal description logic KALC with constant domains. More precisely, we present a tableau procedure that is capable of deciding, given an ALC-formula ' with extra modal operators (which are applied only to concepts and TBox axioms, but not to roles), whether ' is satisfiable in a model with constant domains and arbitrary accessibility relations. Tableau-based

... this paper we study two special cases of this framework: full systems and hypercubes. Both model static situations in which no agent has any information about another agent's state. Full systems and hypercubes are an appropriate model for the initial congurations of many systems of interest. We establish a correspondence between full systems and hypercube systems and certain classes of Kripke frames. We show that these classes of systems correspond to the same logic. Moreover, this logic is also the same as that generated by the larger class of weakly directed frames. We provide a sound and complete axiomatization, S5WDn , of this logic, and study its computational complexity. Finally, we show that under certain natural assumptions, in a model where knowledge evolves over time, S5WDn characterises the properties of knowledge not just at the initial conguration, but also at all later congurations. In particular, this holds for homogeneous broadcast systems, which capture settings in which agents are initially ignorant of each others local states, operate synchronously, have perfect recall, and can communicate only by broadcasting.

abstract. We study and give a summary of the complexity of 15 basic normal monomodal logics under the restriction to the Horn fragment and/or bounded modal depth. As new results, we show that: a) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by k ≥ 2 in the modal logics K 4 and KD4 is PSPACE-complete, in K is NP-complete; b) the satisfiability problem of modal formulas with modal depth bounded by 1 in K 4, KD4, and S4 is NP-complete; c) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by 1 in K, K 4, KD4, and S4 is PTIME-complete. In this work, we also study the complexity of the multimodal logics Ln under the mentioned restrictions, where L is one of the 15 basic monomodal logics. We show that, for n ≥ 2: a) the satisfiability problem of sets of Horn modal clauses in K5n, KD5n, K45n, and KD45n is PSPACE-complete; b) the satisfiability problem of sets of Horn modal clauses with modal depth bounded by k ≥ 2 in Kn, KBn, K5n, K45n, KB5n is NP-complete, and in KDn, Tn, KDBn, Bn,

It is shown that all proper normal extensions of the bi-modal system S5 have the poly-size model property. In fact, every normal proper extension L of S5 complete with respect to a class of finite frames FL . To each such class corresponds a natural number b(L) { the bound of L. For every L, there exists a polynomial P ( ) of degree b(L)+1 such that every L-satisfiable formula ' is satisfiable on an L-frame whose universe is bounded by P (j'j), for j'j the number of subformulas of '. It is shown that this bound is optimal.

We give sound and complete analytic tableau systems for the propositional bimodal logics KB , KB C , KB 5 , and KB 5C . These logics have two universal modal operators K and B , where K stands for knowing and B stands for believing. The logic KB is a combination of the modal logic S5 (for K ) and KD45 (for B ) with the interaction axioms I : K ! B and C : B ! K B . The logics KB C , KB 5 , KB 5C are obtained from KB respectively by deleting the axiom C (for KB C ), the axioms 5 (for KB 5 ), and both of the axioms C and 5 (for KB 5C ). As analytic sequent-like tableau systems, our calculi give simple decision procedures for reasoning about both knowledge and belief in the mentioned logics.