Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We show in detail that H(#; @) is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of Ehrenfeucht-Frasse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that H(#; @) corresponds to the fragment of rst-order logic which is invariant for generated submodels. We then show that H(#; @) enjoys (strong) interpolation, provide counterexamples for its nite variable fragments, and show that weak interpolation holds for the sublanguage H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sh...

Abstract. Hybrid languages are extended modal languages which can refer to (or even quantify over) states. Such languages are better behaved proof theoretically than ordinary modal languages for they internalize the apparatus of labeled deduction. Moreover, they arise naturally in a variety of applications, including description logic and temporal reasoning. Thus it would be useful to have a map of their complexity-theoretic properties, and this paper provides one. Our work falls into two parts. We first examine the basic hybrid language and its multi-modal and tense logical cousins. We show that the basic hybrid language (and indeed, multi-modal hybrid languages) are no more complex than ordinary uni-modal logic: all have pspace-complete K-satisfiability problems. We then show that adding even one nominal to tense logic raises complexity from pspace to exptime. In the second part we turn to stronger hybrid languages in which it is possible to bind nominals. We prove a general expressivity result showing that even the weak form of binding offered by the ↓ operator easily leads to undecidability.

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 their simplest form, hybrid languages are propositional modal languages which can refer to states. They were introduced by Arthur Prior, the inventor of tense logic, and played an important role in his work: because they make reference to specic times possible, they remove the most serious obstacle to developing modal approaches to temporal representation and reasoning. However very little is known about the computational complexity of hybrid temporal logics. In this paper we analyze the complexity of the satisability problem of a number of hybrid temporal logics: the basic hybrid language over transitive frames; nominal tense logic over transitive frames, strict total orders, and transitive trees; nominal Until logic; and referential interval logic. We discuss the eects of including nominals, the @ operator, the somewhere modality E, and the dierence operator D. Adding nominals to tense logic leads for several frame{classes to an increase in complexity of the satisability pro...

We study the complexity of the combination of the Description Logics ALCQ and ALCQI with a terminological formalism based on cardinality restrictions on concepts. These combinations can naturally be embedded into C 2 , the two variable fragment of predicate logic with counting quantiers, which yields decidability in NExpTime. We show that this approach leads to an optimal solution for ALCQI , as ALCQI with cardinality restrictions has the same complexity as C 2 (NExpTime-complete). In contrast, we show that for ALCQ, the problem can be solved in ExpTime. This result is obtained by a reduction of reasoning with cardinality restrictions to reasoning with the (in general weaker) terminological formalism of general axioms for ALCQ extended with nominals . Using the same reduction, we show that, for the extension of ALCQI with nominals, reasoning with general axioms is a NExpTime-complete problem. Finally, we sharpen this result and show that pure concept satisability for A...

The fusion Ll Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halldencompleteness are preserved under forming fusions of normal polyadic polymodal logics. Those problems remained open in [Fine & Schurz [3]] and [Kracht &Wolter [10]]. The paper de nes the fusion `l `r of two classical modal consequence relations and proves that decidability transfers also in this case. Finally, these results are used to prove a general decidability result for modal logics based on superintuitionistic logics. Given two logical system L1 and L2 it is natural to ask whether the fusion (or join) L1 L2 of them inherits the common properties of both L1 and L2. Let us consider some examples: (i) It is known that the rst order theory of one equivalence relation has the nite model property and is decidable. However, the rst order theory of two equivalence relations does not have the nite model property and is in fact undecidable (see Janiczak [7]). This result shows that even if we know the rst order properties of the individual relations of a theory, there may be no algorithm to determine the purely logical consequences of these properties. (ii) Various positive and negative results are known for joins of term rewriting systems (TRSs) whose vocabularies are disjoint. For example, the join of two TRSs is con uent i the two TRSs are con uent but there are complete TRSs whose join is not complete (see e.g. Klop [8]). In fact, the literature on TRSs shows how useful the study of joins of systems can be. (iii) In contrast to rst order theories the join of two decidable equational theories in disjoint languages is decidable as well. This was proved by Pigozzi in [12]. So we observe interesting di erences between logical systems by investigating the behavior of joins. To form the join of two modal logics (in languages with disjoint sets of modal operators) is { in a sense { a generalization of forming the join of two equational theories in disjoint languages. Namely, it is well-known that each modal logic corresponds to an equational theory of boolean algebras with operators. So the join of two modal logics corresponds to

The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satisfiability checking algorithms for these logics. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. Models with both finite and arbitrary constant domains are investigated. We begin by considering description logics with only one modal operator and then prove a general transfer theorem which makes it possible to lift the obtained results to many systems of polymodal description logic.

1 Motivation Since Modal Logics are an extension of Propositional Logic, they provide Boolean operators for constructing complex formulae. However, most Modal Logics do not admit Boolean operators for constructing complex modal parameters to be used in the box and diamond operators. This asymmetry is not present in Boolean Modal Logics, in which box and diamond quantify over arbitrary Boolean combinations of atomic modal parameters [9]. Boolean Modal Logics have been considered in various forms and contexts: 1. "Pure " Boolean Modal Logic has been studied in [9]. Negation and intersection of modal parameters occur in some variants of Propositional Dynamic Logic, see, e.g., [7, 16, 22].

The paper studies many-dimensional modal logics corresponding to products of Kripke frames. It proves results on axiomatisability, the finite model property and decidability for product logics, by applying a rather elaborated modal logic technique: p-morphisms, the finite depth method, normal forms, filtrations. Applications to first order predicate logics are considered too. The introduction and the conclusion contain a discussion of many related results and open problems in the area.

this paper argues for the rich world of representation that lies between these two extremes." Levesque and Brachman (1985) 1 Introduction Time and space belong to those few fundamental concepts that always puzzled scholars from almost all scientific disciplines, gave endless themes to science fiction writers, and were of vital concern to our everyday life and commonsense reasoning. So whatever approach to AI one takes [ Russell and Norvig, 1995 ] , temporal and spatial representation and reasoning will always be among its most important ingredients (cf. [ Hayes, 1985 ] ). Knowledge representation (KR) has been quite successful in dealing separately with both time and space. The spectrum of formalisms in use ranges from relatively simple temporal and spatial databases, in which data are indexed by temporal and/or spatial parameters (see e.g. [ Srefik, 1995; Worboys, 1995 ] ), to much more sophisticated numerical methods developed in computational geom