In this chapter, we introduce and explain the basic notions of Description Logic, including syntax, semantics and reasoning services, and we explain how the latter are used in applications. The concept language of the DL ALCIn this section, we will describe the central notions of Description Logic first on an intuitive level and then on a more precise level. As a running example, we use the domain of university courses and teaching, and we will use a conceptualisation given informally, in graphical form, in Figure 2.1. Please note that this is one way of viewing university teachingwhich might be very different from the reader's way of viewing it. Also, as it is an informal representation, different readers may interpret arrows in different ways; that is, our representation does not come with a well-defined semantics that would inform us in an unambiguous way how to interpret the different arrows. 1 In the next sections, we will describe our way of viewing university teaching in a DL knowledge base, thereby establishing some constraints on the meaning of terms like "Professor" and "teaches" used in Figure 2.1 and throughout this section.In Description Logic, we assume that we want to describe some abstraction of some domain of interest, and that this abstraction is populated by elements. 2 We use three main building blocks to describe these elements:• Concepts represent sets of elements and can be viewed as unary pred-1 Our graphical representation looks somewhat similar to an extended ER diagram, for which such a well-defined semantics has been specified [Che76, CLN94]. 2 We have chosen the term "elements" rather than "individuals" or "objects" to prevent the reader from making false assumptions.

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Conjunctive queries play an important role as an expressive query language for Description Logics (DLs). Although modern DLs usually provide for transitive roles, conjunctive query answering over DL knowledge bases is only poorly understood if transitive roles are admitted in the query. In this paper, we consider unions of conjunctive queries over knowledge bases formulated in the prominent DL SHIQ and allow transitive roles in both the query and the knowledge base. We show decidability of query answering in this setting and establish two tight complexity bounds: regarding combined complexity, we prove that there is a deterministic algorithm for query answering that needs time single exponential in the size of the KB and double exponential in the size of the query, which is optimal. Regarding data complexity, we prove containment in co-NP.

Combining knowledge representation and reasoning formalisms is an important and challenging task. It is important because non-trivial AI applications often comprise different aspects of the world, thus requiring suitable combinations of available formalisms modeling each of these aspects. It is challenging because the computational behavior of the resulting hybrids is often much worse than the behavior of their components.In this paper, we propose a new combination method which is computationally robust in the sense that the combination of decidable formalisms is again decidable, and which, nonetheless, allows non-trivial interactions between the combined components.The new method, called E-connection, is defined in terms of abstract description systems (ADSs), a common generalization of description logics, many logics of time and space, as well as modal and epistemic logics. The basic idea of E-connections is that the interpretation domains of n combined systems are disjoint, and that these domains are connected by means of n-ary 'link relations.' We define several natural variants of E-connections and study in-depth the transfer of decidability from the component systems to their E-connections.Description Logic-Spatial Logic. A description logic L 1 (say, ALC or SHIQ [42]) talks about a domain D 1 of abstract objects. A spatial logic L 2 (say, qualitative S4 u [70,16,66,30] or quantitative MS [69,48]) talks about some spatial domain D 2 . An obvious E-connection is given by the relation E ⊆ D 1 ×D 2 defined by taking (x, y) ∈ E iff y belongs to the spatial extension of x-whenever x occupies some space. Then, given an L 1 -concept, say, river,

Recently, it has been shown that the small DL EL, which allows for conjunction and existential restrictions, has better algorithmic properties than its counterpart FL₀, which allows for conjunction and value restrictions. Whereas the subsumption problem in FL₀ becomes already intractable in the presence of aclyc TBoxes, it remains tractable in EL even w.r.t. general concept inclusion axioms (GCIs). On the one hand, we will extend the positive result for EL by identifying a set of expressive means that can be added to EL without sacrificing tractability. On the other hand, we will show that basically all other additions of typical DL constructors to EL with GCIs make subsumption intractable, and in most cases even EXPTIME-complete. In addition, we will show that subsumption in FL₀ with GCIs is EXPTIME-complete.

Most of the research on temporalized Description Logics (DLs) has concentrated on the case where temporal operators can be applied to concepts, and sometimes additionally to TBox axioms and ABox assertions. The aim of this article is to study temporalized DLs where temporal operators on TBox axioms and ABox assertions are available, but temporal operators on concepts are not. While the main application of existing temporalized DLs is the representation of conceptual models that explicitly incorporate temporal aspects, the family of DLs studied in this article addresses applications that focus on the temporal evolution of data and of ontologies. Our results show that disallowing temporal operators on concepts can significantly decrease the complexity of reasoning. In particular, reasoning with rigid roles (whose interpretation does not change over time) is typically undecidable without such a syntactic restriction, whereas our logics are decidable in elementary time even in the presence of rigid roles. We analyze the effects on computational complexity of dropping rigid roles, dropping rigid concepts, replacing temporal TBoxes with global ones, and restricting the set of available temporal operators. In this way, we obtain a novel family of temporalized DLs whose complexity ranges from 2- ExpTime-complete via NExpTime-complete to ExpTime-complete.

We survey temporal description logics that are based on standard temporal logics such as LTL and CTL. In particular, we concentrate on the computational complexity of the satisfiability problem and algorithms for deciding it.

As fragments of first-order logic, Description logics (DLs) do not provide nonmonotonic features such as defeasible inheritance and default rules. Since many applications would benefit from the availability of such features, several families of nonmonotonic DLs have been developed that are mostly based on default logic and autoepistemic logic. In this paper, we consider circumscription as an interesting alternative approach to nonmonotonic DLs that, in particular, supports defeasible inheritance in a natural way. We study DLs extended with circumscription under different language restrictions and under different constraints on the sets of minimized, fixed, and varying predicates, and pinpoint the exact computational complexity of reasoning for DLs ranging from ALC to ALCIO and ALCQO. When the minimized and fixed predicates include only concept names but no role names, then reasoning is complete for NExp NP . It becomes complete for NP NExp when the number of minimized and fixed predicates is bounded by a constant. If roles can be minimized or fixed, then complexity ranges from NExp NP to undecidability.