Carsten Lutz

Conjunctive query answering for the description logic SHIQ

Conjunctive queries play an important role as an expressive query language for Description Logics (DLs). Although modern DLs usually provide for transitive roles, it was an open problem whether conjunctive query answering over DL knowledge bases is decidable if transitive roles are admitted in the query. In this paper, we consider conjunctive queries over knowledge bases formulated in the popular DL SHIQ and allow transitive roles in both the query and the knowledge base. We show that query answering is decidable and establish the following complexity bounds: regarding combined complexity, we devise 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. Regarding data complexity, we prove co-NP-completeness.

Temporal Description Logics: A Survey

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.

Deciding inseparability and conservative extensions in the description logic EL

We study the problem of deciding whether two ontologies are inseparable w.r.t. a signature @S, i.e., whether they have the same consequences in the signature @S. A special case is to decide whether the extension of an ontology is conservative. By varying the language in which ontologies are formulated and the query language that is used to describe consequences, we obtain different versions of the problem. We focus on the lightweight description logic EL as an ontology language, and consider query languages based on (i) subsumption queries, (ii) instance queries over ABoxes, (iii) conjunctive queries over ABoxes, and (iv) second-order logic. For query languages (i) to (iii), we establish ExpTime-completeness of both inseparability and conservative extensions. Case (iv) is equivalent to a model-theoretic version of inseparability and conservative extensions, and we prove it to be undecidable. We also establish a number of robustness properties for inseparability.

Complexity and succinctness of public announcement logic

There is a recent trend of extending epistemic logic (EL) with dynamic operators that allow to express the evolution of knowledge and induced by knowledge-changing actions. The most basic such extension is public announcement logic (PAL), which is obtained from EL by adding an operator for truthful public announcements. In this paper, we consider the computational complexity of PAL and show that it coincides with that of EL. This holds in the single- and multi-agent case, and also in the presence of common knowledge operators. We also prove that there are properties that can be expressed exponentially more succinct in PAL than in EL. This shows that, despite the known fact that PAL and EL have the same expressive power, there is a benefit in adding the public announcement operator to EL: it exponentially increases the succinctness of formulas without having negative effects on computational complexity.

The complexity of circumscription in description logic

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 NEXPNP. It becomes complete for NPNEXP when the number of minimized and fixed predicates is bounded by a constant. If roles can be minimized or fixed, then complexity ranges from NEXPNP to undecidability.

LTL over description logic axioms

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.

Fusions of description logics and abstract description systems

Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion. Though description logics are closely related to modal logics, they are not necessarily normal. In addition, ABox reasoning in description logics is not covered by the results from modal logics. In this paper, we extend the decidability transfer results from normal modal logics to a large class of description logics. To cover different description logics in a uniform way, we introduce abstract description systems, which can be seen as a common generalization of description and modal logics, and show the transfer results in this general setting.

Keys, nominals, and concrete domains

Many description logics (DLs) combine knowledge representation on an abstract, logical level with an interface to “concrete” domains like numbers and strings with built-in predicates such as <, +, and prefix-of. These hybrid DLs have turned out to be useful in several application areas, such as reasoning about conceptual database models. We propose to further extend such DLs with key constraints that allow the expression of statements like “US citizens are uniquely identified by their social security number”. Based on this idea, we introduce a number of natural description logics and perform a detailed analysis of their decidability and computational complexity. It turns out that naive extensions with key constraints easily lead to undecidability, whereas more careful extensions yield NExpTime-complete DLs for a variety of useful concrete domains.

Complexity of Terminological Reasoning Revisited

TBoxes in their various forms are key components of knowledge representation systems based on description logics (DLs) since they allow for a natural representation of terminologicsd knowledge. Largely due to a classical result given by Nebel [15], complexity analyses for DLs have, until now, mostly failed to take into account the most basic form of TBoxes, so-called acyclic TBoxes. In this paper, we concentrate on DLs for which reasoning without TBoxes is PSPACE-complete, sad show that there exist logics for which the complexity of reasoning remains in PSPACE if acyclic TBoxes are added and also logics for which the complexity increases. This demonstrates that it is necessary to take acyclic TBoxes into account for complexity cinalyses.

CEL: a polynomial-time reasoner for life science ontologies

CEL (Classifier for eL) is a reasoner for the small description logic ∈L + which can be used to compute the subsuinption hierarchy induced by eL + ontologies. The most distinguishing feature of CEL is that, unlike all other modern DL reasoners, it is based on a polynomial-time subsumption algorithm, which allows it to process very large ontologies in reasonable time. In spite of its restricted expressive power, eL+ is well-suited for formulating life science ontologies.