The instance problem and the most specific concept in the description logic EL w.r.t. terminological cycles with descriptive semantics

In two previous reports we have investigated both standard and non-standard inferences in the presence of terminological cycles for the description logic EL, which allows for conjunctions, existential restrictions, and the top concept. Regarding standard inference problems, it was shown there that the subsumption problem remains polynomial for all three types of semantics usually considered for cyclic definitions in description logics, and that the instance problem remains polynomial for greatest fixpoint semantics. Regarding non-standard inference problems, it was shown that, w.r.t. greatest fixpoint semantics, the least common subsumer and the most specific concept always exist and can be computed in ploynomial time, and that, w.r.t. descriptive semantics, the least common subsumer need not exist. The present report is concerned with two problems left open by this previous work, namely the instance problem and the problem of computing most specific concepts w.r.t. descriptive semantics, which is the usual first-order semantics for description logic. We will show that the instance problem is polynomial also in this context. Similar to the case of the least common subsumer, the most specific concept w.r.t. descriptive semantics need not exist, but we are able to characterize the cases in which it exists and give a decidable sufficient condition for the existence of the most specific concept. Under this condition, it can be computed in polynomial time

Subsumption and Instance Problem in ELH w.r.t. General TBoxes

Recently, it was shown for the DL EL that subsumption and instance problem w.r.t. cyclic terminologies can be decided in polynomial time. In this paper, we show that both problems remain tractable even when admitting general concept inclusion axioms and simple role inclusion axioms

Reasoning in ELH w.r.t. General Concept Inclusion Axioms

In the area of Description Logic (DL) based knowledge representation, research on reasoning w.r.t. general terminologies has mainly focused on very expressive DLs. Recently, though, it was shown for the DL EL, providing only the constructors conjunction and existential restriction, that the subsumption problem w.r.t. cyclic terminologies can be decided in polynomial time, a surprisingly low upper bound. In this paper, we show that even admitting general concept inclusion (GCI) axioms and role hierarchies in EL terminologies preserves the polynomial time upper bound for subsumption. We also show that subsumption becomes co-NP hard when adding one of the constructors number restriction, disjunction, and `allsome', an operator used in the DL k-rep. An interesting implication of the first result is that reasoning over the widely used medical terminology snomed is possible in polynomial time

Most specific consequences in the description logic EL

The notion of a most specific consequence with respect to some terminological box is introduced, conditions for its existence in the description logic EL and its variants are provided, and means for its computation are developed. Algebraic properties of most specific consequences are explored. Furthermore, several applications that make use of this new notion are proposed and, in particular, it is shown how given terminological knowledge can be incorporated in existing approaches for the axiomatization of observations. For instance, a procedure for an incremental learning of concept inclusions from sequences of interpretations is developed

Unification in the Description Logic EL

The Description Logic EL has recently drawn considerable attention since, on the one hand, important inference problems such as the subsumption problem are polynomial. On the other hand, EL is used to define large biomedical ontologies. Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The main result of this paper is that unification in EL is decidable. More precisely, EL-unification is NP-complete, and thus has the same complexity as EL-matching. We also show that, w.r.t. the unification type, EL is less well-behaved: it is of type zero, which in particular implies that there are unification problems that have no finite complete set of unifiers.Comment: 31page

On the Computation of Common Subsumers in Description Logics

Description logics (DL) knowledge bases are often build by users with expertise in the application domain, but little expertise in logic. To support this kind of users when building their knowledge bases a number of extension methods have been proposed to provide the user with concept descriptions as a starting point for new concept definitions. The inference service central to several of these approaches is the computation of (least) common subsumers of concept descriptions. In case disjunction of concepts can be expressed in the DL under consideration, the least common subsumer (lcs) is just the disjunction of the input concepts. Such a trivial lcs is of little use as a starting point for a new concept definition to be edited by the user. To address this problem we propose two approaches to obtain "meaningful" common subsumers in the presence of disjunction tailored to two different methods to extend DL knowledge bases. More precisely, we devise computation methods for the approximation-based approach and the customization of DL knowledge bases, extend these methods to DLs with number restrictions and discuss their efficient implementation

Hybrid Unification in the Description Logic EL

Unification in Description Logics (DLs) has been proposed as an inference service that can, for example, be used to detect redundancies in ontologies. For the DL EL, which is used to define several large biomedical ontologies, unification is NP-complete. However, the unification algorithms for EL developed until recently could not deal with ontologies containing general concept inclusions (GCIs). In a series of recent papers we have made some progress towards addressing this problem, but the ontologies the developed unification algorithms can deal with need to satisfy a certain cycle restriction. In the present paper, we follow a different approach. Instead of restricting the input ontologies, we generalize the notion of unifiers to so-called hybrid unifiers. Whereas classical unifiers can be viewed as acyclic TBoxes, hybrid unifiers are cyclic TBoxes, which are interpreted together with the ontology of the input using a hybrid semantics that combines fixpoint and descriptive semantics. We show that hybrid unification in EL is NP-complete and introduce a goal-oriented algorithm for computing hybrid unifiers

Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis

Description Logic (abbrv. DL) belongs to the field of knowledge representation and reasoning. DL researchers have developed a large family of logic-based languages, so-called description logics (abbrv. DLs). These logics allow their users to explicitly represent knowledge as ontologies, which are finite sets of (human- and machine-readable) axioms, and provide them with automated inference services to derive implicit knowledge. The landscape of decidability and computational complexity of common reasoning tasks for various description logics has been explored in large parts: there is always a trade-off between expressibility and reasoning costs. It is therefore not surprising that DLs are nowadays applied in a large variety of domains: agriculture, astronomy, biology, defense, education, energy management, geography, geoscience, medicine, oceanography, and oil and gas. Furthermore, the most notable success of DLs is that these constitute the logical underpinning of the Web Ontology Language (abbrv. OWL) in the Semantic Web. Formal Concept Analysis (abbrv. FCA) is a subfield of lattice theory that allows to analyze data-sets that can be represented as formal contexts. Put simply, such a formal context binds a set of objects to a set of attributes by specifying which objects have which attributes. There are two major techniques that can be applied in various ways for purposes of conceptual clustering, data mining, machine learning, knowledge management, knowledge visualization, etc. On the one hand, it is possible to describe the hierarchical structure of such a data-set in form of a formal concept lattice. On the other hand, the theory of implications (dependencies between attributes) valid in a given formal context can be axiomatized in a sound and complete manner by the so-called canonical base, which furthermore contains a minimal number of implications w.r.t. the properties of soundness and completeness. In spite of the different notions used in FCA and in DLs, there has been a very fruitful interaction between these two research areas. My thesis continues this line of research and, more specifically, I will describe how methods from FCA can be used to support the automatic construction and extension of DL ontologies from data

Standard and Non-standard reasoning in Description Logics

The present work deals with Description Logics (DLs), a class of knowledge representation formalisms used to represent and reason about classes of individuals and relations between such classes in a formally well-defined way. We provide novel results in three main directions. (1) Tractable reasoning revisited: in the 1990s, DL research has largely answered the question for practically relevant yet tractable DL formalisms in the negative. Due to novel application domains, especially the Life Sciences, and a surprising tractability result by Baader, we have re-visited this question, this time looking in a new direction: general terminologies (TBoxes) and extensions thereof defined over the DL EL and extensions thereof. As main positive result, we devise EL++(D)-CBoxes as a tractable DL formalism with optimal expressivity in the sense that every additional standard DL constructor, every extension of the TBox formalism, or every more powerful concrete domain, makes reasoning intractable. (2) Non-standard inferences for knowledge maintenance: non-standard inferences, such as matching, can support domain experts in maintaining DL knowledge bases in a structured and well-defined way. In order to extend their availability and promote their use, the present work extends the state of the art of non-standard inferences both w.r.t. theory and implementation. Our main results are implementations and performance evaluations of known matching algorithms for the DLs ALE and ALN, optimal non-deterministic polynomial time algorithms for matching under acyclic side conditions in ALN and sublanguages, and optimal algorithms for matching w.r.t. cyclic (and hybrid) EL-TBoxes. (3) Non-standard inferences over general concept inclusion (GCI) axioms: the utility of GCIs in modern DL knowledge bases and the relevance of non-standard inferences to knowledge maintenance naturally motivate the question for tractable DL formalism in which both can be provided. As main result, we propose hybrid EL-TBoxes as a solution to this hitherto open question