In Description Logics the reasoning service most specific concept (msc) constructs a concept description that generalizes an ABox individual into a concept description. For the Description Logic EL the msc may not exist, if computed with respect to general EL-TBoxes or cyclic ABoxes. However, it is still possible to find a concept description that is the msc up to a fixed role-depth, i.e. with respect to a maximal nesting of quantifiers. In this report we present a practical approach for computing the roledepth bounded msc, based on the polynomial-time completion algorithm for EL. We extend these methods to Prob-EL⁰¹c , which is a probabilistic variant of EL. Together with the companion report [9] this report devises computation methods for the bottom-up construction of knowledge bases for EL and Prob-EL⁰¹c .

The Description Logic EL is an inexpressive knowledge representation language, which nevertheless has recently drawn considerable attention in the knowledge representation and the ontology community 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. In a recent paper, we have shown that unification in EL is NP-complete, and thus of a complexity that is considerably lower than in other Description Logics of comparably restricted expressive power. In this paper, we introduce a new NP-algorithm for solving unification problem in EL, which is based on a reduction to satisfiability in propositional logic (SAT). The advantage of this new algorithm is, on the one hand, that it allows us to employ highly optimized state of the art SAT solverswhen implementing an EL-unification algorithm. On the other hand, this reduction provides us with a proof of the fact that EL-unification is in NP that is much simpler than the one given in our previous paper on EL-unification.

Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The inexpressive Description Logic EL is of particular interest in this context since, on the one hand, several large biomedical ontologies are defined using EL. On the other hand, unification in EL has recently been shown to be NP-complete, and thus of considerably lower complexity than unification in other DLs of similarly restricted expressive power. However, EL allows the use of the top concept (>), which represents the whole interpretation domain, whereas the large medical ontology SNOMEDCT makes no use of this feature. Surprisingly, removing the top concept from EL makes the unification problem considerably harder. More precisely, we will show that unification in EL without the top concept is PSpace-complete. / This is an updated version of the original report that includes Appendix A on locality of unifiers.

Unification has been investigated both in modal logics and in description logics, albeit with different motivations. In description logics, unification can be used to detect redundancies in ontologies. In this context, it is not sufficient to decide unifiability, one must also compute appropriate unifiers and present them to the user. For the description logic EL, which is used to define several large biomedical ontologies, deciding unifiability is an NP-complete problem. It is known that every solvable EL-unification problem has a minimal unifier, and that every minimal unifier is a local unifier. Existing unification algorithms for EL compute all minimal unifiers, but additionally (all or some) non-minimal local unifiers. Computing only the minimal unifiers would be better since there are considerably less minimal unifiers than local ones, and their size is usually also quite small. In this paper we investigate the question whether the known algorithms for EL-unification can be modified such that they compute exactly the minimal unifiers without changing the complexity and the basic nature of the algorithms. Basically, the answer we give to this question is negative.

Fuzzy description logics can be used to model vague knowledge in application domains. This paper analyses the consistency and satisfiability problems in the description logic SHI with semantics based on a complete residuated De Morgan lattice. The problems are undecidable in the general case, but can be decided by a tableau algorithm when restricted to finite lattices. For some sublogics of SHI, we provide upper complexity bounds that match the complexity of crisp reasoning.

In the work of Baader and Distel, a method has been proposed to axiomatize all general concept inclusions (GCIs) expressible in the description logic EL⊥ and valid in a given interpretation I. This provides us with an effective method to learn EL⊥-ontologies from interpretations, which itself can be seen as a different representation of linked data. In another report, we have extended this approach to handle errors in the data. This has been done by not only considering valid GCIs but also those whose confidence is above a certain threshold 𝑐. In the present work, we shall extend the results by describing another way to compute bases of confident GCIs. We furthermore provide experimental evidence that this approach can be useful for practical applications. We finally show that the technique of unravelling can also be used to effectively turn confident EL⊥gfp-bases into EL⊥-bases.

Axiom pinpointing has been introduced in description logics (DLs) to help the user to understand the reasons why consequences hold and to remove unwanted consequences by computing minimal (maximal) subsets of the knowledge base that have (do not have) the consequence in question. The pinpointing algorithms described in the DL literature are obtained as extensions of the standard tableau-based reasoning algorithms for computing consequences from DL knowledge bases. Although these extensions are based on similar ideas, they are all introduced for a particular tableau-based algorithm for a particular DL. The purpose of this paper is to develop a general approach for extending a tableau-based algorithm to a pinpointing algorithm. This approach is based on a general definition of „tableaux algorithms,' which captures many of the known tableau-based algorithms employed in DLs, but also other kinds of reasoning procedures.

The eight topological RCC8(or Egenhofer-Franzosa)- relations between spatial regions play a fundamental role in spatial reasoning, spatial and constraint databases, and geographical information systems. In analogy with Halpern and Shoham’s modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the RCC8-relations. The semantics is based on region spaces induced by standard topological spaces, in particular the real plane. We investigate the expressive power and computational complexity of the logics obtained in this way. It turns our that, similar to Halpern and Shoham’s logic, the expressive power is rather natural, but the computational behavior is problematic: topological modal logics are usually undecidable and often not even recursively enumerable. This even holds if we restrict ourselves to classes of finite region spaces or to substructures of region spaces induced by topological spaces. We also analyze modal logics based on the set of RCC5relations, with similar results.

We make a first step towards adapting an existing approach for privacypreserving publishing of linked data to Description Logic (DL) ontologies. We consider the case where both the knowledge about individuals and the privacy policies are expressed using concepts of the DL EL, which corresponds to the setting where the ontology is an EL instance store. We introduce the notions of compliance of a concept with a policy and of safety of a concept for a policy, and show how optimal compliant (safe) generalizations of a given EL concept can be computed. In addition, we investigate the complexity of the optimality problem.

In contrast to qualitative linear temporal logics, which can be used to state that some property will eventually be satisfied, metric temporal logics allow to formulate constraints on how long it may take until the property is satisfied. While most of the work on combining Description Logics (DLs) with temporal logics has concentrated on qualitative temporal logics, there has recently been a growing interest in extending this work to the quantitative case. In this paper, we complement existing results on the combination of DLs with metric temporal logics over the natural numbers by introducing interval-rigid names. This allows to state that elements in the extension of certain names stay in this extension for at least some specified amount of time.