In Description Logics (DLs), both tableau-based and automatabased algorithms are frequently used to show decidability and complexity results for basic inference problems such as satisfiability of concepts. Whereas tableau-based algorithms usually yield worst-case optimal algorithms in the case of PSpace-complete logics, it is often very hard to design optimal tableau-based algorithms for ExpTime-complete DLs. In contrast, the automata-based approach is usually well-suited to prove ExpTime upper-bounds, but its direct application will usually also yield an ExpTime-algorithm for a PSpace-complete logic since the (tree) automaton constructed for a given concept is usually exponentially large. In the present paper, we formulate conditions under which an on-the-fly construction of such an exponentially large automaton can be used to obtain a PSpace-algorithm. We illustrate the usefulness of this approach by proving a new PSpace upper-bound for satisfiability of concepts w.r.t. acyclic terminologies in the DL SI, which extends the basic DL ALC with transitive and inverse roles.

Tableau-based decision procedures have been successfully used for solving a wide variety of problems. For some applications, nonetheless, it is desirable not only to obtain a Boolean answer, but also to detect the causes for such a result. In this report, a method for finding explanations on tableau-based procedures is explored, generalizing previous results on the field. The importance and use of the method is shown by means of examples.

Formal Concept Analysis (FCA) can be used to analyze data given in the form of a formal context. In particular, FCA provides efficient algorithms for computing a minimal basis of the implications holding in the context. In this paper, we extend classical FCA by considering data that are represented by relational structures rather than formal contexts, and by replacing atomic attributes by complex formulae defined in some logic. After generalizing some of the FCA theory to this more general form of contexts, we instantiate the general framework with attributes defined in the Description Logic (DL) EL, and with relational structures over a signature of unary and binary predicates, i.e., models for EL. In this setting, an implication corresponds to a so-called general concept inclusion axiom (GCI) in EL. The main technical result of this report is that, in EL, for any finite model there is a finite set of implications (GCIs) holding in this model from which all implications (GCIs) holding in the model follow.

The description logic EL⁺ has recently proved practically useful in the life science domain with presence of several large-scale biomedical ontologies such as Snomed ct. To deal with ontologies of this scale, standard reasoning of classification is essential but not sufficient. The ability to extract relevant fragments from a large ontology and to incrementally classify it has become more crucial to support ontology design, maintenance and reuse. In this paper, we propose a pragmatic approach to module extraction and incremental classification for EL⁺ ontologies and report on empirical evaluations of our algorithms which have been implemented as an extension of the CEL reasoner.

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.

Description Logics (DLs) are a family of knowledge representation formalisms used for terminological reasoning. They have a wide range of applications such as medical knowledge-bases, or the semantic web. Research on DLs has been focused on the development of sound and complete inference algorithms to decide satisfiability and subsumption for increasingly expressive DLs. Non-standard inferences are a group of relatively new inference services which provide reasoning support for the building, maintaining, and deployment of DL knowledge-bases. So far, non-standard inferences are not available for very expressive DLs. In this paper we present first results on non-standard inferences for DLs with transitive roles. As a basis, we give a structural characterization of subsumption for DLs where existential and value restrictions can be imposed on transitive roles. We propose sound and complete algorithms to compute the least common subsumer (lcs).

The paper contains a proof of completeness of a goal-directed inference system for general E-unification with eager Variable Elimination. The proof is based on an analysis of a concept of ground, equational proof. The theory of equational proofs is developed in the first part. Solving variables in a goal is then shown to be reflected in defined transformations of an equational proof. The termination of these transformations proves termination of inferences with eager Variable Elimination.

Previous results for combining decision procedures for the word problem in the non-disjoint case do not apply to equational theories induced by modal logics - which are not disjoint for sharing the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other types of equational theories. In this paper, we present a new approach for combining decision procedures for the word problem in the non-disjoint case that applies to equational theories induced by modal logics, but is not restricted to them. The known fusion decidability results for modal logics are instances of our approach. However, even for equational theories induced by modal logics our results are more general since they are not restricted to so-called normal modal logics. / This report has also appeared as Report No. 03-03, Department of Computer Science, The University of Iowa.

In two previous papers we have investigates the problem of computing the least common subsumer (lcs) and the most specific concept (msc) for the description logic EL in the presence of terminological cycles that are interpreted with descriptive semantics, which is the usual first-order semantics for description logics. In this setting, neither the lcs nor the msc needs to exist. We were able to characterize the cases in which the lcs/msc exists, but it was not clear whether this characterization yields decidability of the existence problem. In the present paper, we develop a common graph-theoretic generalization of these characterizations, and show that the resulting property is indeed decidable, thus yielding decidability of the existence of the lcs and the msc. This is achieved by expressing the property in monadic second-order logic on infinite trees. We also show that, if it exists, then the lcs/msc can be computed in polynomial time.

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.