Matching of concepts with variables (concept patterns) is a relatively new operation that has been introduced in the context of concept description languages (description logics), originally to help discard unimportant aspects of large concepts appearing in industrial-strength knowledge bases. This paper proposes a new approach to performing matching, based on a "concept-centered" normal form, rather than the more standard "structural subsumption" normal form for concepts. As a result, matching can be performed (in polynomial time) using arbitrary concept patterns of the description language FL: , thus removing restrictions from previous work. The paper also addresses the question of matching problems with additional "side conditions", which were motivated by practical experience. 1 Introduction The traditional inference problems for Description Logic (DL) systems (like subsumption) are now wellinvestigated. This means that algorithms are available for solving the subsumption proble...

Abstract. Systems of language equations of the form {ϕ(X1,..., Xn) = ∅, ψ(X1,..., Xn) � = ∅} are studied, where ϕ, ψ may contain set-theoretic operations and concatenation; they can be equivalently represented as strict inequalities ξ(X1,..., Xn) ⊂ L0. It is proved that the problem whether such an inequality has a solution is Σ2-complete, the problem whether it has a unique solution is in (Σ3 ∩Π3)\(Σ2 ∪Π2), the existence of a regular solution is a Σ1-complete problem, while testing whether there are finitely many solutions is Σ3-complete. The class of languages representable by their unique solutions is exactly the class of recursive sets, though a decision procedure cannot be algorithmically constructed out of an inequality, even if a proof of solution uniqueness is attached. 1

In a previous work, we describe a method to combine decision procedures for the word problem for theories sharing constructors. One of the requirements of our combination method is that constructors be collapse-free. This paper removes that requirement by modifying the method so that it applies to non-collapse-free constructors as well. This broadens the scope of our combination results considerably, for example in the direction of equational theories corresponding to modal logics.

Uni cation considers concept patterns, i.e., concept descriptions with variables, and tries to make these descriptions equivalent by replacing the variables by appropriate concept descriptions. Baader and Kusters have shown that uni cation in FL reg , a description logic that allows for the concept constructors top concept, concept conjunction, and value restrictions as well as the role constructors union, composition, and transitive closure, is an ExpTime-complete problem and that solvable FL reg - uni cation problems always have least uni ers. In the present paper, we generalize these results to a DL which extends FL reg by the bottom concept.

We give a resolution-based procedure for deciding unifiability in the variety of bounded distributive lattices. The main idea is to use a structure-preserving translation to clause form to reduce the problem of testing the satisfiability of a unification problem S to the problem of checking the satisfiability of a set Φ_S of (constrained) clauses. These ideas can be used for unification with free constants and for unification with linear constant restrictions. Complexity issues are also addressed.

Abstract. 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, we show that EL-unification is NP-complete, and thus has the same complexity as EL-matching. 1

paper [6] describing the evolution of the current web from a human-processable environment to a machine-processable one. The basic idea was to annotate web resources and give them a machine-processable meaning; the Semantic Web was born.

Language equations are equations where both the constants occurring in the equations and the solutions are formal languages. They have first been introduced in formal language theory, but are now also considered in other areas of computer science. In the present paper, we restrict the attention to language equations with one-sided concatenation, but in contrast to previous work on these equations, we allow not just union but all Boolean operations to be used when formulating them. In addition, we are not just interested in deciding solvability of such equations, but also in deciding other properties of the set of solutions, like its cardinality (finite, infinite, uncountable) and whether it contains least/greatest solutions. We show that all these decision problems are ExpTime-complete.

Matching of concepts against patterns is a new inference task in Description Logics, which was originally motivated by applications of the Classic system. Consequently, the work on this problem was until now mostly concerned with sublanguages of the Classic language, which does not allow for existential restrictions. Motivated by an application in chemical process engineering, which requires a description language with existential restrictions, this paper investigates the matching problem in Description Logics with existential restrictions. It turns out that existential restrictions make matching more complex in two respects. First, whereas matching in sublanguages of Classic is polynomial, deciding the existence of matchers is an NP-complete problem in the presence of existential restrictions. Second, whereas in sublanguages of Classic solvable matching problems have a unique least matcher, this is not the case for languages with existential restrictions. Thus, it is not...

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