ABSTRACT. We introduce a very general variety of path description dependencies (PDDs) for an expressive dialect of the FunDL family of description logics called structural PDDs. In general, PDDs enable capturing equality generating dependencies for an ontology in a progressively more fine-grained manner, starting with equality implied by simple alignment of facts about entities through to new structural PDDs in which equality only follows according to a structured alignment of non-empty sets of facts about an entity. We show that logical consequence for this new FunDL dialect is decidable if a given ontology appeals to an exclusive use of structural PDDs, but that logical consequence becomes undecidable when more course grained varieties of PDDs are also allowed in the ontology. An extension to a referring expression type language for defining concepts in this description logic to serve as referring expressions that depend on structural identification is also presented and is tied to a diagnosis of a singularity condition for such concepts to logical consequence of PDDs for an ontology.

ABSTRACT. Standard Description Logics (DLs) can encode quantitative aspects of an application domain through either number restrictions, which constrain the number of individuals that are in a certain relationship with an individual, or concrete domains, which can be used to assign concrete values to individuals using so-called features. These two mechanisms have been extended towards very expressive DLs, for which reasoning nevertheless remains decidable. Number restrictions have been generalized to more powerful comparisons of sets of role successors in ALCSCC, while the comparison of feature values of different individuals in ALC(D) has been studied in the context of ω-admissible concrete domains D. In this paper, we combine both formalisms and investigate the complexity of reasoning in the thus obtained DL ALCOSCC(D), which additionally includes the ability to refer to specific individuals by name. We show that, in spite of its high expressivity, the consistency problem for this DL is ExpTime-complete, assuming that the constraint satisfaction problem of D is also decidable in exponential time. It is thus not higher than the complexity of the basic DL ALC. At the same time, we show that many natural extensions to this DL, including a tighter integration of the concrete domain and number restrictions, lead to undecidability.

ABSTRACT. The EL family of description logics facilitates efficient polynomial-time reasoning and has been standardized as the profile OWL 2 EL of the Web Ontology Language. EL can represent and reason not only with symbolic knowledge but also with concrete knowledge expressed by numbers, strings, and other concrete datatypes. Such concrete domains must be convex to avoid introducing disjunctions “through the backdoor.” However, the hitherto existing concrete domains provide only limited utility. In order to overcome this issue, we introduce a novel form of concrete domains based on semi-lattices. They are convex by design and can thus be integrated into Horn-DLs such as EL. Moreover, they allow for FBoxes to express dependencies between concrete features. We describe four instantiations concerned with real intervals, 2D-polygons, regular languages, and graphs. This is an extended abstract of an article accepted at the 15th International Symposium on Frontiers of Combining Systems (FroCoS 2025).

ABSTRACT. We propose an extension of Description Logics (DLs) with generic concepts and conditional axioms. Inspired by object-oriented languages, generic concepts allow a compact definition of concepts with similar structures. For example, one can define a generic concept Owner[X] to describe objects that own another object from X, and later use a specific replacement of the parameter X, such as Owner[Pet] representing pet owners. Conditional axioms can be used to set bounds on the values that replace the generic parameters. For example, we could restrict replacements of X in a concept Feeder[X] to only subconcepts of Pet. As the set of possible parameter replacements can be infinite and even uncountable, the generic extensions are, in general, undecidable. To identify decidable generic DLs, we focus on the case of terminologies, requiring that variables are only used in definitions of generic concepts. We formulate syntactic restrictions that allow reducing generic to classical entailment and further conditions that ensure decidability.

ABSTRACT. Description Logics (DLs) excel at representing structured knowledge in several application domains, but fall very short when it comes to reasoning about their numeric aspects. We consider the expressive DL ALCHOIQ with closed predicates and extend it with features ranging over user-specified finite numeric intervals, feature assertions, and local additive constraints on feature values. We illustrate the power of this language for describing problems that involve ontological and numeric reasoning and study reasoning problems that go beyond satisfiability, such as finding models that minimize some costs. We show that these additional numeric modeling and reasoning capabilities can be accommodated by extending a standard reasoning technique for ALCHOIQ using linear inequalities, and the extension does not necessarily increase the worst-case computational cost.