Abduction is a central approach to explain missing entailments from a knowledge base by providing a hypothesis, that would, if added to the knowledge base, make the missing entailment become true. Abduction under repair semantics has recently been investigated in detail, where several desirable properties and optimality criteria were considered, such as signature-restrictions and minimality in size and of introduced conflicts. Naturally, hypotheses that satisfy more than one of these properties or combine a property with an optimality criterion would be even more desirable for applications. So far, such hypotheses have not been investigated in the literature. In the present paper, we consider the ABox abduction problem for hypotheses satisfying more than one property or additional optimality criteria, for EL_bot under brave and AR semantics. Our main observation is that often requiring additional properties for hypotheses does not lead to an increase of complexity.

Given a knowledge base (KB) with a non-entailed fact, the ABox abduction problem asks for possible extensions of the KB that would entail this fact. This problem has many applications, ranging from diagnosis to explainability and repair. ABox abduction has been well-investigated for consistent KBs and classical semantics, but little is known for the case of inconsistent KBs, which can be caused by erroneous data. In this paper we define suitable notions of abduction in this setting and propose criteria that guide abduction towards "useful" hypotheses. To regain meaningful reasoning in the presence of inconsistencies, we use well-established repair semantics. We provide a comprehensive landscape of the complexity of ABox abduction under repair semantics, treating different variants of the abduction problem for the light-weight description logics DL-Lite and EL_bot.

We introduce the notion of contrastive ABox explanations to answer questions of the type "Why is a an instance of C, but b is not?". While there are various approaches for explaining positive entailments (why is C(a) entailed by the knowledge base) as well as missing entailments (why is C(b) not entailed) in isolation, contrastive explanations consider both at the same time, which allows them to focus on the relevant commonalities and differences between a and b. We develop an appropriate notion of contrastive explanations for the special case of ABox reasoning with description logic ontologies, and analyze the computational complexity for different variants under different optimality criteria, considering lightweight as well as more expressive description logics. We implemented a first method for computing one variant of contrastive explanations, and evaluated it on generated problems for realistic knowledge bases.

Abduction is the task of computing a sufficient extension of a knowledge base (KB) that entails a conclusion not entailed by the original KB. It serves to compute explanations, or hypotheses, for such missing entailments. While this task has been intensively investigated for perfect data and under classical semantics, less is known about abduction when erroneous data results in inconsistent KBs. In this paper we define a suitable notion of abduction under repair semantics, and propose a set of minimality criteria that guides abduction towards `useful' hypotheses. We provide initial complexity results on deciding existence of and verifying abductive solutions with these criteria, under different repair semantics and for the description logics DL-Lite and EL_bot.

We introduce Craig interpolation and related notions such as uniform interpolation, Beth definability, and theory decomposition in classical propositional logic. We present four approaches to computing interpolants: via quantifier elimination, from formulas in disjunctive normal form, and by extraction from resolution or tableau refutations. We close with a discussion of the size of interpolants and links to circuit complexity.

Craig interpolation and uniform interpolation have many applications in knowledge representation, including explainability, forgetting, modularization and reuse, and even learning. At the same time, many relevant knowledge representation formalisms do in general not have Craig or uniform interpolation, and computing interpolants in practice is challenging. We have a closer look at two prominent knowledge representation formalisms, description logics and logic programming, and discuss theoretical results and practical methods for computing interpolants.

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.

Coalition Logic (CL) is a well-known formalism for reasoning about the strategic abilities of groups of agents in multi-agent systems. Coalition Logic with Common Knowledge (CLC) extends CL with operators from epistic logics, and thus with the ability to model the individual and common knowledge of agents. We have formalized the syntax and semantics of both logics in the interactive theorem prover Lean 4, and used it to prove soundness and completeness of its axiomatization. Our formalization uses the type class system to generalize over different aspects of CLC, thus allowing us to reuse some of to prove properties in related logics such as CL and CLK (CL with individual knowledge).

We introduce ontology-mediated planning, in which planning problems are combined with an ontology. Our formalism differs from existing ones in that we focus on a strong separation of the formalisms for describing planning problems and ontologies, which are only losely coupled by an interface. Moreover, we present a black-box algorithm that supports the full expressive power of OWL DL. This goes beyond what existing approaches combining automated planning with ontologies can do, which only support limited description logics such as DL-Lite and description logics that are Horn. Our main algorithm relies on rewritings of the ontology-mediated planning specifications into PDDL, so that existing planning systems can be used to solve them. The algorithm relies on justifications, which allows for a generic approach that is independent of the expressivity of the ontology language. However, dedicated optimizations for computing justifications need to be implemented to enable an efficient rewriting procedure. We evaluated our implementation on benchmark sets from several domains. The evaluation shows that our procedure works in practice and that tailoring the reasoning procedure has significant impact on the performance.

Concrete domains have been introduced in the context of Description Logics to allow references to qualitative and quantitative values. In particular, the class of ω-admissible concrete domains, which includes Allen's interval algebra, the region connection calculus (RCC8), and the rational numbers with ordering and equality, has been shown to yield extensions of 𝒜ℒ𝒞 for which concept satisfiability w.r.t. a general TBox is decidable. In this paper, we present an algorithm based on type elimination and use it to show that deciding the consistency of an 𝒜ℒ𝒞(𝔇) ontology is ExpTime-complete if the concrete domain 𝔇 is ω-admissible and its constraint satisfaction problem is decidable in exponential time. While this allows us to reason with concept and role assertions, we also investigate feature assertions f(a,c) that can specify a constant c as the value of a feature f for an individual a. We show that, under conditions satisfied by all known ω-admissible domains, we can add feature assertions without affecting the complexity.

We introduce ontology-mediated planning, in which planning problems are combined with an ontology. Our formalism differs from existing ones in that we focus on a strong separation of the formalisms for describing planning problems and ontologies, which are only losely coupled by an interface. Moreover, we present a black-box algorithm that supports the full expressive power of OWL DL. This goes beyond what existing approaches combining automated planning with ontologies can do, which only support limited description logics such as DL-Lite and description logics that are Horn. Our main algorithm relies on rewritings of the ontology-mediated planning specifications into PDDL, so that existing planning systems can be used to solve them. The algorithm relies on justifications, which allows for a generic approach that is independent of the expressivity of the ontology language. However, dedicated optimizations for computing justifications need to be implemented to enable an efficient rewriting procedure. We evaluated our implementation on benchmark sets from several domains. The evaluation shows that our procedure works in practice and that tailoring the reasoning procedure has significant impact on the performance.

One of the advantages of formalizing domain knowledge in OWL ontologies is that one can use reasoning systems to infer implicit information automatically. However, it is not always straightforward to understand why certain entailments are inferred, and others are not. The popular ontology editor Protégé offers two explanation services to deal with this issue: justifications for OWL 2 DL ontologies, and proofs generated by the reasoner ELK for lightweight OWL 2 EL ontologies. Since justifications are often insufficient for explaining inferences, there is thus only little tool support for more comprehensive explanations in expressive ontology languages, and there is no tool support at all to explain why something was not derived. In this paper, we present Evee, a Java library and a collection of plug-ins for Protégé that offers advanced explanation services for both inferred and missing entailments. Evee explains inferred entailments using proofs in description logics up to ALCH. Missing entailments can be explained using counterexamples and abduction. We evaluated the effectiveness and the interface design of our plug-ins with description logic experts, ontology engineers, and students in two user studies. In these experiments, we were able to not only validate the tool but also gather feedback and insights to improve the existing designs.

Ontologies based on Description Logics may contain errors, which are usually detected when reasoning produces consequences that follow from the ontology, but do not hold in the modelled application domain. In previous work, we have introduced repair approaches for E L ontologies that are optimal in the sense that they preserve a maximal amount of consequences. In this paper, we will, on the one hand, review these approaches, but with an emphasis on motivation rather than on technical details. On the other hand, we will describe new results that address the problems that optimal repairs may become very large or need not even exist unless strong restrictions on the terminological part of the ontology apply. We will show how one can deal with these problems by introducing concise representations of optimal repairs.

Logic-based approaches to AI have the advantage that their behavior can in principle be explained with the help of proofs of the computed consequences. For ontologies based on Description Logic (DL), we have put this advantage into practice by showing how proofs for consequences derived by DL reasoners can be computed and displayed in a user-friendly way. However, these methods are insufficient in applications where also numerical reasoning is relevant. The present paper considers proofs for DLs extended with concrete domains (CDs) based on the rational numbers, which leave reasoning tractable if integrated into the lightweight DL $\mathcal{E}\hspace{-0.1em}\mathcal{L}_\bot$. Since no implemented DL reasoner supports these CDs, we first develop reasoning procedures for them, and show how they can be combined with reasoning approaches for pure DLs, both for $\mathcal{E}\hspace{-0.1em}\mathcal{L}_\bot$ and the more expressive DL $\mathcal{ALC}$. These procedures are designed such that it is easy to extract proofs from them. We show how the extracted CD proofs can be combined with proofs on the DL side into integrated proofs that explain both the DL and the CD reasoning.

Understanding logical entailments derived by a description logic reasoner is not always straight-forward for ontology users. For this reason, various methods for explaining entailments using justifications and proofs have been developed and implemented as plug-ins for the ontology editor Prot\'eg\'e. However, when the user expects a missing consequence to hold, it is equally important to explain why it does not follow from the ontology. In this paper, we describe a new version of $\rm E{\scriptsize VEE}$, a Prot\'eg\'e plugin that now also provides explanations for missing consequences, via existing and new techniques based on abduction and counterexamples.