Research on knowledge graph embeddings has recently evolved into knowledge base embeddings, where the goal is not only to map facts into vector spaces but also constrain the models so that they take into account the relevant conceptual knowledge available. This paper examines recent methods that have been proposed to embed knowledge bases in description logic into vector spaces through the lens of their geometric-based semantics. We identify several relevant theoretical properties, which we draw from the literature and sometimes generalize or unify. We then investigate how concrete embedding methods fit in this theoretical framework.

In this paper, we explore a quantitative approach to querying inconsistent description logic knowledge bases. We consider weighted knowledge bases in which both axioms and assertions have (possibly infinite) weights, which are used to assign a cost to each interpretation based upon the axioms and assertions it violates. Two notions of certain and possible answer are defined by either considering interpretations whose cost does not exceed a given bound or restricting attention to optimal-cost interpretations. Our main contribution is a comprehensive analysis of the combined and data complexity of bounded cost satisfiability and certain and possible answer recognition, for description logics between ELbot and ALCO.

The Shapley value, originally introduced in cooperative game theory for wealth distribution, has found use in KR and databases for the purpose of assigning scores to formulas and database tuples based upon their contribution to obtaining a query result or inconsistency. In the present paper, we explore the use of Shapley values in ontology-mediated query answering (OMQA) and present a detailed complexity analysis of Shapley value computation (SVC) in the OMQA setting. In particular, we establish a PF/#P-hard dichotomy for SVC for ontology-mediated queries (T,q) composed of an ontology T formulated in the description logic ELHI_\bot and a connected constant-free homomorphism-closed query q. We further show that the #P-hardness side of the dichotomy can be strengthened to cover possibly disconnected queries with constants. Our results exploit recently discovered connections between SVC and probabilistic query evaluation and allow us to generalize existing results on probabilistic OMQA.

We introduce a constructive method applicable to a large number of description logics (DLs) for establishing the concept-based Beth definability property (CBP) based on sequent systems. Using the highly expressive DL RIQ as a case study, we introduce novel sequent calculi for RIQ-ontologies and show how certain interpolants can be computed from sequent calculus proofs, which permit the extraction of explicit definitions of implicitly definable concepts. To the best of our knowledge, this is the first sequent-based approach to computing interpolants and definitions within the context of DLs, as well as the first proof that RIQ enjoys the CBP. Moreover, due to the modularity of our sequent systems, our results hold for any restriction of RIQ, and are applicable to other DLs by suitable modifications.

The quality of Machine Learning (ML) models strongly depends on the input data, as such Feature Engineering (FE) is often required in ML. In addition, with the proliferation of ML-powered systems, especially in critical contexts, the need for interpretability and explainability becomes increasingly important. Since manual FE is time-consuming and requires case specific knowledge, we propose KRAFT, an AutoFE framework that leverages a knowledge graph to guide the generation of interpretable features. Our hybrid AI approach combines a neural generator to transform raw features through a series of transformations and a knowledge-based reasoner to evaluate features interpretability using Description Logics (DL). The generator is trained through Deep Reinforcement Learning (DRL) to maximize the prediction accuracy and the interpretability of the generated features. Extensive experiments on real datasets demonstrate that KRAFT significantly improves accuracy while ensuring a high level of interpretability.

Graph Neural Networks (GNNs) are effective for node classification in graph-structured data, but they lack explainability, especially at the global level. Current research mainly utilizes subgraphs of the input as local explanations or generates new graphs as global explanations. However, these graph-based methods are limited in their ability to explain classes with multiple sufficient explanations. To provide more expressive explanations, we propose utilizing class expressions (CEs) from the field of description logic (DL). Our approach explains heterogeneous graphs with different types of nodes using CEs in the EL description logic. To identify the best explanation among multiple candidate explanations, we employ and compare two different scoring functions: (1) For a given CE, we construct multiple graphs, have the GNN make a prediction for each graph, and aggregate the predicted scores. (2) We score the CE in terms of fidelity, i.e., we compare the predictions of the GNN to the predictions by the CE on a separate validation set. Instead of subgraph-based explanations, we offer CE-based explanations.

Planning and reasoning about actions and processes, in addition to reasoning about propositions, are important issues in recent logical and computer science studies. The widespread use of actions in everyday life such as IoT, semantic web services, etc., and the limitations and issues in the action formalisms are two factors that lead us to study how actions are represented. Since 2007, there have been some ideas to integrate Description Logic (DL) and action formalisms for representing both static and dynamic knowledge. Meanwhile, time is an important factor in dynamic situations, and actions change states over time. In this study, on the one hand, we examined related logical structures such as extensions of description logics (DLs), temporal formalisms, and action formalisms. On the other hand, we analyzed possible tools for designing and developing the Knowledge and Action Base (KAB). For representation and reasoning about actions, we embedded actions into DLs (such as Dynamic-ALC and its extensions). We propose a terminable algorithm for action projection, planning, checking the satisfiability, consistency, realizability, and executability, and also querying from KAB. Actions in this framework were modeled with SPIN and added to state space. This framework has also been implemented as a plugin for the Prot\'eg\'e ontology editor. During the last two decades, various algorithms have been presented, but due to the high computational complexity, we face many problems in implementing dynamic ontologies. In addition, an algorithm to detect the inconsistency of actions' effects was not explicitly stated. In the proposed strategy, the interactions of actions with other parts of modeled knowledge, and a method to check consistency between the effects of actions are presented. With this framework, the ramification problem can be well handled in future works.

Relational concept analysis (RCA) is an extension of formal concept analysis allowing to deal with several related contexts simultaneously. It has been designed for learning description logic theories from data and used within various applications. A puzzling observation about RCA is that it returns a single family of concept lattices although, when the data feature circular dependencies, other solutions may be considered acceptable. The semantics of RCA, provided in an operational way, does not shed light on this issue. In this report, we define these acceptable solutions as those families of concept lattices which belong to the space determined by the initial contexts (well-formed), cannot scale new attributes (saturated), and refer only to concepts of the family (self-supported). We adopt a functional view on the RCA process by defining the space of well-formed solutions and two functions on that space: one expansive and the other contractive. We show that the acceptable solutions are the common fixed points of both functions. This is achieved step-by-step by starting from a minimal version of RCA that considers only one single context defined on a space of contexts and a space of lattices. These spaces are then joined into a single space of context-lattice pairs, which is further extended to a space of indexed families of context-lattice pairs representing the objects manippulated by RCA. We show that RCA returns the least element of the set of acceptable solutions. In addition, it is possible to build dually an operation that generates its greatest element. The set of acceptable solutions is a complete sublattice of the interval between these two elements. Its structure and how the defined functions traverse it are studied in detail.

One way that the current state of the art measures the reasoning ability of transformer-based models is by evaluating accuracy in downstream tasks like logical question answering or proof generation over synthetic contexts expressed in natural language. However, most of the contexts used are in practice very simple; in most cases, they are generated from short first-order logic sentences with only a few logical operators and quantifiers. In this work, we seek to answer the question how well a transformer-based model will perform reasoning over expressive contexts. For this purpose, we construct a synthetic natural language question-answering dataset, generated by description logic knowledge bases. For the generation of the knowledge bases, we use the expressive language $\mathcal{ALCQ}$. The resulting dataset contains 384K examples, and increases in two dimensions: i) reasoning depth, and ii) length of sentences. We show that the performance of our DeBERTa-based model, DELTA$_M$, is marginally affected when the reasoning depth is increased and it is not affected at all when the length of the sentences is increasing. We also evaluate the generalization ability of the model on reasoning depths unseen at training, both increasing and decreasing, revealing interesting insights into the model's adaptive generalization abilities.

Maurice Funk, and co-authors Balder ten Cate, Jean Christoph Jung and Carsten Lutz, won a distinguished paper award at the 32nd International Joint Conference on Artificial Intelligence (IJCAI) for their work SAT-Based PAC Learning of Description Logic Concepts. In this interview, Maurice tells us more about knowledge bases and querying, why this is an interesting area for study, and their methodology and results. Our research is in the area of knowledge representation, or more specifically knowledge bases and querying. A knowledge base contains facts like a traditional database e.g. "Bob is a fish" and "Amelia is a dog", but also background knowledge formulated in some formal language e.g.