The study of architectures and methods for artificial neural networks so that they can learn and perform tasks from the realm of logic-based knowledge representation and reasoning has a long-standing tradition Besold et al. [2017]. This research area is sometimes referred to as "neuro-symbolic integration" (or "neural-symbolic integration") and there are at least two primary rationales that can be found in the literature on the subject. The first is the desire to arrive at systems that combine the robustness and trainability of artificial neural networks with the transparency and interpretability of knowledge-based systems, while at the same time making use of structured background knowledge. The second rationale is more prevalent in cognitive science and lies in addressing the fundamental gap between symbolic and subsymbolic representation and processing, based on the observation that humans perceive much of their own thinking, introspectively, as symbolic, while the physical structure of the brain gives rise to artificial neural networks as a mathematical and computational abstraction. Many of the earlier lines of research on neuro-symbolic integration, discussed primarily from a cognitive science perspective, can be found in Besold et al. [2017]. Of particular interest is the integration of deep learning with logics that are not propositional in nature, since propositional logic is of limited applicability to knowledge representation and reasoning tasks. In the wake of deep learning breakthroughs, fundamental issues around neuro-symbolic integration have recently received increased attention with some progress being made as new approaches emerge. In particular, there has been progress in developing neural networks that can learn to reason.

Neuro-Symbolic Artificial Intelligence -- the combination of symbolic methods with methods that are based on artificial neural networks -- has a long-standing history. In this article, we provide a structured overview of current trends, by means of categorizing recent publications from key conferences. The article is meant to serve as a convenient starting point for research on the general topic.

Pattern-based, modular ontologies have several beneficial properties that lend themselves to FAIR data practices, especially as it pertains to Interoperability and Reusability. However, developing such ontologies has a high upfront cost, e.g. reusing a pattern is predicated upon being aware of its existence in the first place. Thus, to help overcome these barriers, we have developed MODL: a modular ontology design library. MODL is a curated collection of well-documented ontology design patterns, drawn from a wide variety of interdisciplinary use-cases. In this paper we present MODL as a resource, discuss its use, and provide some examples of its contents.

Concept Induction refers to the problem of creating complex Description Logic class descriptions (i.e., TBox axioms) from instance examples (i.e., ABox data). In this paper we look particularly at the case where both a set of positive and a set of negative instances are given, and complex class expressions are sought under which the positive but not the negative examples fall. Concept induction has found applications in ontology engineering, but existing algorithms have fundamental performance issues in some scenarios, mainly because a high number of invokations of an external Description Logic reasoner is usually required. In this paper we present a new algorithm for this problem which drastically reduces the number of reasoner invokations needed. While this comes at the expense of a more limited traversal of the search space, we show that our approach improves execution times by up to several orders of magnitude, while output correctness, measured in the amount of correct coverage of the input instances, remains reasonably high in many cases. Our approach thus should provide a strong alternative to existing systems, in particular in settings where other systems are prohibitively slow.

Semantic Web knowledge representation standards, and in particular RDF and OWL, often come endowed with a formal semantics which is considered to be of fundamental importance for the field. Reasoning, i.e., the drawing of logical inferences from knowledge expressed in such standards, is traditionally based on logical deductive methods and algorithms which can be proven to be sound and complete and terminating, i.e. correct in a very strong sense. For various reasons, though, in particular, the scalability issues arising from the ever-increasing amounts of Semantic Web data available and the inability of deductive algorithms to deal with noise in the data, it has been argued that alternative means of reasoning should be investigated which bear high promise for high scalability and better robustness. From this perspective, deductive algorithms can be considered the gold standard regarding correctness against which alternative methods need to be tested. In this paper, we show that it is possible to train a Deep Learning system on RDF knowledge graphs, such that it is able to perform reasoning over new RDF knowledge graphs, with high precision and recall compared to the deductive gold standard.

It has been argued that it is much easier to convey logical statements using rules rather than OWL (or description logic (DL)) axioms. Based on recent theoretical developments on transformations between rules and DLs, we have developed ROWLTab, a Protege plugin that allows users to enter OWL axioms by way of rules; the plugin then automatically converts these rules into OWL 2 DL axioms if possible, and prompts the user in case such a conversion is not possible without weakening the semantics of the rule. In this paper, we present ROWLTab, together with a user evaluation of its effectiveness compared to entering axioms using the standard Protege interface. Our evaluation shows that modeling with ROWLTab is much quicker than the standard interface, while at the same time, also less prone to errors for hard modeling tasks.

Once the conceptual overview, in terms of a somewhat informal class diagram, has been designed in the course of engineering an ontology, the process of adding many of the appropriate logical axioms is mostly a routine task. We provide a Protege plugin which supports this task, together with a visual user interface, based on established methods for ontology design pattern modeling.

In our experience, some ontology users find it much easier to convey logical statements using rules rather than OWL (or description logic) axioms. Based on recent theoretical developments on transformations between rules and description logics, we develop ROWL, a Protege plugin that allows users to enter OWL axioms by way of rules; the plugin then automatically converts these rules into OWL DL axioms if possible, and prompts the user in case such a conversion is not possible without weakening the semantics of the rule.

We provide a detailed example for modular ontology modeling based on ontology design patterns.

Conjunctive query answering over expressive Horn Description Logic ontologies is a relevant and challenging problem which, in some cases, can be addressed by application of the chase algorithm. In this paper, we define a novel acyclicity notion which provides a sufficient condition for termination of the restricted chase over Horn-SRIQ TBoxes. We show that this notion generalizes most of the existing acyclicity conditions (both theoretically and empirically). Furthermore, this new acyclicity notion gives rise to a very efficient reasoning procedure. We provide evidence for this by providing a materialization based reasoner for acyclic ontologies which outperforms other state-of-the-art systems.