Knowledge base embeddings are a widely applied technique, used for instance to improve link prediction tasks on knowledge graphs by using the geometric regularities occurring during learning. Techniques where ontological concepts are interpreted as boxes have shown to be particularly useful in this context, as they are both suitably expressive and of low computational complexity. However, to use those regularities for learning, it is necessary to determine and understand the possible biases in the approach: how do we distinguish what is learned due to regularities in the data from what is simply based on the representational limitations of the embedding? In this paper, we establish that there are some severe limitations in expressivity when modeling description logic ontologies with box embeddings in intended target languages such as $\mathcal{ELHO}(\circ)^\bot$. We illustrate that, under some weak assumptions, box semantics always satisfy Helly’s Property, and is thus too weak to capture semantically $\mathcal{ELHO}(\circ)^\bot$ in an adequate way. We then characterize how so-called Helly-satisfiable $\mathcal{ELHO}(\circ)^\bot$ ontologies can be adequately determined. We discuss the implications of this result with respect to existing box embedding approaches and real-world use cases.

Smart Materials (SMat) promise to open new opportunities in the area of Intelligent Environments (IE), whether as part of dedicated smart devices or as the fabric constituting everyday appliances and building infrastructure. Through the use of ontologies both IE engineers and the IEs themselves can be aware of, and predict, how novel configurable and changing materials react under different conditions. In contrast to conventional Smart Objects, however, as computational software/hardware-systems, lending themselves to the object-oriented perspective of conventional ontology specification languages, SMat and IE in the wider sense require a perspective focussing on extended spaces and numerical domains. Both are known to be problematic in terms of usability and computational complexity for the traditional object-oriented languages, with even very basic notions already leading into undecidability. Context Logic (CL), in contrast, is a formalism specialized for these domains. This paper demonstrates how terminology from this area involving extended spaces and numerical domains can be modeled in CL.

This paper is concerned with knowledge graph embedding with background knowledge, taking the formal perspective of logics. In knowledge graph embedding, knowledge— expressed as a set of triples of the form (a R b) (“a is R-related to b”)—is embedded into a real-valued vector space. The embedding helps exploiting geometrical regularities of the space in order to tackle typical inductive tasks of machine learning such as link prediction. Recent embedding approaches also consider incorporating background knowledge, in which the intended meanings of the symbols a, R, b are further constrained via axioms of a theory. Of particular interest are theories expressed in a formal language with a neat semantics and a good balance between expressivity and feasibility. In that case, the knowledge graph together with the background can be considered to be an ontology. This paper develops a cone-based theory for embedding in order to advance the expressivity of the ontology: it works (at least) with ontologies expressed in the description logic ALC, which comprises restricted existential and universal quantifiers, as well as concept negation and concept disjunction. In order to align the classical Tarskian Style semantics for ALC with the sub-symbolic representation of triples, we use the notion of a geometric model of an ALC ontology and show, as one of our main results, that an ALC ontology is satisfiable in the classical sense iff it is satisfiable by a geometric model based on cones. The geometric model, if treated as a partial model, can even be chosen to be faithful, i.e., to reflect all and only the knowledge captured by the ontology. We introduce the class of axis-aligned cones and show that modulo simple geometric operations any distributive logic (such as ALC) interpreted over cones employs this class of cones. Cones are also attractive from a machine learning perspective on knowledge graph embeddings since they give rise to applying conic optimization techniques.

This paper presents an embedding of ontologies expressed in the ALC description logic into a real-valued vector space, comprising restricted existential and universal quantifiers, as well as concept negation and concept disjunction. Our main result states that an ALC ontology is satisfiable in the classical sense iff it is satisfiable by a partial faithful geometric model based on cones. The line of work to which we contribute aims to integrate knowledge representation techniques and machine learning. The new cone-model of ALC proposed in this work gives rise to conic optimization techniques for machine learning, extending previous approaches by its ability to model full ALC.