Cones, Negation, and All That

This paper summarizes results on embedding 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. The 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. This is an extended abstract of the paper “Cone Semantics for Logics with Negation” to be published in the proceedings of the 29th International Joint Conference on Artificial Intelligence (IJCAI 2020).