Foundations of Formal Reasoning over Knowledge Bases Combining Symbolic and Sub-Symbolic Knowledge

More and more organizations are relying on Machine Learning (ML) models to support internal decision-making processes. To better support such processes, it would be highly beneficial to contextualize the inductively acquired knowledge encoded in these models and enable formal reasoning over it. Despite significant progress in Neural-Symbolic AI, this specific challenge remains largely under-explored. We propose a framework that allows to integrate the knowledge induced by ML classifiers with the knowledge specified by logic-based formalisms. The framework is based on the novel notion of Hybrid Knowledge Base (HKB), consisting of two components: an ontology and a set of ML binary classifiers. As usual, the ontology provides an intensional representation of the modeled domain through logic-based axioms, while the binary classifiers implicitly encode the extensional knowledge. Specifically, a HKB associates to each concept and role mentioned in the ontology a classifier based on a set of features deemed to be relevant for the application domain, thereby virtually populating the concepts and roles with the instances and pairs of instances from the feature space. Besides the definition of the new framework, as a more technical contribution we show how to reason in this framework by studying query answering over HKBs. In particular, we investigate the computational complexity of query answering in a rich language over HKBs in which the ontology is specified in (the Description Logic counterpart of) RDFS, while the binary classifiers are represented by Multi-Layer Perceptrons.