  Many ontologies, i.e., Description Logic (DL) knowledge bases, have been
developed to provide rich knowledge about various domains. An ontology consists
of an ABox, i.e., assertion axioms between two entities or between a concept
and an entity, and a TBox, i.e., terminology axioms between two concepts.
Neural logical reasoning (NLR) is a fundamental task to explore such knowledge
bases, which aims at answering multi-hop queries with logical operations based
on distributed representations of queries and answers. While previous NLR
methods can give specific entity-level answers, i.e., ABox answers, they are
not able to provide descriptive concept-level answers, i.e., TBox answers,
where each concept is a description of a set of entities. In other words,
previous NLR methods only reason over the ABox of an ontology while ignoring
the TBox. In particular, providing TBox answers enables inferring the
explanations of each query with descriptive concepts, which make answers
comprehensible to users and are of great usefulness in the field of applied
ontology. In this work, we formulate the problem of neural logical reasoning
across TBox and ABox (TA-NLR), solving which needs to address challenges in
incorporating, representing, and operating on concepts. We propose an original
solution named TAR for TA-NLR. Firstly, we incorporate description logic based
ontological axioms to provide the source of concepts. Then, we represent
concepts and queries as fuzzy sets, i.e., sets whose elements have degrees of
membership, to bridge concepts and queries with entities. Moreover, we design
operators involving concepts on top of fuzzy set representation of concepts and
queries for optimization and inference. Extensive experimental results on two
real-world datasets demonstrate the effectiveness of TAR for TA-NLR.
