  Query embedding approaches answer complex logical queries over incomplete
knowledge graphs (KGs) by computing and operating on low-dimensional vector
representations of entities, relations, and queries. However, current query
embedding models heavily rely on excessively parameterized neural networks and
cannot explain the knowledge learned from the graph. We propose a novel query
embedding method, AConE, which explains the knowledge learned from the graph in
the form of SROI^{-} description logic axioms while being more
parameter-efficient than most existing approaches. AConE associates queries to
a SROI^{-} description logic concept. Every SROI^{-} concept is embedded as a
cone in complex vector space, and each SROI^{-} relation is embedded as a
transformation that rotates and scales cones. We show theoretically that AConE
can learn SROI^{-} axioms, and defines an algebra whose operations correspond
one to one to SROI^{-} description logic concept constructs. Our empirical
study on multiple query datasets shows that AConE achieves superior results
over previous baselines with fewer parameters. Notably on the WN18RR dataset,
AConE achieves significant improvement over baseline models. We provide
comprehensive analyses showing that the capability to represent axioms
positively impacts the results of query answering.
