  Counting answers to a query is an operation supported by virtually all
database management systems. In this paper we focus on counting answers over a
Knowledge Base (KB), which may be viewed as a database enriched with background
knowledge about the domain under consideration. In particular, we place our
work in the context of Ontology-Mediated Query Answering/Ontology-based Data
Access (OMQA/OBDA), where the language used for the ontology is a member of the
DL-Lite family and the data is a (usually virtual) set of assertions. We study
the data complexity of query answering, for different members of the DL-Lite
family that include number restrictions, and for variants of conjunctive
queries with counting that differ with respect to their shape (connected,
branching, rooted). We improve upon existing results by providing a PTIME and
coNP lower bounds, and upper bounds in PTIME and LOGSPACE. For the latter case,
we define a novel query rewriting technique into first-order logic with
counting.
