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 PTIME and coNP lower bounds, and upper bounds in PTIME and LOGSPACE. For the LOGSPACE case, we have devised a novel query rewriting technique into first-order logic with counting.

SHACL (SHape Constraint Language) is a W3C recommendation for validating graph-based data against a set of constraints (called shapes). Importantly, SHACL allows to define recursive shapes, i.e. a shape may refer to itself, directly of indirectly. The recommendation left open the semantics of recursive shapes, but proposals have emerged recently to extend the official semantics to support recursion. These proposals are based on the principle of possibility (or non-contradiction): a graph is considered valid against a schema if one can assign shapes to nodes in such a way that all constraints are satisfied. This semantics is not constructive, as it does not provide guidelines about how to obtain such an assignment, and it may lead to unfounded assignments, where the only reason to assign a shape to a node is that it allows validating the graph.