First-Order Query Evaluation with Cardinality Conditions

We study an extension of first-order logic FO that allows to express cardinality conditions in a similar way as SQLu0027s COUNT operator. The corresponding logic FOC(P) was introduced by Kuske and Schweikardt, who showed that query evaluation for this logic is fixed-parameter tractable on classes of databases of bounded degree. In this paper, we first show that the fixed-parameter tractability of FOC(P) cannot even be generalised to very simple classes of databases of unbounded degree such as unranked trees or strings with a linear order relation. Then, we identify a fragment FOC1(P) of FOCP which is still extends FO and is sufficiently strong to express standard applications of SQLu0027s COUNT operator. Our main result shows that query evaluation for FOC1(P) is fixed-parameter tractable on nowhere dense classes of databases. This, in particular, implies that the counting problem for first-order queries on nowhere dense classes is fixed-parameter tractable.