First-Order Definable Counting-Only Queries

For several practical queries on bags of sets of objects, the answer does not depend on the precise composition of these sets, but only on the number of sets to which each object belongs. This is the case k = 1 for the more general situation where the query answer only depends on the number of sets to which each group of at most k objects belongs. We call such queries k-counting-only. Here, we focus on k$$k$$-SyCALC, k-counting-only queries that are first-order definable. As k$$k$$-SyCALC is semantically defined, however, it is not surprising that it is already undecidable whether a first-order query is in 1-SyCALC. Therefore, we introduce SimpleCALC-k$$k$$, a syntactically defined (strict) fragment of k$$k$$-SyCALC. It turns out that many practical queries in k$$k$$-SyCALC can already be expressed in SimpleCALC-k$$k$$. We prove that the k-counting-only queries form a non-collapsing hierarchy: for every k, there exist (k+1)-counting-only queries that are not k-counting-only. This result specializes to both SimpleCALC-k$$k$$ and k$$k$$-SyCALC. Finally, we establish a strong dichotomy between 1-SyCALC and SimpleCALC-k$$k$$ on the one hand and 2-SyCALC on the other hand by showing that satisfiability, validity, query containment, and query equivalence are decidable for the former two languages, but not for the latter one.