Enumerating teams in first-order team logics

We start the study of the enumeration complexity of different satisfiability problems in first-order team logics. Since many of our problems go beyond DelP, we use a framework for hard enumeration analogous to the polynomial hierarchy, which was recently introduced by Creignou et al. (Discret. Appl. Math. 2019). We show that the problem to enumerate all satisfying teams of a fixed formula in a given first-order structure is DelNP-complete for certain formulas of dependence logic and independence logic. For inclusion logic formulas, this problem is even in DelP. Furthermore, we study the variants of this problem where only maximal or minimal solutions, with respect to cardinality or inclusion, are considered. For the most part these share the same complexity as the original problem. One exception is the cardinality minimum-variant for inclusion logic, which is DelNP-complete, the other is the inclusion maximal-variant for dependence and independence logic, which is in Del⁺NP and DelNP-hard.