Counting of Teams in First-Order Team Logics.

We study descriptive complexity of counting complexity classes in the range from $#$P to $#cdot$NP. A corollary of Faginu0027s characterization of NP by existential second-order logic is that $#$P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. In this paper we extend this study to classes beyond $#$P and extensions of first-order logic with team semantics. These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of FO in Tarskiu0027s semantics. Our results show that the class $#cdot$NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of $#cdot$NP and $#$P , respectively. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean $Sigma_1$-formulae is $#cdot$NP-complete as well as complete for the function class generated by dependence logic.