Counting of Teams in First-Order Team Logics
We study descriptive complexity of counting complexity classes in the range from #P to #·NP. A corollary of Fagin's 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
... order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of FO in Tarski's semantics. Our results show that the class #·NP can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of #·NP and #P , respectively. Our main technical result shows that the problem of counting satisfying assignments for monotone Boolean Σ_1-formulae is #·NP-complete as well as complete for the function class generated by dependence logic.