Counting of Teams in First-Order Team Logics

We study descriptive complexity of counting complexity classes in the range from #P to \({\#\cdot \text{NP}}\) . The proof of Fagin’s characterization of NP by existential second-order logic generalizes to the counting setting in the following sense: The class #P can be logically described as the class of functions counting satisfying assignments to free relation variables in first-order formulae. This was first observed by Saluja et al. In this article, 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 first-order logic in Tarski’s semantics. Our results show that the class \({\#\cdot \text{NP}}\) can be logically characterized by independence logic and existential second-order logic, whereas dependence logic and inclusion logic give rise to subclasses of \({\#\cdot \text{NP}}\) and #P, respectively. We further relate the class obtained from inclusion logic to the complexity class \({ \text{TotP}}\subseteq \text{#P}\) .