# Counting of Teams in First-Order Team Logics

## TL;DR

This paper explores the descriptive complexity of counting classes from #P to #·NP using first-order and team semantics logics, establishing new logical characterizations and complexity results for these classes.

## Contribution

It extends the logical characterization of counting complexity classes beyond #P to classes like #·NP using team-based logics such as independence, dependence, and inclusion logic.

## Key findings

- #·NP characterized by independence logic and existential second-order logic
- Dependence and inclusion logic relate to subclasses of #·NP and #P
- Counting satisfying assignments for monotone Boolean Σ₁-formulae is #·NP-complete

## Abstract

We study descriptive complexity of counting complexity classes in the range from #P to #$\cdot$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 second-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 #$\cdot$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.