On the complexity of team logic and its two-variable fragment

We study the logic FO(∼), the extension of first-order logic with team semantics by unrestricted Boolean negation. It was recently shown to be axiomatizable, but otherwise has not yet received much attention in questions of computational complexity. In this paper, we consider its two-variable fragment FO2(∼) and prove that its satisfiability problem is decidable, and in fact complete for the recently introduced non-elementary class TOWER(poly). Moreover, we classify the complexity of model checking of FO(∼) with respect to the number of variables and the quantifier rank, and prove a dichotomy between PSPACE- and ATIME-ALT(exp, poly)-complete fragments. For the lower bounds, we propose a translation from modal team logic MTL to FO2(∼) that extends the well-known standard translation from modal logic ML to FO2. For the upper bounds, we translate FO(∼) to fragments of second-order logic with PSPACE-complete and ATIME-ALT(exp, poly)-complete model checking, respectively