We study the logic FO(~), the extension of first-order logic with team
semantics by unrestricted Boolean negation. It was recently shown
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)-completeness.
To achieve 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 to a fragment of second-order
logic