Data-Complexity of the Two-Variable Fragment with Counting Quantifiers

Let ϕ be a sentence (i.e. a formula with no free variables) in some logical fragment, ψ(ȳ) a formula with free variables ȳ, a set of ground, function-free literals, and ā a tuple of individual constants with the same arity as ȳ. We are to think of as being a body of data, ϕ a background theory, and ψ(ā) a query which we wish to answer. That answer should be positive just in case {ϕ} entails ψ(ā). What is the computational complexity of our task? A fair reply depends on what, precisely, we take the inputs to our problem to be. For, in practice, the background theory ϕ is static, and the query ψ(ȳ) small: only the database, which is devoid of logical complexity, is large and indefinitely extensible.