First-Order Model Checking Problems Parameterized by the Model

We study the complexity of the model checking problem, for fixed models A, over certain fragments $\mathcal{L}$ of first-order logic, obtained by restricting which of the quantifiers and boolean connectives we permit. These are sometimes known as the expression complexities of $\mathcal{L}$. We obtain various full and partial complexity classification theorems for these logics $\mathcal{L}$ as each ranges over models A, in the spirit of the dichotomy conjecture for the Constraint Satisfaction Problem --- which itself may be seen as the model checking problem for existential conjunctive positive first-order logic.