  We study the complexity of the model checking problem, for fixed model A,
over certain fragments L of first-order logic. These are sometimes known as the
expression complexities of L. We obtain various complexity classification
theorems for these logics 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.
