Preservation Theorems in Finite Model Theory

We develop various aspects of the finite model theory of Lk(there exists) and Lk∞omega(there exists). We establish the optimality of normal forms for Lk∞omega(there exists) over the class of finite structures and demonstrate separations over the class of finite structures and demonstrate separations among descriptive complexity classes within Lk∞omega(there exists). We establish negative results concerning preservation theorems for Lk(there exists) an Lk∞omega(there exists). We introduce a generalized notion of preservation theorem and establish some positive results concerning "generalized preservation theorems" for first-order definable classes of finite structures which are closed under extensions.