Finite Model Theory and Finite Variable Logics

In this dissertation, I investigate some questions about the model theory of finite structures. One goal is to better understand the expressive power of various logical languages, including first order logic (FO), over this class. A second, related, goal is to determine which results from classical model theory remain true when relativized to the class, F, of finite structures. As it is well known that many such results become false, I also consider certain weakened generalizations of classical results. I prove some basic results about the languages Lk∃ and Lk∞ω∃, the existential fragments of the finite variable logics Lk and Lk∞ω. I show that there are finite models whose Lk(∃)-theories are not finitely axiomatizable. I also establish the optimality of a normal form for Lk∞ω∃, and separate certain fragments of this logic. I introduce a notion of a "generalized preservation theorem", and establish certain partial positive results. I then show that existential preservation fails for the language Lk∞ω, both over F and over the class of all structures. I also examine other preservation properties, e.g. for classes closed under homomorphisms. In the final chapter, I investigate the finite model theory of propositional modal logic. I show that, in contrast to more expressive logics, model logic is "well behaved" over F. In particular, I establish that various theorems that are true over the class of all structures also hold over F. I prove that, over F a class of models is FO-definable and closed under bisimulations if it is defined by a modal FO sentence. In addition, I prove that, over F, a class is defined by a modal sentence and closed under extensions if it is defined by a ◊-modal sentence.