Institute for Research in Cognitive Science Finite Model Theory and Finite Variable Logics Ph

Finite Model Theory and Finite Variable Logics Eric Rosen Supervisor: Scott Weinstein In this dissertation, I investigate some questions about the model theory of nite structures. One goal is to better understand the expressive power of various logical languages, including rst-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 nite 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(9) and Lk1!(9), the existential fragments of the nite variable logics Lk and Lk1! . I show that there are nite models whose Lk(9)-theories are not nitely axiomatizable. I also establish the optimality of a normal form for Lk1!(9), 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 L!1! , 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 nal chapter, I investigate the nite model theory of propositional modal logic. I show that, in contrast to more expressive logics, modal 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-de nable and closed under bisimulations i it is de ned by a modal FO sentence. In addition, I prove that, over F , a class is de ned by a modal sentence and closed under extensions i it is de ned by a 3-modal sentence. vi