University of Pennsylvania ScholarlyCommons IRCS Technical Reports Series Institute for Research in Cognitive Science October 1995 Finite Model Theory and Finite Variable Logics Eric Rosen University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/ircs_reports Rosen, Eric, "Finite Model Theory and Finite Variable Logics " (1995). IRCS Technical Reports Series. 139. https://repository.upenn.edu/ircs_reports/139 University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-95-28. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ircs_reports/139 For more information, please contact
[email protected]. Finite Model Theory and Finite Variable Logics Abstract 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. k k I prove some basic results about the languages L ∃ and L ∞ω∃, the existential fragments of the finite k k k variable logics L and L ∞ω. I show that there are finite models whose L (∃)-theories are not finitely k axiomatizable. I also establish the optimality of a normal form for L ∞ω∃, and separate certain fragments of this logic. I introduce a notion of a "generalized preservation theorem", and establish certain partial k positive results.