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Finite Model Theory and Finite Variable Logics 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. I then show that existential preservation fails for the language L ∞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 yb a modal sentence and closed under extensions if it is defined by a ◊-modal sentence. Comments University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-95-28. This thesis or dissertation is available at ScholarlyCommons: https://repository.upenn.edu/ircs_reports/139 Institute for Research in Cognitive Science Finite Model Theory and Finite Variable Logics Ph.D. Dissertation Eric Rosen University of Pennsylvania 3401 Walnut Street, Suite 400C Philadelphia, PA 19104-6228 October 1995 Site of the NSF Science and Technology Center for Research in Cognitive Science IRCS Report 95-28 Finite Model Theory and Finite Variable Logics Eric Rosen A Dissertation in Philosophy Presented to the Faculties of the UniversityofPennsylvania in Partial Fulllment of the Requirements for the Degree of Do ctor of Philosophy Scott Weinstein Sup ervisor of Dissertation Samuel Freeman Graduate Group Chairp erson c Copyright by Eric Rosen Tomyparents iii Acknowledgments I am extremely grateful to my advisor Scott Weinstein for all that he has taughtme and for the help and encouragement that he has provided so graciously over the last four years It has b een an immense pleasure working with him he made this pro ject fun and exciting Scott originally suggested that I pursue nite mo del theory a sub ject which has develop ed into an active and fertile area of research and he continues to guide me in the right direction This dissertation owes a great deal to his ability to nd and ask the most interesting questions Manyofmy results are attempts to answer problems that he originally p osed I could not haveasked for a b etter advisor Iwould like to thank Steven Lindell for the manyways in whichhehelpedmewith this work Scott Steven and I have met regularly to discuss a variety of topics in nite mo del theory during which time I learned much of what I know ab out descriptive com plexity theorySteven also listened patiently to manyofmy ideas and provided useful feedback Finally asamember of my dissertation committee he read mywork carefully and suggested many improvements I am also grateful to Maria Bonet Anuj Dawar Yuri Gurevich and James Rogers for valuable discussions on topics covered in my dissertation Marco Hollenb ergs detailed umber comments on the nal chapter help ed me improve the presentation and eliminate a n of errors Iowe a sp ecial debt of gratitude to Prof Johan van Benthem In March of he visited Penn to give a series of lectures on his research on mo dal logic At that time I hadanumber of very interesting discussions with b oth him and Scott during which they suggested lo oking at this work on mo dal logic from the p ersp ective of nite mo del theory Since then Prof van Benthem has made more many useful recommendations whichhave iv b een very helpful Chapter contains some results in this area Long b efore myadventures in logic b egan I entered the Penn philosophy department intending to pursue myinterest in Wittgenstein and the philosophy of language Although things have turned out dierentlyIwould like to thank Professors Thomas Ricketts and Gary Ebbs for what they taught me ab out analytic philosophy and for what they showed me ab out how to do philosophy Thank you also to my friends here who made gradu ate scho ol more interesting and enjoyable Peter Dillard Amy Herzel Scott Kimbrough Rukesh Korde Shari RudavskyPeter Schwartz Lisa Shab el and Larry Shapiro Alisa and Alan my sister and brother have also b een go o d friends From the very b eginning my parents instilled in me a love of learning For all their love and supp ort I dedicate this dissertation to them v ABSTRACT Finite Model Theory and Finite Variable Logics Eric Rosen Sup ervisor Scott Weinstein In this dissertation I investigate some questions ab out the mo del theory of nite struc tures One goal is to b etter understand the expressivepower of various logical languages including rstorder logic FO over this class A second related goal is to determine which results from classical mo del theory remain true when relativized to the class F of nite structures As it is wellknown that many such results b ecome false I also consider certain weakened generalizations of classical results k k Iprove some basic results ab out the languages L and L the existential k k fragments of the nite variable logics L and L I show that there are nite mo dels k whose L theories are not nitely axiomatizable I also establish the optimalityofa k and separate certain fragments of this logic I intro duce a notion normal form for L of a generalized preservation theorem and establish certain partial p ositive results I then show that existential preservation fails for the language L bothover F and over the class of all structures I also examine other preservation prop erties eg for classes closed under homomorphisms In the nal chapter I investigate the nite mo del theory of prop ositional mo dal logic Ishow that in contrast to more expressive logics mo dal logic is wellb ehaved 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 mo dels is FOdenable and closed under bisimulations i it is dened byamodalFOsentence In addition I prove that over F a class is dened by a mo dal sentence and closed under extensions i it is dened bya mo dal sentence vi Contents Acknowledgments iv Abstract vi Intro duction Preservation theorems Preliminaries Logical equivalence and EhrenfeuchtFraisse games k k Basic nite mo del theory for L and L Existential preservation Generalized preservation theorems The failure of existential preservation for L Other generalized preservation theorems The class HOM Generalized preservation theorems for HOM and MON Mo dal logic over nite structures Background Preservation theorems Conclusion Bibliography vii Chapter Intro duction Finite mo del theory investigates the mo del theory of nite structures This sub ject in teracts with a variety of elds from math logic and computer science including classical mo del theory graph theory and complexity theory The dierent areas and asp ects of nite mo del theory are unied byaninterest in the expressivepower of logical languages In this dissertation I pursue mo del theoretic questions p ertaining to denabilitypaying particular attention to preservation theorems Some of these problems are just nite ver sions of results from classical mo del theory That is we can ask whether a classical theorem remains true when restricted to the class F of nite mo dels Other questions are varia tions on standard ideas Chapters and for example examine preservation theorems involving languages other than rstorder logic It is well known that many theorems from classical mo del theory b ecome false over the class of nite mo dels see For example the LosTarski theorem states that a rstorder sentence denes a class of mo dels that is closed under extensions if and only if it is equivalent to an existential sentence Tait showed that this prop
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