Finite Models, Logic and Complexity

Finite model theory deals with the model theory of finite structures. As a branch of model theory it is concerned with the analysis of structural properties in terms of logics. The attention to finite structures is not so much a restriction in scope as a shift in perspective. The main parts of classical model theory (the model theory related to first-order logic) as well as of abstract model theory (the comparative model theory of other logics) almost exclusively concern infinite structures; finite models are disregarded as trivial in some respects and as intractable in others. In fact, the most successful tools of classical model theory fail badly in restriction to finite structures. The compactness theorem in particular, which is one of the corner stones of classical model theory, does not hold in the realm of finite structures. Several examples of other important theorems from classical model theory that are no longer true in the finite case are discussed in [Gur84].

There are on the other hand specific new issues to be considered in the finite. These issues mainly account for the growing interest in finite model theory and promote its development into a theory in its own right. One of the specific issues in a model theory of finite structures is complexity. Properties and transformations of finite structures can be considered under algebraic and combinatorial aspects, under the aspect of logical definability, and also under the aspect of computational complexity. Issues of computational complexity form one of the main links also between finite model theory and theoretical computer science.

In this introduction I merely intend to indicate selectively some main ideas and lines of research that motivate the present investigations. There are a number of surveys that also cover various other aspects of finite model theory — see for instance [Fag90, Gur84, Gur88, Imm87a, Imm89]. A general reference is the new textbook on finite model theory by Ebbinghaus and Flum [EF95].

The model theory of fields is a fascinating subject stretching from Tarski's work on the decidability of the theories of the real and complex fields to Hrushovksi's recent proof of the Mordell-Lang conjecture for function fields. Our goal in this volume is to give an introduction to this fascinating area concentrating on connections to stability theory.

The first paper Introduction to the model theory of fields begins by introducing the method of quantifier elimination and applying it to study the definable sets in algebraically closed fields and real closed fields. These first sections are aimed for beginning logic students and can easily be incorporated into a first graduate course in logic. They can also be easily read by mathematicians from other areas. Algebraically closed fields are an important examples of ω-stable theories. Indeed in section 5 we prove Macintyre's result that that any infinite cj-stable field is algebraically closed. The last section surveys some results on algebraically closed fields motivated by Zilber's conjecture on the nature of strongly minimal sets. These notes were originally prepared for a two week series of lecture scheduled to be given in Bejing in 1989. Because of the Tinnanmen square massacre these lectures were never given.

The second paper Model theory of differential fields is based on a course given at the University of Illinois at Chicago in 1991. Differentially closed fields provide a fascinating example for many model theoretic phenomena (Sacks referred to differentially closed fields as the “least misleading example”). This paper begins with an introduction to the necessary differential algebra and elementary model theory of differential fields. Next we examine types, ranks and prime models, proving among other things that differential closures are not minimal and that for K > N0 there are 2k non-isomorphic models. We conclude with a brief survey of differential Galois theory including Poizat's model theoretic proof of Kolchin's result that the differential Galois group of a strongly normal extension is an algebraic group over the constants and the Pillay-Sokolovic result that any superstable differential field has no proper strongly normal expansions. Most of this article can be read by a beginning graduate student in model theory. At some points a deeper knowledge of stability theory or algebraic geometry will be helpful.