Some connections between finite and infinite model theory

Abstract

Most of the work in model theory has, so far, considered in nite structures and the methods and results that have been worked out in this context can usually not be transferred to the study of nite structures in an obvious way. In addition, some basic results from in nite model theory fail within the context of nite models. The theory about nite structures has largely developed in connection with theoretical computer science, in particular complexity theory [12]. The question arises whether these two "worlds", the study of in nite structures and the study of nite structures, can be weaved together in some way and enrich each other. In particular, one may ask if it is possible to adapt notions and methods which have played an important role in in nite model theory to the context of nite structures, and in this way get a better understanding of fairly large and su ciently well-behaved classes of nite structures. If we are to study structures in relation to some formal language, then the question arises which one to choose. Most of in nite model theory considers rst-order logic. Within nite model theory various restrictions and extensions of rst-order logic have been considered, since rst-order logic may be considered as being both too strong and too weak (in different senses) for the study of nite structures. A reasonable candidate for studying nite structures, with a viewpoint from in nite model theory, is the language Ln, rst order logic L restricted to formulas in which at most n variables occur, whether free or bound. Theories consisting of only Ln-formulas, even those which are "complete" within Ln, may have both nite and in nite models, or only nite models, or only in nite models. The language Ln has the nice properties of being closed under subformulas, quanti cation and negation. Also, there is a pebble game which distinguishes whether two structures satisfy exactly the same Ln-sentences or not ([23] and implicitly in [29]). The notion of a type plays an important role in in nite model theory. In nite model theory the notion of an Ln-type, i.e. a type restricted to Ln-formulas, has been used; the number of di erent Ln-types of an Ln-theory can be seen as a measure of the complexity of the theory. Dawar observed [5] that for every Ln-theory T with nite models there is an upper bound, depending only on the number of Ln-types (in n free variables) of T , of the size of the smallest model of T . Later Grohe proved that this upper bound is not recursive [17]. The language Ln has also been considered in the context of (only) in nite models in the work of Hedman [19] where complete theories (within full rst-order logic) which are axiomatizable by Ln-sentences are studied. For a general overview about interactions (and di erences) between nite and in nite model theory, see [30]. For a survey about the use of nite variable logics in nite model theory, see [16]. Within in nite model theory the area of stability theory has had great in uence. It studies a quite large class of "managable" (in nite) structures and their complete rstorder theories. Work in the direction of developing the basics of a similar theory for nite structures was rst carried out by Hyttinen [21]. Then, from a di erent viewpoint, the author developed some results, inspired by stability theory, aiming at understanding when an Ln-theory with in nite models also must have arbitrarily large nite models [9, 8].