On Finite Model Theory

The subject of this paper is the part of finite model theory intimately related to the classical model theory. In the very beginning of our career in computer science, we attended a few lectures on database theory where databases were inconspicuously allowed to be infinite and then classical model-theoretical theorems were applied. The use of infinite databases aroused our suspicion and prompted us to investigate the status of some most famous modeltheoretical theorems in the case of finite structures [Gu84]. The theorems miserably fail. One theorem (a theorem of Roger Lyndon: Every sentence monotone in a predicate P is logically equivalent to a sentence positive in P [Ly59]) resisted the attack and was refuted by Miklos Ajtai and ourselves later [AG87]. In Section 1, we give some old and new counter-examples to classical model-theoretic theorems in the finite case. Of course, some classical theorems survive the transition to the finite case. For example, Ehrenfeucht’s game-theoretic characterization [Eh61] of the indistinguishability of two structures by sentences of a bounded depth remains valid in a very obvious way. For a while, it seemed however that no classical model-theoretic theorems (especially those whose proof involves the compactness theorem in one way or another) survive the transition to the finite case in a non-trivial way. The first, to our knowledge, positive result into this direction was published by Miklos Ajtai and ourselves [AG89]. The result is explained in Section 2. The second positive result was proven recently by Saharon Shelah and ourself [GS90]; this is the subject of Section 3. This article reflects a lecture given to Workshop on Feasible Mathematics that took place in June 1989 in Cornell University. (It was a wonderful, very well organized workshop with many interesting lectures and plenty time for discussions.) The reflection is not very faithful though. First, we skipped the part on logics tailored for computational complexity ∗Final version in: “Feasible Mathematics, Workshop, Cornell University, June, 1989” (ed. S.R. Buss and P.J. Scott), Perspectives in Computer Science, Birkhauser, 1990. †Address: EECS Department, University of Michigan, Ann Arbor, MI 48109-2122, USA. The work was partially supported by NSF grants DCR 85-03275 and CCR 89-04728.