Then 5 Conclusion Deenition 5.1 (positive Borel Point-set) Lemma 4.2 Let M Be a Model of the Signature 0 = Hh; <i Such That M 6 J

an immediate corollary to the main result in 2] was the theorem of Furst, Saxe, and Sipser 6] and Ajtai 1] that there is a uniform sequence of polynomial size constant depth monotone Boolean circuits that is not equivalent to whatever non-uniform sequence of polynomial size constant depth positive Boolean circuits. The thing is, we can simply translate the point-sets deenable in nite models by a monotone rst-order formula into monotone Boolean circuits (whose depth is basically the logical depth of the formula). This provides for uniformity in a very strong sense. Theorem 5.4 above shows that positive rst order logic fails however to capture even strongly uniform ((rst order deenable) positive Boolean circuits of constant depth and polynomial size, and therefore rst order logic seems to be a bad candidate to the role of uniform version of positive Boolean circuits of constant depth and polynomial size. Indeed, both monotone (Ajtai and Gurevich) and positive (present paper) rst order formulas fail to adequately capture uniform classes of positive Boolean circuits of constant depth and polynomial size: the former express too much, and the latter too little. Michael Taitslin made some very valuable comments to earlier versions of the paper, and I would like to express my sincere thanks to them. I am especially grateful to an anonymous referee who, among other things, suggested a simpliication of the proof of Lemma 3.2. So we refuted Lyndon's Lemma for nite case, by showing that there exists a rst order formula of a certain signature nitely monotone in a (binary) predicate and not nitely equivalent to any rst order formula positive in this predicate. To achieve this result, we introduced a new characterization of expressibility by positive formulas, namely, positive version of Fra ss e-Ehrenfeucht games, and showed that these games exactly characterize pairs of ((nite or innnite) models that preserve truth of positive rst order formulas. Then we came up with an example of a sequence of pairs of nite models, that we call grids and reduced grids, respectively, and that preserve truth of positive sentences (as demonstrated by the proof of existence of a winning strategy in positive games), and showed that, however, a monotone sentence is capable of telling grids from reduced grids. It turns out that the technical results of this paper provide for a much ner distinction between positive and monotone formulas, as compared to the technique in …