Random models and the Maslov class

Extract

In [GS] Gurevich and Shelah introduce a novel method for proving that every satisfiable formula in the Gödel class has a finite model (the Gödel class is the class of prenex formulas of pure quantification theory with prefixes ∀∀∃ … ∃). They dub their method “random models”: it proceeds by delineating, given any F in the Gödel class and any integer p, a set of structures for F with universe {1, …, p} that can be treated as a finite probability space S. They then show how to calculate an upper bound on the probability that a structure chosen at random from S makes F false; from this bound they are able to infer that if p is sufficiently large, that probability will be less than one, so that there will exist a structure in S that is a model for F. The Gurevich-Shelah proof is somewhat simpler than those known heretofore. In particular, there is no need for the combinatorial partitionings of finite universes that play a central role in the earlier proofs (see [G] and [DG, p. 86]). To be sure, Gurevich and Shelah obtain a larger bound on the size of the finite models, but this is relatively unimportant, since searching for finite models is not the most efficient method to decide satisfiability.

Gurevich and Shelah note that the random model method can be used to treat the Gödel class extended by initial existential quantifiers, that is, the prefix-class ∀…∀∃…∃; but they do not investigate further its range of applicability to syntactically specified classes.