Probabilities on finite models

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Abstract
Let be a finite set of (nonlogical) predicate symbols. By an -structure, we mean a relational structure appropriate for . Let be the set of all -structures with universe {1, …, n}. For each first-order -sentence σ (with equality), let μn(σ) be the fraction of members of for which σ is true. We show that μn(σ) always converges to 0 or 1 as n → ∞, and that the rate of convergence is geometrically fast. In fact, if T is a certain complete, consistent set of first-order -sentences introduced by H. Gaifman [6], then we show that, for each first-order -sentence σ, μn(σ) →n 1 iff T ⊩ ω. A surprising corollary is that each finite subset of T has a finite model. Following H. Scholz [8], we define the spectrum of a sentence σ to be the set of cardinalities of finite models of σ. Another corollary is that for each first-order -sentence a, either σ or ˜σ has a cofinite spectrum (in fact, either σ or ˜σ is “nearly always“ true).Let be a subset of which contains for each in exactly one structure isomorphic to . For each first-order -sentence σ, let νn(σ) be the fraction of members of which a is true. By making use of an asymptotic estimate [3] of the cardinality of and by our previously mentioned results, we show that vn(σ) converges as n → ∞, and that limn νn(σ) = limn μn(σ).If contains at least one predicate symbol which is not unary, then the rate of convergence is geometrically fast.

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