Spectra of Monadic Second-Order Formulas with One Unary Function

We establish the eventual periodicity of the spectrum of any monadic second-order formula where: (i) all relation symbols, except equality, are unary, and (ii) there is only one function symbol and that symbol is unary

On the very weak 0 − 1 law for random graphs with orders

(463) revision:2008-10-30 modified:2008-10-30 Abstract: Let us draw a graph R on {0, 1,..., n − 1} by having an edge {i, j} with probability p |i−j|, where ∑ i pi < ∞, and let Mn = (n, <, R). For a first order sentence ψ let an ψ be the probability of Mn | = ψ. We know that the sequence a1 ψ, a2 ψ,..., an ψ,... does not necessarily converge. But here we find a weaker substitute which we call the very weak 0-1 law. We prove that limn→∞(an ψ − an+1 ψ) = 0. For this we need a theorem on the (first order) theory of distorted sum of models. Saharon: 1) Check: line before rest of the proof 3.4A, should it be ̸=? 2) Check the numerical bounds