Zero-One Laws for Existential First-Order Sentences of Bounded Quantifier Depth

Abstract

For any fixed positive integer k , let α k denote the smallest α ∈ (0,1) such that the random graph sequence { G ( n, n -α )} n does not satisfy the zero-one law for the set ε k of all existential first-order sentences that are of quantifier depth at most k . This article finds upper and lower bounds on α k , showing that as k → ∞, we have α k = ( k - 2 - t ( k )) -1 for some function t ( k ) = Θ ( k -2 ). We also establish the precise value of α k when k = 4.