0–1 Laws for Fragments of Existential Second-Order Logic: A Survey

Abstract

The probability of a property on the collection of all finite relational structures is the limit as n → ∞ of the fraction of structures with n elements satisfying the property, provided the limit exists. It is known that the 0-1 law holds for every property expressible in first-order logic, i.e., the probability of every such property exists and is either 0 or 1. Moreover, the associated decision problem for the probabilities is solvable.In this survey, we consider fragments of existential second-order logic in which we restrict the patterns of first-order quantifiers. We focus on fragments in which the first-order part belongs to a prefix class. We show that the classifications of prefix classes of first-order logic with equality according to the solvability of the finite satisfiability problem and according to the 0-1 law for the corresponding Σ1 1 fragments are identical, but the classifications are different without equality.KeywordsDecision ProblemRandom GraphRelation SymbolExtension AxiomCountable StructureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.