12/01/1990 09:00 AM
Computer Engineering
The probability of a property on the collection of all finite relational structures is the limit as n --> infinity 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 any property expressible in first-order logic, i.e., the probability of any such property exists and is either 0 or 1. Moreover, the associated decision problem for the probabilities is solvable. We investigate here 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 Sigma_1^1 fragment are identical.
