Zero-One Laws

Definition

A query language is said to have the 0–1 law if every Boolean query that contains no constants (i.e., the query does not mention any particular element from the domain of potential data values) is almost surely true or almost surely false. The notions of being “almost surely true,” respectively, “almost surely false” are defined as follows: Let σ be a fixed database schema. For each natural number n, let DB_n (σ) be the set of all database instances of schema σ whose active domain is a subset of {1,…,n} (i.e., all database entries belong to {1,…,n}). For a Boolean query q of schema σ let μ_n (q) be the probability that a database D chosen uniformly at random from DB_n (σ) is a “yes”‐instance of query q. In other words, μ_n (q) is the number of databases in DB_n (σ) on which q evaluates to “yes,” divided by the number of all databases in DB_n (σ). Query q is said to be almost surely true (respectively, almost surely false), if the limit μ(q) := lim_n→∞μ_n (q) exists and is equal to...