A zero-one law for logic with a fixed-point operator

The logic obtained by adding the least-fixed-point operator to first-order logic was proposed as a query language by Aho and Ullman (in “Proc. 6th ACM Sympos. on Principles of Programming Languages,” 1979, pp. 110–120) and has been studied, particularly in connection with finite models, by numerous authors. We extend to this logic, and to the logic containing the more powerful iterative-fixed-point operator, the zero-one law proved for first-order logic in (Glebskii, Kogan, Liogonki, and Talanov (1969), Kibernetika 2, 31–42; Fagin (1976), J. Symbolic Logic 41, 50–58). For any sentence φ of the extend logic, the proportion of models of φ among all structures with universe {1,2,…, n} approaches 0 or 1 as n tends to infinity. We also show that the problem of deciding, for any φ, whether this proportion approaches 1 is complete for exponential time, if we consider only φ's with a fixed finite vocabulary (or vocabularies of bounded arity) and complete for double-exponential time if φ is unrestricted. In addition, we establish some related results.