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

Andreas Blass, Yuri Gurevich, Dexter Kozen
1985 Information and Control
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-fixedpoint operator, the zero-one law proved for first-order logic in (Glebskii, Kogan, Liogonki, and Talanov
more » ... (1969) , Kibernetika 2, 31-42; Fagin (1976), J. Symbolic Logic 41, 50-58). For any sentence q~ of the extended logic, the proportion of models of q~ 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 cp, whether this proportion approaches 1 is complete for exponential time, if we consider only q)'s with a fixed finite vocabulary (or vocabularies of bounded arity) and complete for double-exponential time if ~0 is unrestricted. In addition, we establish some related results. 71 Glebskii, Kogan, Liogonki, and Talanov (1969) and independently Fagin (1976) showed that every first-order sentence satisfies the zero-one law. Grandjean (1982) showed that the problem of deciding which of the two limit values is correct for a given first-order sentence is PSPACE complete. (We state these results precisely and review their proofs in Sect. 1.) Kaufmann and Shelah have shown that the zero-one law is violated badly within monadic second-order logic. We extend the zero-one law to sentences in the logic obtained by adding to first-order logic the least-fixed-point operator studied in (Aho and Ullman, 1979; Chandra and Harel, 1982; Immerman, 1982a; Kozen, 1982; Vardi, 1982) or the more powerful iterative fixed point operator (Gurevich, 1984; Livchak, 1983) . We show that any formula in these extended logics is equivalent, in random structures (i.e., with probability approaching 1 as the structures get larger), to a first-order formula. This result, which immediately implies the zero-one law, contrasts with the well-known fact that the least-fixed-point operator greatly increases the expressive power of first-order logic. Contrary to what one might expect, our equivalence result does not allow us to transfer PSPACE completeness of the theory of random structures from first-order logic to the fixed-point operators. The difficulty is that the translation process, from the extended logics to first-order logic, can vastly increase the length of formulas. This difficulty cannot be overcome without proving PSPACE=EXPTIME, for we show that the decision problem for the theory of random structures in logic with a fixedpoint operator is EXPTIME hard.