The 0-1 law fails for the class of existential second order Godel sentences with equality

P. Kolaitis and M. Vardi (see Proc. 19th ACM Symp. on Theory of Computing, p.425-35 (1987), and Proc. 3rd Ann. Symp. on Logic in Computer Science, p.2-11 (1988)) proved that the 0-1 law holds for the second-order existential sentences whose first-order parts are formulas of Bernays-Schonfinkel or Ackermann prefix classes. They also provided several examples of second-order formulas for which the 0-1 law does not hold and noticed that the classification of second-order sentences for which the 0-1 law holds resembles the classification of decidable cases of prenex first-order sentences. The only cases they have not settled were the cases of Godel classes with and without equality. The authors confirm the conjecture of Kolaitis and Vardi that the 0-1 law does not hold for the existential second-order sentences whose first-order part is in the godel prenex form with equality.<
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