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A geometric zero-one law

  • Robert H. Gilman (a1), Yuri Gurevich (a2) and Alexei Miasnikov (a3)


Each relational structure X has an associated Gaifman graph, which endows X with the properties of a graph. If x is an element of X, let Bn(x) be the ball of radius n around x. Suppose that X is infinite, connected and of bounded degree. A first-order sentence ϕ in the language of X is almost surely true (resp. a.s. false) for finite substructures of X if for every xX, the fraction of substructures of Bn(x) satisfying ϕ approaches 1 (resp. 0) as n approaches infinity. Suppose further that, for every finite substructure, X has a disjoint isomorphic substructure. Then every ϕ is a.s. true or a.s. false for finite substructures of X. This is one form of the geometric zero-one law. We formulate it also in a form that does not mention the ambient infinite structure. In addition, we investigate various questions related to the geometric zero-one law.



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[1]Benjamini, I. and Schramm, O., Percolation beyond Zd, many questions and a few answers, Electronic Communications in Probability, vol. 1 (1996), no. 8, pp. 7182, electronic.
[2]Benjamini, I., Recent progress on percolation beyond Zd,
[3]Blass, A. and Gurevich, Y., Zero-one laws: Thesauri and parametric conditions, Bulletin of European Association for Theoretical Computer Science, vol. 91 (2007).
[4]Compton, K.J., A logical approach to asymptotic combinatorics I. First order properties, Advances in Math., vol. 65 (1987), pp. 6596.
[5]Compton, K.J., 0-1 laws in logic and combinatorics, Nato Advanced Study Institute on Algorithms and Order (Rival, I., editor), D. Reidel, 1989, pp. 353383.
[6]H-D. Ebbinghaus, and Flum, J., Finite model theory, Springer, 1995.
[7]Fagin, R., Probabilities on finite models, this Journal, vol. 41 (1976), pp. 5058.
[8]Gaifman, H., On local and nonlocal properties, Proceedings of the Herbrand symposium (Marseilles, 1981) (Amsterdam), Studies in Logic and the Foundations of Mathematic, vol. 107, North-Holland, 1982, pp. 105135.
[9]Glebski, Y., Kogan, V., Liogonkij, M.I., and Talanov, V.A., The extent and degree of satisfiability of formulas of the restricted predicate calculus, Kibernetika, vol. 2 (1969), pp. 1727.
[10]Gurevich, Y., Zero-one laws, The logic in computer science column, current trends in theoretical computer science (Rozenberg, G. and Salomaa, A., editors), Series in Computer Science, vol. 40, World Scientific, p. 1993.
[11]Jajcay, R. and Siráñ, J., A construction of vertex-transitive non-Cayley graphs, Austalas. J. Combin., vol. 10 (1994), pp. 105114.
[12]Kolaitis, Ph.G., Promel, H.J., and Rotschild, B.L., Kl+1-free graphs: asymptotic structure and a 0−1 law, Transactions of the American Mathematical Society, vol. 303 (1987), pp. 637671.
[13]Winkler, P., Random structures and zero-one laws, Finite and infinite combinatorics in sets and logic (Sauer, N.Wet al., editor), NATO Advanced Science Institutes Series, Kluver, 1993, pp. 399420.



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