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A geometric zero-one law

Published online by Cambridge University Press:  12 March 2014

Robert H. Gilman ,
Yuri Gurevich  and
Alexei Miasnikov
Robert H. Gilman
Affiliation:
Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, Nj 07030, USA, E-mail: rgilman@stevens.edu
Yuri Gurevich
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, Wa 98052, USA E-mail: gurevich@microsoft.com
Alexei Miasnikov
Affiliation:
Department of Mathematics and Statistics, Mcgill University, Montreal, Quebec H3A 2K6, Canada, E-mail: alexeim@math.mcgill.ca
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Abstract

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 x ∈ X, 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.

Keywords

Finite structurezero-one lawpercolation
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 929 - 938
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[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. 71–82, electronic.CrossRefGoogle Scholar
[2]Benjamini, I., Recent progress on percolation beyond Zd, http://research.microsoft.com/~sehramm/pyondrep/.Google Scholar
[3]Blass, A. and Gurevich, Y., Zero-one laws: Thesauri and parametric conditions, Bulletin of European Association for Theoretical Computer Science, vol. 91 (2007).Google Scholar
[4]Compton, K.J., A logical approach to asymptotic combinatorics I. First order properties, Advances in Math., vol. 65 (1987), pp. 65–96.CrossRefGoogle Scholar
[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. 353–383.CrossRefGoogle Scholar
[6]H-D. Ebbinghaus, and Flum, J., Finite model theory, Springer, 1995.Google Scholar
[7]Fagin, R., Probabilities on finite models, this Journal, vol. 41 (1976), pp. 50–58.Google Scholar
[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. 105–135.CrossRefGoogle Scholar
[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. 17–27.Google Scholar
[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.Google Scholar
[11]Jajcay, R. and Siráñ, J., A construction of vertex-transitive non-Cayley graphs, Austalas. J. Combin., vol. 10 (1994), pp. 105–114.Google Scholar
[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. 637–671.Google Scholar
[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. 399–420.CrossRefGoogle Scholar