### A geometric zero-one law

Robert H. Gilman, Yuri Gurevich, and Alexei Miasnikov
Source: J. Symbolic Logic Volume 74, Issue 3 (2009), 929-938.

#### 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.

First Page:
Primary Subjects: 03C13
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1245158092
Digital Object Identifier: doi:10.2178/jsl/1245158092
Zentralblatt MATH identifier: 05609397
Mathematical Reviews number (MathSciNet): MR2548469

### References

I. Benjamini and O. Schramm, Percolation beyond $Z^ d$, many questions and a few answers, Electronic Communications in Probability, vol. 1 (1996), no. 8, pp. 71--82, electronic.
Mathematical Reviews (MathSciNet): MR1423907
Digital Object Identifier: doi:10.1049/ecej:19890012
--------, Recent progress on percolation beyond $Z^d$, http://research.microsoft.com/~schramm/pyondrep/.
A. Blass and Y. Gurevich, Zero-one laws: Thesauri and parametric conditions, Bulletin of European Association for Theoretical Computer Science, vol. 91 (2007).
Mathematical Reviews (MathSciNet): MR2313497
K.J. Compton, A logical approach to asymptotic combinatorics I. First order properties, Advances in Math., vol. 65 (1987), pp. 65--96.
Mathematical Reviews (MathSciNet): MR893471
Digital Object Identifier: doi:10.1016/0001-8708(87)90019-3
Zentralblatt MATH: 0646.60012
--------, $0$--$1$ laws in logic and combinatorics, Nato Advanced Study Institute on Algorithms and Order (I. Rival, editor), D. Reidel, 1989, pp. 353--383.
Mathematical Reviews (MathSciNet): MR1037789
H-D. Ebbinghaus and J. Flum, Finite model theory, Springer, 1995.
Mathematical Reviews (MathSciNet): MR1409813
R. Fagin, Probabilities on finite models, Journal of Symbolic Logic, vol. 41 (1976), pp. 50--58.
Mathematical Reviews (MathSciNet): MR476480
Digital Object Identifier: doi:10.2307/2272945
Zentralblatt MATH: 0341.02044
H. Gaifman, 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.
Mathematical Reviews (MathSciNet): MR757024
Y. Glebski, V. Kogan, M.I. Liogonkij, and V.A. Talanov, The extent and degree of satisfiability of formulas of the restricted predicate calculus, Kibernetika, vol. 2 (1969), pp. 17--27.
Y. Gurevich, Zero-one laws, The logic in computer science column. current trends in theoretical computer science (G. Rozenberg and A. Salomaa, editors), Series in Computer Science, vol. 40, World Scientific, p. 1993.
R. Jajcay and J. Siráň, A construction of vertex-transitive non-Cayley graphs, Austalas. J. Combin., vol. 10 (1994), pp. 105--114.
Mathematical Reviews (MathSciNet): MR1296944
Ph.G. Kolaitis, H.J. Promel, and B.L. Rotschild, $K_l+1$--free graphs: asymptotic structure and a $0$--$1$ law, Transactions of the American Mathematical Society, vol. 303 (1987), pp. 637--671.
Mathematical Reviews (MathSciNet): MR902790
Digital Object Identifier: doi:10.2307/2000689
Zentralblatt MATH: 0641.05025
P. Winkler, Random structures and zero-one laws, Finite and infinite combinatorics in sets and logic (N.W. Sauer et al., editor), NATO Advanced Science Institutes Series, Kluver, 1993, pp. 399--420.
Mathematical Reviews (MathSciNet): MR1261219