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.
Full-text: Access denied (no subscription detected)
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
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.
--------, 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).
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
--------, $0$--$1$ laws in logic and combinatorics, Nato Advanced Study Institute on Algorithms and Order (I. Rival, editor), D. Reidel, 1989, pp. 353--383.
H-D. Ebbinghaus and J. Flum, Finite model theory, Springer, 1995.
R. Fagin, Probabilities on finite models, Journal of Symbolic Logic, vol. 41 (1976), pp. 50--58.
Mathematical Reviews (MathSciNet):
MR476480
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.
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
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.
