TY - JOUR
T1 - A geometric zero-one law
AU - Gilman, Robert H.
AU - Gurevich, Yuri
AU - Miasnikov, Alexei
PY - 2009/9
Y1 - 2009/9
N2 - 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.
AB - 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.
KW - Finite structure
KW - Percolation
KW - Zero-one law
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U2 - 10.2178/jsl/1245158092
DO - 10.2178/jsl/1245158092
M3 - Article
AN - SCOPUS:70349436878
SN - 0022-4812
VL - 74
SP - 929
EP - 938
JO - Journal of Symbolic Logic
JF - Journal of Symbolic Logic
IS - 3
ER -