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September 2009 A geometric zero-one law
Robert H. Gilman, Yuri Gurevich, Alexei Miasnikov
J. Symbolic Logic 74(3): 929-938 (September 2009). DOI: 10.2178/jsl/1245158092


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.


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Robert H. Gilman. Yuri Gurevich. Alexei Miasnikov. "A geometric zero-one law." J. Symbolic Logic 74 (3) 929 - 938, September 2009.


Published: September 2009
First available in Project Euclid: 16 June 2009

zbMATH: 1181.03037
MathSciNet: MR2548469
Digital Object Identifier: 10.2178/jsl/1245158092

Primary: 03C13

Keywords: Finite structure , percolation , Zero-one law

Rights: Copyright © 2009 Association for Symbolic Logic


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Vol.74 • No. 3 • September 2009
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