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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: [email protected]
Yuri Gurevich
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, Wa 98052, USA E-mail: [email protected]
Alexei Miasnikov
Affiliation:
Department of Mathematics and Statistics, Mcgill University, Montreal, Quebec H3A 2K6, Canada, E-mail: [email protected]
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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

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