A geometric zero-one law

Robert H. Gilman, Yuri Gurevich, Alexei Miasnikov

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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.

Original languageEnglish
Pages (from-to)929-938
Number of pages10
JournalJournal of Symbolic Logic
Volume74
Issue number3
DOIs
StatePublished - Sep 2009

Keywords

  • Finite structure
  • Percolation
  • Zero-one law

Fingerprint

Dive into the research topics of 'A geometric zero-one law'. Together they form a unique fingerprint.

Cite this