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For a fixed infinite structure $\Gamma$ with finite signature tau, we study the following computational problem: input are quantifier-free first-order tau-formulas phi_0,phi_1,...,phi_n that define relations R_0,R_1,\dots,R_n over Gamma. The question is whether the relation R_0 is primitive positive definable from R_1,...,R_n, i.e., definable by a… (More)
In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on M X , where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ… (More)
We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are p-divisible for infinitely many primes p, or groups of the form L p∈I Z(p ∞), where I is an infinite set of primes.
Consider a standard probability space (X, µ), i.e., a space isomorphic to the unit interval with Lebesgue measure. We denote by Aut(X, µ) the au-tomorphism group of (X, µ), i.e., the group of all Borel automorphisms of X which preserve µ (where two such automorphisms are identified if they are equal µ-a.e.). A Borel equivalence relation E ⊆ X 2 is called… (More)
Consider a measure-preserving action Γ (X, µ) of a countable group Γ and a measurable cocycle α : X × Γ → Aut(Y) with countable image , where (X, µ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ X is non-amenable, then there does not exist a countable-to-one Borel… (More)
We develop Fraïssé theory, namely the theory of Fraïssé classes and Fraïssé limits, in the context of metric structures. We show that a class of finitely generated structures is Fraïssé if and only if it is the age of a separable approximately homogeneous structure, and conversely, that this structure is necessarily the unique limit of the class, and is… (More)