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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)
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)
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)
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