We study probabilistic team semantics which is a semantical framework allowing the study of logical and probabilistic dependencies simultaneously. We examine and classify the expressive power of logical formalisms arising by different probabilistic atoms such as conditional independence and different variants of marginal distribution equivalences. We also relate the framework to the first-order theory of the reals and apply our methods to the open question on the complexity of the implication problem of conditional independence.

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V=L, then many of them are Σ11‐complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in sans-serifZFC whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is Σ11‐complete (it is, if V=L, but can be forced not to be).

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We show that although the Galvin-Prikry Theorem does not hold on generalized Baire space with the standard topol-ogy, there are similar theorems which do hold on generalized Baire space with certain coarser topologies.

We start by giving a survey to the theory of Borel * (κ) sets in the generalized Baire space Baire(κ) = κ κ . In particular we look at the relation of this complexity class to other complexity classes which we denote by Borel(κ), ∆ 1 1 (κ) and Σ 1 1 (κ) and the connections between Borel * (κ) sets and the infinitely deep language M κ + κ . In the end of the paper we will prove the consistency of Borel * (κ) = Σ 1 1 (κ).