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arXiv · 2506.06187

Computable presentations of randomizations

Nicolás Cuervo Ovalle·Isaac Goldbring
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Abstract

We initiate the effective metric structure theory of Keisler randomizations. We show that a classical countable structure $\mathcal{M}$ has a decidable presentation if and only if its Borel randomization $\mathcal{M}^{[0,1)}$ has a computable presentation for which the constant functions are uniformly computable points. We determine a sufficient condition for which the uniform computability of the constant functions can be dropped. We show that when $\mathcal{M}$ is effectively $\omega$-categorical, then $\mathcal{M}^{[0,1)}$ is computably categorical, that is, has a unique computable presentation up to computable isomorphism. A special case of this result is that the unique separable atomless probability algebra is computably categorical. Finally, we show that all randomizations admit effective quantifier elimination.

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Nicolás Cuervo Ovalle, Isaac Goldbring. 2025-06-06. Computable presentations of randomizations. https://arxiv.org/abs/2506.06187

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