Computable presentations of randomizations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913881750765568 |
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| author | Ovalle, Nicolás Cuervo Goldbring, Isaac |
| author_facet | Ovalle, Nicolás Cuervo Goldbring, Isaac |
| contents | 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 $ω$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06187 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computable presentations of randomizations Ovalle, Nicolás Cuervo Goldbring, Isaac Logic 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 $ω$-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. |
| title | Computable presentations of randomizations |
| topic | Logic |
| url | https://arxiv.org/abs/2506.06187 |