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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2406.19513 |
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| _version_ | 1866917735976402944 |
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| author | Hrushovski, Ehud |
| author_facet | Hrushovski, Ehud |
| contents | We study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover's normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim-Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations.
We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov-Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric.
For sequences of approximate equivalence relations with an 'approximately unique' probability logic 1-type, we obtain a structure theorem generalizing the `Lie model' theorem for approximate subgroups. The models here are Riemannian homogeneous spaces, fibered over a locally finite graph.
Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin.
For NIP theories, pursuing a question of Pillay's, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19513 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximate equivalence relations Hrushovski, Ehud Logic We study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover's normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim-Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations. We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov-Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric. For sequences of approximate equivalence relations with an 'approximately unique' probability logic 1-type, we obtain a structure theorem generalizing the `Lie model' theorem for approximate subgroups. The models here are Riemannian homogeneous spaces, fibered over a locally finite graph. Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin. For NIP theories, pursuing a question of Pillay's, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets. |
| title | Approximate equivalence relations |
| topic | Logic |
| url | https://arxiv.org/abs/2406.19513 |