Computable presentations of randomizations

TL;DR

This paper develops an effective metric structure theory for Keisler randomizations, establishing conditions for decidability and computable categoricity of randomized structures, and proving that all admit effective quantifier elimination.

Contribution

It introduces a framework for the effective metric structure theory of Keisler randomizations, linking decidability and categoricity properties of structures and their randomizations.

Findings

A countable structure has a decidable presentation iff its Borel randomization has a computable presentation with uniformly computable constant functions.

Effective $oldsymbol{ extomega}$-categoricity of a structure implies the computable categoricity of its randomization.

All randomizations admit effective quantifier elimination.

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.