PreprintCC BY 4.0arXiv
Non-aware computable presentations of randomizations
Abstract
Let 2 be the two-element structure in the empty signature. We build a computable presentation Q of the Borel randomization 2[0,1) in which neither constant random variable is a computable point. Consequently, Q is not aware. This gives a negative answer to a question of Cuervo Ovalle and Goldbring \cite{CG}. We also construct a countable structure M with no computable presentation but M[0,1) has a computable presentation. This answers another question and establishes the necessity of the awareness condition stated in one of the main theorems in \cite{CG}.
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BibTeX
@misc{zhao2026nonaware,
title = {{Non-aware computable presentations of randomizations}},
author = {Xuanheng Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30569},
eprint = {2610.30569},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30569},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-aware computable presentations of randomizations
AU - Xuanheng Zhao
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30569
UR - https://arxiv.org/abs/2610.30569
AB - Let 2 be the two-element structure in the empty signature. We build a computable presentation Q of the Borel randomization 2^[0,1) in which neither constant random variable is a computable point. Consequently, Q is not aware. This gives a negative answer to a question of Cuervo Ovalle and Goldbring \cite{CG}. We also construct a countable structure M with no computable presentation but M^[0,1) has a computable presentation. This answers another question and establishes the necessity of the awareness condition stated in one of the main theorems in \cite{CG}.
ER -