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The universal measure of probabilistically nonrandom objects

  • Vladimir Vovk
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Abstract

A finite object is called probabilistically random if it is an algorithmically random element of a finite set whose Kolmogorov complexity is relatively small. This note studies the amount of probabilistically nonrandom objects (those that are not probabilistically random) as gauged by the universal measure and without assuming any structure on the objects. The main results imply that the universal measure of probabilistically nonrandom objects is vanishingly small and establish the dependence of their universal measure on the required degree of probabilistic randomness.

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Cite

BibTeXRIS
BibTeX
@misc{vovk2026universal,
  title = {{The universal measure of probabilistically nonrandom objects}},
  author = {Vladimir Vovk},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13513},
  eprint = {2609.13513},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13513},
}
RIS
TY  - UNPB
TI  - The universal measure of probabilistically nonrandom objects
AU  - Vladimir Vovk
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.13513
UR  - https://arxiv.org/abs/2609.13513
AB  - A finite object is called probabilistically random if it is an algorithmically random element of a finite set whose Kolmogorov complexity is relatively small. This note studies the amount of probabilistically nonrandom objects (those that are not probabilistically random) as gauged by the universal measure and without assuming any structure on the objects. The main results imply that the universal measure of probabilistically nonrandom objects is vanishingly small and establish the dependence of their universal measure on the required degree of probabilistic randomness.
ER  -