Algorithmic Randomness: A General Framework: Difference between revisions
Created page with "Title: Algorithmic Randomness: A General Framework Research Question: How can we extend the algorithmic theory of randomness to arbitrary distributions and non-compact spaces while maintaining its key properties? Methodology: The study uses the framework of constructive topology and lower semicomputable functions to define a uniform test of randomness. This test is used to measure the deficiency of randomness, which is the difference between the original measure and th..." |
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Title: Algorithmic Randomness: A General Framework | Title: Algorithmic Randomness: A General Framework | ||
Research Question: How can we | Research Question: How can we develop a framework for studying algorithmic randomness in a general space, allowing for non-compact spaces and arbitrary distributions? | ||
Methodology: The study uses the | Methodology: The study uses the concept of constructive topological spaces and measures over these spaces. It defines a uniform test, a lower semicomputable function that measures the deficiency of randomness. The paper also introduces neutral measures, which are measures that do not decrease randomness. | ||
Results: The | Results: The research shows that there is a universal test that is lower semicomputable and has strong properties of randomness conservation. It also demonstrates that there is a lower semicomputable semimeasure that is neutral. Furthermore, the paper provides an expression for mutual information in terms of the deficiency of independence. | ||
Implications: | Implications: This study extends the algorithmic theory of randomness to arbitrary Bernoulli distributions and arbitrary distributions, removing previous restrictions. It provides a general framework for studying these questions and related ones, such as statistics for a family of distributions. The results have implications for the understanding of randomness in general spaces, particularly in continuous spaces like the space of continuous functions. | ||
Link to Article: https://arxiv.org/abs/ | Link to Article: https://arxiv.org/abs/0312039v3 | ||
Authors: | Authors: | ||
arXiv ID: | arXiv ID: 0312039v3 | ||
[[Category:Computer Science]] | [[Category:Computer Science]] | ||
[[Category:Randomness]] | [[Category:Randomness]] | ||
[[Category: | [[Category:Spaces]] | ||
[[Category: | [[Category:General]] | ||
[[Category: | [[Category:Distributions]] | ||
[[Category: | [[Category:Measures]] | ||
Latest revision as of 15:06, 24 December 2023
Title: Algorithmic Randomness: A General Framework
Research Question: How can we develop a framework for studying algorithmic randomness in a general space, allowing for non-compact spaces and arbitrary distributions?
Methodology: The study uses the concept of constructive topological spaces and measures over these spaces. It defines a uniform test, a lower semicomputable function that measures the deficiency of randomness. The paper also introduces neutral measures, which are measures that do not decrease randomness.
Results: The research shows that there is a universal test that is lower semicomputable and has strong properties of randomness conservation. It also demonstrates that there is a lower semicomputable semimeasure that is neutral. Furthermore, the paper provides an expression for mutual information in terms of the deficiency of independence.
Implications: This study extends the algorithmic theory of randomness to arbitrary Bernoulli distributions and arbitrary distributions, removing previous restrictions. It provides a general framework for studying these questions and related ones, such as statistics for a family of distributions. The results have implications for the understanding of randomness in general spaces, particularly in continuous spaces like the space of continuous functions.
Link to Article: https://arxiv.org/abs/0312039v3 Authors: arXiv ID: 0312039v3