A Characterization of Constructive Dimension | Zendy

Satyadev Nandakumar | Zendy

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A Characterization of Constructive Dimension

Author(s) -

Satyadev Nandakumar

Publication year - 2008

Publication title -

electronic notes in theoretical computer science

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.242

H-Index - 60

ISSN - 1571-0661

DOI - 10.1016/j.entcs.2008.03.025

Subject(s) - constructive , impossibility , mathematics , characterization (materials science) , hausdorff dimension , martingale (probability theory) , discrete mathematics , dimension (graph theory) , class (philosophy) , martingale difference sequence , context (archaeology) , measure (data warehouse) , hausdorff space , pure mathematics , computer science , artificial intelligence , paleontology , materials science , process (computing) , database , political science , law , biology , nanotechnology , operating system

In the context of Kolmogorov's algorithmic approach to the foundations of probability, Martin-Löf defined the concept of an individual random sequence using the concept of a constructive measure 1 set. Alternate characterizations use constructive martingales and measures of impossibility. We prove a direct conversion of a constructive martingale into a measure of impossibility and vice versa, such that their success sets, for a suitably defined class of computable probability measures, are equal. The direct conversion is then generalized to give a new characterization of constructive dimensions, in particular, the constructive Hausdorff dimension and the constructive packing dimension

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