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A recipe for randomness

S Pincus, B H Singer

    Proceedings of the National Academy of Sciences of the United States of America
    |September 2, 1998
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces novel constructive methods for generating random sequences, addressing a gap in current techniques. It utilizes approximate entropy (ApEn) to create irregular finite and normal infinite sequences.

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    Area of Science:

    • Information Theory
    • Computational Complexity
    • Number Theory

    Background:

    • Existing theories on randomness (e.g., probability, algorithmic complexity) lack constructive methods for generating certified random outputs.
    • Approximate entropy (ApEn) provides a computable measure of sequential irregularity, offering a foundation for new generation techniques.

    Purpose of the Study:

    • To develop explicit, constructive methods for producing highly irregular finite and normal infinite sequences.
    • To refine the concept of normality for infinite sequences by classifying numbers based on initial segment irregularity.

    Main Methods:

    • Leveraging constructions and properties derived from approximate entropy (ApEn).
    • Developing techniques for generating diverse classes of normal numbers with quantifiable deviations from maximal irregularity.

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    Main Results:

    • Successful demonstration of explicit techniques for generating broad sets of highly irregular finite sequences.
    • Introduction of refined methods for constructing diverse classes of normal infinite sequences and normal numbers.

    Conclusions:

    • The proposed methods fill a critical void in constructive randomness generation.
    • The work provides a computable framework for understanding and generating sequences with desired levels of irregularity and normality.