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Math Help - Randomness

  1. #1
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    Randomness

    Something tells me I made a similar post before but I did not find it.
    What is the rigorous mathematical definition of randomness? I was thinking like this: Assume you have an algorithm which produces numbers then we can define it as random if the limit of the correlation coefficient approaches zero as n \rightarrow \infty.
    I am talking about concepts (probability and algorithms) which I have never studied thus you might not understand what I am trying to say. But I am trying to say for example an algorithm is defined in such a way it produces only one's. Then we can create a plot with the x-axis being the number of times the algorithm used and y-axis is its result output. Then by the correlation coefficient formula that coefficient is always one. Thus, the limit as increasing the number of times using the algorithm is 1 not zero thus that algorithm is not random. Does anyone understand what I am trying to ask?
    CaptainBlack this is what you know best Computability Theory, help me.
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  2. #2
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    One good definition is that of algorithmic incompressibility: a sequence of bits is incompressible if the size S(n) of the shortest computer program (Turing machine) which produces the first n bits satisfies S(n)/n -> 1: that is, you cannot write a shorter program than one which simply copies the bits from a file.
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  3. #3
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    I have no idea what you just said. No need to explain it to me, just asking what you think about my definition?
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  4. #4
    hpe
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    Quote Originally Posted by ThePerfectHacker
    What is the rigorous mathematical definition of randomness? I was thinking like this: Assume you have an algorithm which produces numbers then we can define it as random ...
    If you produce a sequence of numbers with an algorithm, that sequence is not random.
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  5. #5
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    You might want to look at Gregory Chaitin's pages, lots of articles there
    on this stuff. If you can you might as well consult the horses mouth

    The most recent of these may be the best place to start, as it is pitched
    at a semi-popular level.

    RonL
    Last edited by CaptainBlack; January 14th 2006 at 01:58 PM.
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    Rigorous definition of randomness

    Guess what? There is no such thing! We postulate a space of possibilities and a subset of events and assign probabilities to them. Some people will say that a selection in accordance with those probabilities is a "random" selection, others will insist that only "equiprobable events" can provide random selection.

    Actually, mathematicians don't discuss randomness very much. Probability measures, yes, randomness, no.
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  7. #7
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    A rigorous definition of randomness

    I think that the only problem with PerfectHacker's definition is his use of the word algorithm.
    Otherwise I think it is a good starting point to work form. It seems to me that some definition of the notion of onservation or measurment is necessary -consider quantum mechanics, where a physical process is completley non-random or deterministic (unitary evolution), until somebody decides to measure an observable of the system.

    Any thoughts?
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