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The genes of computation

  1. Dec 8, 2013 #1
    I have been considering the idea for awhile that all mathematics could be broken down into atomic pieces that are indivisible, and therefore any formula could be determined through a process of elimination by trying every possible mathematical formula.

    Here is what I have come up with so far, meant to be processed by a computer, but since I don't know everything about mathematics I don't know if every possible mathematical formula on the set of real numbers can be expressed with this method.

    Every atom would be made up of the following 3 units

    1. scaler {-∞,...,∞)
    2. type { constant, variable, group/absolute value, floor/ceil value}
    3. operation { + , * , ^ (power) }

    In the case of a constant the scaler is that constant
    In the case of a variable the scaler is multiplied by the variable
    In the case of a group the absolute value of the scaler is the number of items in the group and the sign determines if it is a simple group or an absolute value
    In the case of a floor the scaler is the number of items in the group and the sign determines if it is a floor or ceil

    The reasoning behind the operations is that subtraction is expressed by a negative scaler, division can be represented by x^-1 * Y, and square roots can also be represented by powers.

    The method of executing these operations would be by calculating the inner-most groups first and disregarding the operation from the first atom in each group such that "x + 1" could be represented by the 3x2 matrix

    1 variable *
    1 constant +

    The operation on the first row of the matrix would therefore always be ignored, and in theory, a 3xN matrix could then define every possible mathematical formula.

    I haven't included any calculus methods since sums can be expanded out and I believe integrals and derivatives can still be expressed with these atomic functions, but I could be wrong which is why I'm asking.

    Using only standard variables (ie. no vectors, matrix, tensors, etc.) are there any other operations or features which should be included that are computable with finite computational resources?

    [edit] Please note, a computer would solve this problem as a genetic algorithm since it wouldn't be possible to try every possible combination of scalars since there are infinite values that can be selected. Fitness values could be used to extract probable solutions and optimizations could be run to find the optimal constant values.
     
    Last edited: Dec 8, 2013
  2. jcsd
  3. Dec 8, 2013 #2
    Google theory of computation - the Wikipedia article is an OK start.
     
  4. Dec 8, 2013 #3
    @MrAnchovy,

    Thanks I think that was exactly what I was looking for. It looks like I need to do more research into the turing machine also. What I was able to gather from the wikipedia article though is that there are problems that are unsolvable so before going down a rabbit hole to never return I'll also need to learn more about determining if a problem is solvable. There is no point in devoting computational resources to problems that can't be solved.
     
  5. Dec 9, 2013 #4
    You should also look at Mathematica/Wolfram Alpha, Maple, MATLAB etc. which have had many lifetimes of effort put into them.
     
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