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Given a function (functional actually) $f(x,g(x))$, can a notion of simplicity be attached with respect to the function $g(x)$? (all functions and args are real).

Specifically, intuitively one could say that the function $f(x,0)$ is simpler than the general function $f(x,g(x))$ - notice that $g(x)$ is null- because it eliminates terms. However, how could one quantify this using some measure of $f$? For example, energy arguments could be used. Is there a way to do this?

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Consider function $f(x,y)=\frac{1}{1+\sqrt{x}}(1+y)$. Obviously $f(x,0)$ is not simpler then $f(x,\sqrt{x})$ – Norbert Dec 31 '11 at 12:09
Nice example, however your argument points to the simplest, up to a constant, function. Using your function, $f(x,0)=\frac{1}{1+\sqrt{x}}$ is simpler than $f(x,y)=\frac{1}{1+\sqrt{x}}(1+y(x))$, for arbitrary $y(x)$. If $y=\sqrt{x}$ then $f(x,\sqrt{x})$ is the simplest. – Jorge Dec 31 '11 at 13:25
I think you should look for kolmogorov complexity. Also V. Arnold made some efforts in this question. The problem is that they considered more discrete then continuous functions. – Norbert Dec 31 '11 at 13:30
Yes i was aware of Kolmogorov complexity, and various such notions, but they are not defined for continuous functions (to the best of my knowledge, that is). – Jorge Dec 31 '11 at 13:35

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