Artificial Intelligence and Natural Man by Margaret A. Boden

By Margaret A. Boden

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3 Approximation in the Maximum Norm All approximation results in the previous section only address the interpolation of a finite number of points or the approximation of general mappings in probability. Hence the height of the trees that are to be considered is limited. Consequently the recurrence for which approximation takes place is restricted. In term classification tasks this is appropriate in many situations: In automated theorem proving a term with more than 500 symbols will rarely occur, for example.

Xk of g, adding one layer with k + 1 multiplying units, and one layer with two units with identical activation function, g can be computed in an MLP with O(lg k) layers each containing O(k) units. Since x . y = ((x + y)2 _ x2 _ y2)/2 we can substitute the multiplying units by units with square activation functions. The order of the bounds remains the same. Since a is locally C 2 with a " ~ 0, points x0 and xl exist with lim a(xo + e x ) - a ( x o ) = x lim a(xx + ex) + O ' ( X l - - e X ) ,-,0 e2 "(Zl) -- and 2 a ( x l ) = x2 for all x.

Xm, y) = g(gl ( x l , . . , xm), y), respectively, this yields a prefix representation of the trees such that the images of the trees ti are mutually different, gl can be computed by one hidden layer with the perceptron activation function and linear neurons copying Yl, ... or y, respectively, and one linear output neuron: for this purpose, lx~=~ is substituted by H(xi - ai) + H ( - x i + aj) - 1. The linear outputs of gl can be integrated into the first layer of g. Since we are dealing with a finite number of points, the biases in the perceptron neurons can be slightly changed such that no activation coincides with 0 on the inputs t~.

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