Learning exact functions of one variable from data by GMDH manifold

It has been the traditional Al that tried to find out the exact functions of one variable when given only certain number of correspondences between input data and output data. However, their way of doing it was not a formalized one but with the help of an empirical set of heuristic rules that experts in mathematics made for each type of problems [Langley et al.,1983;Lenat.19781. Their researches have often gone astray, in the actual circumstances. into a too audacious direction in which they try to find out the exact functions in spite of noise, [Langley,1990; Schaffer.19901. As it is quite difficult even when there is no noise among the data, their approach comes nearer and nearer to the finite element method [P. C. Dune. 19681 which generally does not find out the exact function . but try only to ecclectically approximate it with a linear summation of some known functions, with a help of empirical heuristic rules. The GeNeuVon system introduced here is a mathematically formal learning system. It is capable of finding out the exact function which can be any combination of differentiable functions used by humans today,i.e. polynomials, fractional functions, functions having square roots, exponential -functions, logarithmic functions, and trigonometric functions. if only it is given a sufficient number of data. GeNeuVon is also capable of finding out the inverse function for any of these differentiable functions. In order to enable all these, Taylor series development and learning of coefficients by extended GMDH (Group Method of Data Handling) are built in the GeNeuVon system. It is astonishing that nobody has discovered this methodology before.