where y* is the desired output.
It has been shown by a number of people that the weights of such an
element are assured of arriving at a set of values which produce the
correct output-input relation after a sufficient number of errors,
provided that such a set exists. An upper bound on the number of
possible errors can be given which depends only on the initial weight
values and the logical function to be learned. This does not, however,
solve our network problem for two reasons.
First, as the number, n, of inputs gets large, the number of errors
to be expected for most functions which can be learned increases to
unreasonable values. For example, for n = 6, most such functions
result in 500 to 1000 errors compared to an average of 32 errors to be
expected in a perfect learning device.
Second, and more important, the fraction of those logical functions
which can be generated in a single element becomes vanishingly small as
n increases. For example, at n = 6 less than one in each three trillion
logical functions is so obtainable.