SINGLE ELEMENTS

To simplify the problem it is assumed that the network receives a set of two-valued inputs, x₁, x₂, ..., xₙ, and is required to produce only a single two-valued output, y. It is convenient to assign the numerical quantities +1 and -1 to the two values of each variable.

The simplest network would consist of a single linear threshold element with a set of weights, c₀, c₁, c₂, ..., cₙ. These determine the output-input relation or function so that y is +1 or -1 according as the quantity, c₀ + c₁x₁ + c₂x₂ + ... + cₙxₙ, is positive or not, respectively. It is possible for such a single element to exhibit an adaptive behavior as follows. If, for a given set, x₁, x₂, ..., xₙ, the output, y, is correct, then make no changes to the weights. Otherwise change the weights according to the equations

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.