It should be noted that this restriction does not seriously affect the logical capabilities of a network. In fact, if a certain logical function can be achieved in a network with the use of weights of unrestricted sign, then the same function can be generated in another network with only positive interconnecting weights and, at worst, twice the number of elements. In the worst case this is done by generating in the restricted network both the output and its complement for each element of the unrestricted network. (It is assumed that there are no loops in the network.) A Variable Bias The central problem in network learning is that of determining, for a given input, the set of elements whose outputs can be altered so as to correct the final element, and which will do the least amount of damage to previous adaptations to other inputs. Once this set has been determined, the incrementing rule given for a single element will apply in this case as well (subject to the restriction of leaving interconnecting weights positive), since the desired final output coincides with that desired for each of the elements to be changed (because of positive interconnecting weights). In the process of arriving at such a decision three factors need to be considered. Elements selected for change should tend to be those whose output would thereby be affected for a minimum number of other possible inputs. At the same time it should be ascertained that a change in each of the elements in question does indeed contribute significantly towards correcting the final output. Finally, a minimum number of such elements should be used. It would appear at first that this kind of decision is impossible to achieve if the complexity of the decision apparatus is kept comparable to that of the basic input-output network as mentioned earlier. However, in the method to be described it is felt that a reasonable approximation to these requirements will be achieved without an undue increase in complexity. It is assumed that in addition to its normal inputs, each element receives a variable input bias which we can call b. The output of every element should then be determined by the sign of the usual weighted sum of its inputs plus this bias quantity. This bias is to be the same for each element of the network. If b = 0 the network will behave as before. However, if b is increased gradually, various elements throughout the network will commence changing from -1 to +1, with one or a few changing at any one time as a rule. If b is decreased, the opposite will occur.