This paper describes an effort at UC Berkeley and the International Computer Science Institute to develop a supercomputer for artificial neural network applications. Our perspective has been strongly influenced by earlier experiences with the construction and use of a simpler machine. In particular, we have observed Amdahl's Law in action in our designs and those of others. These observations inspire attention to many factors beyond fast multiply-accumulate arithmetic. We describe a number of these factors along with rough expressions for their influence and then give the applications targets, machine goals and the system architecture for the machine we are currently designing.