Are Rosenblatt multilayer perceptrons more powerfull than sigmoidal multilayer perceptrons? From a counter example to a general result

In the eighties the problem of the lack of an efficient algorithm to train multilayer Rosenblatt perceptrons was solved by sigmoidal neural networks and backpropagation. But should we still try to find an efficient algorithm to train multilayer hardlimit neuronal networks, a task known as a NP-Complete problem? In this work we show that this would not be a waste of time by means of a counter example where a two layer Rosenblatt perceptron with 21 neurons showed much more computational power than a sigmoidal feedforward two layer neural network with 300 neurons trained by backpropagation for the same classification problem. We show why the synthesis of logical functions with threshold gates or hardlimit perceptrons is an active research area in VLSI design and nanotechnology and we review some of the methods to synthesize logical functions with a multilayer hardlimit perceptron and we propose the search for an efficient method to synthesize any classification problem with analogical inputs with a two layer hardlimit perceptron as a near future objective. Nevertheless we recognize that with hardlimit multilayer perceptrons we cannot approximate continuous functions as we can easily do with multilayer sigmoidal neural networks, with multilayer hardlimit perceptrons we can only solve any classification problem, as we plan to demonstrate in a near future.