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This paper deals with the foundation of a Non-linear Threshold Logic as a significative extension of the classical Threshold Logic initiated by McCulloch and Pitts 50 years ago. The mathematical theory of this new logic is based on Heaviside a Non-linear argument of the Function. The determination of the parameters is made by an algebraic learning. For the first time, we introduced the so-called Heaviside Fixed Function.
The models of the Dubois' Fractal Neural Network and the Fukushima et al Neocognitron give the original framework of the new Non-linear Threshold Logic.
The Threshold Logic, using linear argument of the Heaviside function, is not capable of resolving all Boolean functions with only one neuron.
It is demonstrated that our Non-linear Threshold Logic resolved all the Boolean functions with one single neuron for any number of inputs, ie even the XOR problem.
This is a good test for our method. Moreover it is shown that the very difficult Parity problem has also a very elegant solution for obtaining the parity of any Boolean patterns from one single neuron with any number of inputs.
In the technology of neural networks, the back-propagation methods are based on hidden neurons to resolve the Boolean functions. There are some difficulties to obtain the global minimum as the number of neurons increases and the combinatorial explosion gives many problems for practical applications.
The Non-linear Threshold Logic will permit to diminish a great quantity of neurons in the neural networks with the back-propagation.
The possibilities of bigger practical applications are so given by this Non-linear Threshold Logic in the technology of neural machines due to the diminution of the necessary number of neurons and so to avoid some combinatorial explosion problems. Turbo Brain 1.0, a package for Personal Computers, has been developped from this Non-linear Threshold Logic with a Symbolic software.