(These are excerpts from my book "Intelligence is not Artificial")
Analog Computation
In 1992 Hava Siegelmann of BarIlan University in Israel and
Eduardo Sontag of Rutgers University developed "analogic recurrent neural networks"
("Analog Computation via Neural Networks", paper submitted in 1992 but published only in 1994).
For 60 years it was assumed that no computing device can be more powerful than a Universal Turing Machine. Hava Siegelmann proved mathematically that analog RNNs can achieve superTuring computing ("On the Computational Power of Neural Nets", 1992). Alan Turing himself had tried to imagine a way to extend the computational power of his universal machine ("Systems of Logic Based on Ordinals", 1938), but his idea cannot be implemented in practice. Siegelmann's system was not the first system to break the Turing limit using real numbers, and nobody has built a computer yet that can perform operations on real numbers in a single step.
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