Overview: Humanity‘s most basic intellectual quest to decipher nature and master it has led to numerous efforts to build machines that simulate the world or communi cate with it [Bus70, Tur36, MP43, Sha48, vN56, Sha41, Rub89, NK91, Nyc92]. The computational power and dynamic behavior of such machines is a central question for mathematicians, computer scientists, and occasionally, physicists. Our interest is in computers called artificial neural networks. In their most general framework, neural networks consist of assemblies of simple processors, or "neurons," each of which computes a scalar a
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