This paper reasons about the need to seek for particular kinds of models of computation that imply stronger computability than the classical models. A possible such model, constituting a chaotic dynamical system, is presented. This system, which we term as the analog shift map, when viewed as a computational model has super-Turing power and is equivalent to neural networks and the class of analog machines. This map may be appropriate to describe idealized physical phenomena. A straightforward method of measuring the area of a surface is by counting the number of atoms there. One may be able to develop smart algorithms to group atoms together in sets, and thus speed up the counting time. A totally different approach is by assuming continuous rather than quantized/discretized universe and calculating the relevant integral. Such a continuous algorithm should ideally be implemented on an analog machine, but it can also be approximated by a digital computer that allows for finite precision only. Although the actual hardware is discrete, the core assumption of continuity allows the development of inherently different algorithms to evaluate areas. It is possible that in the theory of computation, we are still at the stage of developing algorithms to count faster. Maybe just by assuming an analog media (although not really having it), we would be able to do much better for some tasks. A more fundamental reason to look for analog computation models stems from recent advances in the field of physics and the aim to simulate idealized physical phenomena on computers. Already in the 18th century, PoincarC realized that the orbits of simple dynamical systems may be extremely unpredictable, and mathematicians have been dealing with this phenomenon since. However, since 1975 “chaos” has been realized by physicists to occur in many systems of scientific interest [7]. Turing machines are indeed able to simulate a large class of systems, but seem not to capture the whole
['Hava T. Siegelmann']
The simple dynamics of super Turing theories