Speculative Discussion of Super‑Turing Machines and Hyper‑Computing Concepts

280 Are the Androids Dreaming Yet? David Malament and Mark Hogarth of the University of California, Irvine have proposed a form of space-time called the Kerr Metric. This allows a machine to break the Turing limit, but has the drawback that as it does so it falls through the event horizon and is sucked into the black hole. We might discover new information but are now trapped inside the event horizon unable to communicate it — a form of cosmic censorship. Candidates for a hyper-computer that could fit inside a human brain include mathematical curiosities which stretch the concept of infinity. The easiest to understand is the Zeno machine. In a Zeno machine a computer runs each successive step of a calculation in half the time of the previous step. The computer can pack an infinite quantity of computation into each finite time interval and can therefore outperform a Turing machine. This theory fails at a practical level because we simply cant build such a machine. There are numerous weird suggestions for mathematical super- Turing machines, and many are described on the Internet. They all fit broadly within the two models above: modifications to space-time or peculiar mathematical paradoxes. The inspiration for the true solution to super-Turing thought may lay in there somewhere, but there are some more plausible proposals to look at next. Plausible Ideas I have characterized the next set of ideas as plausible, but they may still be highly controversial. My only criteria for plausibility are that the mechanism must outperform a machine limited to counting numbers, and it might fit inside our skulls. No black holes allowed. One interesting proposal for a super-Turing machine that could fit inside our skulls is the Adaptive Recurrent Neural Network, ‘ARNN’ proposed by Hava Siegelmann of the University of Massachusetts, Amherst. An ARNN is a neural network with real number weights. As you recall, real numbers are equivalent to the continuum infinity, a larger infinity than that of counting numbers. This is the infinity that defeats a Turing machine, and Siegelmann harnesses it as the basis of her computing machine. She argues that, although the machine cannot be programmed as it is impossible to write real numbers down, once it is running, the weights diverge and real numbers will be used within the network. These real numbers allow the machine to compute using numbers that are not, themselves, computable HOUSE_OVERSIGHT_015970