X-Message-Number: 15386 Date: Thu, 18 Jan 2001 22:56:18 -0700 From: Mike Perry <> Subject: Numbers Again; How to Post Thomas Donaldson, #15375, says: >Given N neurons, the number of possible connections between them goes up >like N!. This is greater than exponential. I think you mean the number of possible configurations, or patterns of connections. Given that the number of (1-way) connections itself is N^2 (again allowing a neuron to connect to itself, mainly because it simplifies the formula here though increasing the number slightly), each pattern of connections will amount to a subset of the possible connections, i.e. any given connection will be either present or absent. So how many configurations is that? The number of subsets of a set of N^2 elements, or 2^(N^2), which indeed is greater than exponential. (Actually it's greater than factorial too.) However, this applies to artificial neural networks too. A probabilistic, 1-tape Turing machine will also have much more than a polynomial number of *possible* computations it could perform as a function of time. It is more powerful than many people seem willing to credit. >The fundamental issue here seems to be that of whether the neurons are >modified when they lose a connection or make a new one. Whatever the case may be, I see no reason in principle why a neuron could not be imitated by a (probabilistic) computer based on a Turning machine. If nothing else, a neuron is a system of particles at the quantum level, and that can be so imitated. And connections too, are systems of particles, as are whole brains, for that matter. So all must be imitatable by a universal quantum simulator, which itself could be imitated, albeit not very efficiently, by a single, humble, classical Turing machine. How to Post: David C. Johnson, who says (#15377), "Please advise me on what I need to do to post," has obviously found the answer. Welcome to CryoNet! Mike Perry Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=15386