**From:** Hector Zenil (*hzenilc@gmail.com*)

**Date:** Fri Nov 23 2007 - 23:53:56 MST

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I did one of my theses on Recurrent Analog Neural Networks (ARNN) and

gave a talk at the University of Liverpool two years ago, the

proceedings were recently published by World Scientific[1] with my

paper under the title "On the possible Computational Power of the

Human Mind". An arXiv version is available online:

http://arxiv.org/abs/cs/0605065

and the presentation from the conference:

http://complexity.vub.ac.be/phil/presentations/Zenil.pdf

ARNNs are basically neural networks, first analyzed by Siegelmann and

Sontag, able to allow arbitrary real numbers as weights.

Traditionally, neural networks have been considered to have rational

or computable weights. From Kleene we know that neural networks with

integer weights are equivalent in computational power to finite

automata. Siegelmann made several contributions in her seminal book[2]

such as the construction of a very small universal neural network with

only 9 neurons (published before in joint with Maurice

Margenstern[3]). She also gave several interesting computational

complexity results related to the ARNN's in general and provided a

proof for neural networks with rational numbers as equivalent to

Turing machines in computational power. Then extended the model to

allow any arbitrary real number that would then automatically allow

the ARNN to compute beyond the Turing limit (e.g. put as weight the

value of $\Omega$ the halting set coded as a real number). Of course

one question is from where one can get that non-computable number in

order to put it in the net configuration to reach hypercomputation

(all known artificial neural networks run basically over Turing

machines), although if the system already has it (e.g. one already

operating in nature or in our head) then it automatically would reach

hypercomputation. But from it one cannot draw any conclusion about its

feasibility. Martin Davis's excellent article [4], "The Myth of

Hypercomputation," clearly articulates several criticisms, including

jumps to conclusions from Siegelmann's work.

Hava Siegelmann is not claiming that the human brain is an ARNN, we

suggest that it can be studied through an ARNN model though, because

of its generality. Of course a brain with a neural network with

arbitrary real number would go beyond the Turing limit, but that is

unlikely, even when there are other elements traditionally taken as

"continuous" variables, such as neural spikes, also covered in [1].

Hector Zenil

[1] Hector Zenil & Francisco Hernandez-Quiroz (2007), Worldviews, On

the possible Computational Power of the Human Mind. Worldviews,

Science and Us: Philosophy and Complexity by Carlos Gershenson, et. al

(ed.), World Scientific, pp. 315-337.

from Amazon: http://www.amazon.com/Worldviews-Science-Us-Philosophy-Complexity/dp/9812705481/ref=sr_1_1?ie=UTF8&s=books&qid=1195886116&sr=8-1Yf

[2] Hava T. Siegelmann (1998), Neural Networks and Analog Computation:

Beyond the Turing Limit, Springer.

[3] Hava T. Siegelmann, Maurice Margenstern (1999), Nine switch-affine

neurons suffice for Turing universality. Neural Networks 12(4-5):

593-600.

[4] Davis, M. (2004) The Myth of Hypercomputation. Alan Turing: Life

and Legacy of. a Great Thinker, Teuscher, C. (ed.) Springer, pp.

195–212.

On Nov 24, 2007 6:39 AM, <sl4.20.pris@spamgourmet.com> wrote:

*> Is a human brain or biological neural network superior to a computer
*

*> or Turing machine?
*

*>
*

*> Has this question ever been answered? There are some scientists that
*

*> claim that the brain has more power than any Turing machine.
*

*> One example would be the following book:
*

*> Hava T. Siegelmann
*

*> Neural Networks and Analog Computation: Beyond the Turing Limit
*

*>
*

*> http://www.amazon.ca/Neural-Networks-Analog-Computation-Beyond/dp/0817639497
*

*>
*

*> Comments?
*

*>
*

*> Roland
*

*>
*

-- Hector Zenil-Chavez hector.zenil-chavez@malix.univ-paris1.fr Université de Lille I (Laboratoire d'Informatique Fondamentale) Université Pantheon-Sorbonne -Paris 1- (IHPST) -------------------------------- zenil.mathrix.org animaexmachina.com --------------------------------- Fondation Suisse Cité Internationale Universitaire de Paris 7, bd Jourdan - chr. 114 75014 Paris France --------------------------------------------------------------

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