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Article 4630 of comp.ai.philosophy:
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>From: clarke@acme.ucf.edu (Thomas Clarke)
Newsgroups: comp.ai.philosophy
Subject: Re: Infinite Minds?
Message-ID: <1992Mar20.133924.19215@cs.ucf.edu>
Date: 20 Mar 92 13:39:24 GMT
References: <1992Mar19.100550.10019@husc3.harvard.edu>
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In article <1992Mar19.100550.10019@husc3.harvard.edu>  
zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:
| In article <1992Mar18.183651.26822@cs.ucf.edu> 
| clarke@acme.ucf.edu (Thomas Clarke) writes:
| 
| >> [...]
| 
| >Wild conclusions?  I merely try to make the point that it is plausible
| >that the brain may not function digitally.  Until the Church-Turing thesis
| >is rigorously proved or disproved, the question remains open.   
| 
| Pardon my intrusion, but what the fuck does Church's thesis (Turing, a
| student of his, was rather a latecomer) have to do with the claim of
| digital brain functioning?  
I'll see if I can make the connection clear.

Church's thesis, as amplified by Turing, says that any precisely specifiable  
computation is equivalent to a Turing machine calculation.  Given a complete  
physics and reducibility, all future observations in the world can be  
calculated in a precisely specifiable manner.  Church's thesis thus applies to  
these calculations of physical computations [A little difficulty with reals,  
but rationals are dense in the reals, and rationals are pairs of integers ...  
see "Computability in Analysis and Physics", Pour-El and Richards, Springer,  
1987]

Therefore, the future actions of organisms, and networks in particular, are  
computable. May be some problem with finding a complete physics, or with the  
program of reducing all to physics...

With regard to networks simulating Turing machines see the article "On the  
Computational Power of Neural Nets", by Siegelman and Sontag (available as  
siegelman.turing.ps.Z on the neuroprose archive site  
archive.cis.ohio-state.edu).  They outline the construction of a neural net  
with <1000 nodes that implements a universal Turing machine.  No tape is used,  
rather an unbounded tape is simulated by using what amounts to a Bernoulli  
shift operation on a selected network to implement a stack.  A pair of stacks  
makes a tape.  That is, if 0<=q<=2 is the variable then b+q/2->q pushes the bit  
b into q, and {int(q)->b, q/2->q} pops the variable b from q.  

This approach is not physically realistic, noise and quantum effects would  
rapidly spoil the information stored in the rational number q, but I think it  
is an interesting construction.  Siegelman and Sontage claim that when real  
numbers are used, they can get a network that computes the uncomputable.  They  
promise publication real soon now.  I am very interested in how this works  
since such a net would cause problems for the Turing modified Church thesis if  
you believe in physics and reductionism.  My guess is it may have to do with  
Chaitin's uncomputable real number which codes the halting probablility of the  
universal Turing machine. 


