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From: franka@mmintl.UUCP (Frank Adams)
Newsgroups: net.philosophy,net.math,net.physics
Subject: Re: Mind as Turing Machine: a proof *and* a disproof!
Message-ID: <774@mmintl.UUCP>
Date: Tue, 5-Nov-85 11:16:19 EST
Article-I.D.: mmintl.774
Posted: Tue Nov  5 11:16:19 1985
Date-Received: Fri, 8-Nov-85 08:24:50 EST
References: <1996@umcp-cs.UUCP> <667@hwcs.UUCP> <2031@umcp-cs.UUCP> <509@klipper.UUCP> <1096@jhunix.UUCP>
Reply-To: franka@mmintl.UUCP (Frank Adams)
Organization: Multimate International, E. Hartford, CT
Lines: 40
Keywords: minds, Turing machines
Xref: linus net.philosophy:2779 net.math:2120 net.physics:3253
Summary: Fallacies refuted


In article <1096@jhunix.UUCP> ins_apmj@jhunix.ARPA (Patrick M Juola) writes:
>In article <509@klipper.UUCP> biep@klipper.UUCP (J. A. "Biep" Durieux) writes:
>>	                                        Psycholinguistics has
>>	found that humans can search their memory in < log n time, n
>>	being the number of items. Turing machines clearly can not do
>>	better than order n time. Proof that humans are not Turing machines.

A human mind has a finite limit on the amount of information it can store.
If you put an upper limit on the amount of information, searching only takes
a finite amount of time.  The behavior of algorithms with small amounts of
data does not necessarily resemble their limiting behavior.

>	I'm sure that a Turing machine can search its memory faster than order
>n : all it would have to do is store the stuff in its memory in some sort of 
>order.  I'm thinking specifically of the structure called a binary tree, where
>everything in the right sub-tree is > the root and the left is < the root.
>Program the machine to start at some designated root (call it position 1) on
>the tape.  If the item to be searched for is < position n, shift left (for 
>example) to position n*2.  If the item is >, shift left to position n*2+1.
>This, on the average, will find any item in memory in log(base 2)n
>comparisons,
>and you've still got an infinite amount of tape to the right for storage of
>other items.

By definition, a Turing machine can only move left or right one frame at a
time.  You can't find the data in less than O(t) time because you can't get
to it any faster than that.

(To be a little more formal, the total information potentially available in
time t is O(t).  A binary search requires potential access to O(2**t)
information.)

>Any data structure that would let
>you perform a binary search would work, and I would be *fascinated* by a proof
>that no such structure exists.  (Yes, J.A., that was a challenge :-)

Is that proof fascinating?  I thought not.

Frank Adams                           ihpn4!philabs!pwa-b!mmintl!franka
Multimate International    52 Oakland Ave North    E. Hartford, CT 06108