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From: amb@duke.UUCP (A. Michael Berman)
Newsgroups: net.philosophy,net.math,net.physics
Subject: Re: Mind as Turing Machine: a proof *and* a disproof!
Message-ID: <6541@duke.UUCP>
Date: Tue, 5-Nov-85 09:33:10 EST
Article-I.D.: duke.6541
Posted: Tue Nov  5 09:33:10 1985
Date-Received: Thu, 7-Nov-85 05:53:44 EST
References: <1996@umcp-cs.UUCP> <667@hwcs.UUCP> <2031@umcp-cs.UUCP> <509@klipper.UUCP> <1096@jhunix.UUCP>
Reply-To: amb@duke.UUCP (A. Michael Berman)
Organization: Duke University
Lines: 44
Xref: watmath net.philosophy:3045 net.math:2482 net.physics:3503

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.
>
>	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.
    ...
>This, on the average, will find any item in memory in log(base 2)n 
>comparisons,

Three comments:

1. re the original comment: the question of whether a Turing machine can
compute something is normally separated from its complexity.  If we could
build a Turing machine that simulated human intelligence at any speed,
then the presumption would surely be that we could bring it 'up to
speed'.  (The suggestion in another note for parallel Turing machines,
for example, might be a way to do it.)  In fact, a Turing machine tends
to be a slow way to do most operations, which leads to the next comment:

2. True, if you count comparisons, a binary search takes log n time
(worst case too, not just on average).  However, while comparison counting is
a reasonable method of analysis on the computation model called a RAM,
or Random Access Machine (which more closely models modern computers
than does a Turing Machine), it is the wrong metric to use with a TM.
Clearly the time for a TM to compute something is proportional to the
number of tape squares processed.  Thus we can't avoid counting all the
work moving back and forth.  In fact, the lower bound for searching on a
TM is O(n).

3. A RAM can search memory in O(1) expected time, by using hashing.
(Can your brain do better?)

The question 'can a TM think' could just as well be 'can a RAM think';
since a TM can simulate a RAM and vice versa, the questions are
equivalent, modulo the time of computation.  The TM is the more
traditional formulation.

Mike Berman, Duke University