Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site umcp-cs.UUCP
Path: utzoo!linus!philabs!cmcl2!seismo!lll-crg!gymble!umcp-cs!mangoe
From: mangoe@umcp-cs.UUCP (Charley Wingate)
Newsgroups: net.philosophy,net.math
Subject: Re: Mind as Turing Machine: a proof *and* a disproof!
Message-ID: <2081@umcp-cs.UUCP>
Date: Sun, 3-Nov-85 14:12:22 EST
Article-I.D.: umcp-cs.2081
Posted: Sun Nov  3 14:12:22 1985
Date-Received: Tue, 5-Nov-85 08:40:52 EST
References: <509@klipper.UUCP> <1096@jhunix.UUCP>
Distribution: net
Organization: U of Maryland, Computer Science Dept., College Park, MD
Lines: 48
Xref: linus net.philosophy:2748 net.math:2112

In article <1096@jhunix.UUCP> ins_apmj@jhunix.ARPA (Patrick M Juola) 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.  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.

Unfortunately, this fails to work because in Turing machines, it is the
number of *steps* which costs you; each step from one cell to the next costs
at least on step.  Now there are two cases to consider:

  1) Suppose we can make the comarison without having to return to a
     reference value (which is reasonable for a small enough range of
     values).  Then it's clear that the tree search is indeed order(n);
     since it will take K shifts to get to the Kth position, no matter what.

  2) Suppose we do have to return to a reference value that is an extra M
     units away, and it takes F*(M+K) to make a comparison with an element
     at position K (i.e., we have to make that many "passes" between the
     two).  For the sequential search this is clearly O(N**2).  For the tree
     version, we get something like

       FM+F(M+2)+F(M+4)+ ... F(M+K/2) which is

       FMlogK+FK   giving us O(N).

The reason why we get these unusual results is that ordinarily the costs of
seeking are negligible.  In this case, they are quite important.

Nevertheless, the original "proof" is still flawed.  The brain is quite
obviously not a turing machine; on the other hand, neither are current
computers.  The important question is whether or not it can be modelled by a
turing machine.  Considering performance, this little "proof" tends one to
believe that it can't; but ignoring performance, this "proof" gets us
nowhere.

Charley Wingate