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Article 1588 of comp.ai.philosophy:
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>From: centaur@terminus.gatech.edu (Anthony G. Francis)
Newsgroups: comp.ai.philosophy
Subject: Re: Daniel Dennett
Message-ID: <centaur.691103968@cc.gatech.edu>
Date: 25 Nov 91 21:19:28 GMT
References: <1991Nov23.214707.1663@cc.gatech.edu> <39701@dime.cs.umass.edu> <centaur.691029638@cc.gatech.edu> <39729@dime.cs.umass.edu>
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yodaiken@chelm.cs.umass.edu (victor yodaiken) writes:

>In article <centaur.691029638@cc.gatech.edu> centaur@terminus.gatech.edu (Anthony G. Francis) writes:
>>yodaiken@chelm.cs.umass.edu (victor yodaiken) writes:
>>AF:
>>On the other hand, claiming that a realizable TM can be reduced to an 
>>unrealizable FSA is not only theoretically shaky, it also dodges
>>the question of what real-world systems actually use. Of course a theory of
>>notation and denotaion can't be implemented on a FSA. But a subset of that
>>theory - that might apply to finite human beings - might be implemented
>>on a Turing Machine.

>Consider the following two abstract machines. T is a turing machine
>that, when started on a tape containing an input of k or fewer symbols from
>a r element alphabet will compute some function, never using more than
>r+j squares of tape. M is a FSA with state set  of all the triples of the
>form (inputing,p,q)  and (working,s,p,q)  where p ranges over all integers
>between 1 and r+j and q ranges over all r+j tuples over the alphabet and
>a blank character. We interpret p as the position of the head, and q as the
>contents of the tape. The initial state of M is (inputing,1,blank-tuple)
>In what sense if M less realizable than T?  

T and M recognize the same set of languages. T and M are not equivalent
machines because they have different space requirements. T requires the
space for its FSA and r+j elements of tape. M requires at least r(r+j)(r+j)
separate states. It is vastly larger, and not realizable in the same set
of space constraints that you have placed on the Turing Machine.

>Mathematically, both are well-defined. In what sense is it "theoretically 
>shaky" to consider the two machines as the same?

It is not theoretically shaky to consider them to represent the same language
in the abstract. It is theoretically shaky to make statements about what
real systems can do by noting their limitations and then ignoring those 
limitations to prove equivalence to a theoretical class of machines that
can't really exist. Turing Machines with finite tapes do _not_ have all
the same properties as FSA's, and you can prove that.

>As for your example of a Sun computer, constructing a FSA representation is
>probably less hard than constructing a pure TM representatation. At least
>for the FSA we have a theory of automata products which will allow us
>to construct a FSA for memory, one for disk .... and combine them. TM
>products are quite a bit messier --- because of the tape. 

First of all, we can't construct a FSA for the _memory_ subsystem, because
of the physical constraints of this universe. Essentially, my point is that
Turing Machines require unbounded amounts of storage, and for a vast collection
of problems, actual physical storage is sufficient for an implementation of a
Turing Machine to solve those problems. A direct implementation of a FSA for
that same set of problems, however, is _impossible_. So, the alternatives that
I see are to simply restrict our discussions to theoretical machines, in
which case Turing Machines are more powerful, or to restrict ourselves to
machines with space constraints, in which case Turing Machines are more
powerful.

For an even better argument on the nonequivalence of TM's and FSA's, check
one of the related threads. Someone else wrote another argument for the
nonequivalence of bus architectures and FSA's, which I thought was quite good.
--
Anthony G. Francis, Jr.  - Georgia Tech {Atl.,GA 30332}
Internet Mail Address: 	 - centaur@cc.gatech.edu
UUCP Address:		 - ...!{allegra,amd,hplabs,ut-ngp}!gatech!prism!gt4864b
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