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Article 1542 of comp.ai.philosophy:
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>From: yodaiken@chelm.cs.umass.edu (victor yodaiken)
Newsgroups: comp.ai.philosophy
Subject: Re: Daniel Dennett
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Date: 24 Nov 91 05:16:54 GMT
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In article <1991Nov23.214707.1663@cc.gatech.edu> centaur@terminus.gatech.edu (Anthony G. Francis) writes:
>>AGF:
>>>[argument about FSA < PDA < TM deleted]
>>MZ:
>>A brain has no infinite tape; nor have you.  Sorry, but you are limited to
>>the finite state automata.  Also note that an infinite table is still a ...
>
>The universe is, as far as we can tell, finite, and so your statement
>(and the statement of another poster) is trivially true, since any FSA we
>could build must have a finite number of states, any PDA must have a finite
>stack size, and any Turing Machine must have a finite sized tape. Ok, that's
>a given. But even finite-sized FSA's have limitations that finite-sized
>PDA's and TM's don't.
>

This argument seems to be quite widespread, and I'm curious to see if you
can shed some light on it. PErhaps you could define "FSA" in such a way
as to make the distinction between FSAs and "finite-sized TM's" clear.
To my, no doubt naive, understanding, a Turing machine limited to a finite
tape *is* a FSA, or, at the very least, is a representation of an FSA.

>Given a space constraint, I can design a fairly small PDA to accept the string 
>(a^i)(b^i), that is, i a's followed by i b's. A finite PDA can only accept 
>a string _of this form_ twice its length of its stack; a FSA can only accept
>a string of size n, where n is its number of states. If we construct a
>FSA with one state for every element of the PDA's stack plus the total
>states of the PDA (oh, three or four tops) then it can only accept a string
>of size i+4 - a string of little more than half the length of the PDA's string,
>and hence it accepts a language little more than half the size. As the
>complexity of the language rises, the limitations of the FSA become greater.

This argument seems to me to confuse notation with denotation. 
A similar argument would show that base 2 notation is "less powerful" than
base 10 notation. 


