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Article 4587 of comp.ai.philosophy:
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>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech
Subject: Re: A rock implements every FSA
Message-ID: <1992Mar18.185350.10004@husc3.harvard.edu>
Date: 18 Mar 92 23:53:48 GMT
References: <1992Mar18.045939.3084@bronze.ucs.indiana.edu> <1992Mar18.095140.9984@husc3.harvard.edu> <centaur.700937150@cc.gatech.edu>
Organization: Dept. of Math, Harvard Univ.
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In article <centaur.700937150@cc.gatech.edu> 
centaur@terminus.gatech.edu (Anthony G. Francis) writes:

>zeleny@zariski.harvard.edu (Mikhail Zeleny) writes:

DC:
>>>Of course C and D will be found in the table.  That's my point.  They
>>>won't be found in the rock, though.  Putnam's construction doesn't
>>>even *define* any physical states corresponding to C and D, let alone
>>>ensure that the right transition relation holds between them.
>>>Therefore the claim that the system in question implements the FSA is
>>>groundless.  At best, it implements a "trace" of a particular run
>>>of the FSA, as Joseph O'Rourke nicely put it.

MZ:
>>This is easy: first, you interpret the states of Putnam's automaton as
>>ordered pairs <state, input> of a FSA (cf. the relevant comments on p.124);
>>follow this by running through enough input/state combinations to exhaust
>>the finite combinatorial possibilities afforded by the machine's table.
>>Finally, you do the mapping.  In this way, there will be no counterfactual
>>possibilities left unaccounted for.

AGF:
>Impossible. As a quick check to any basic automata text will show you,
>a FSA can support input strings of unbounded (e.g., in layman's terms, 
>infinite) size.

Unbounded equals infinite only in idiot's terms. Still, your comment is 
basically valid. 

AGF:
>		 The pumping lemma (that nasty little theorem that proves
>that no FSA can accept strings longer than its number of states without
>accepting all similiar strings containing some common repeating substring)
>won't buy your way out of this one; the set of strings is still infinite
>in size and therefore the number of ordered pairs in the above example
>is infinite as well and cannot be mapped to any real rock.

But the pumping lemma was supposed to be my move!  OK, so let's cut to the
chase.  

AGF:
>If you're willing to devise another mapping, go ahead; however, any
>such mapping of Putnam's type would, of course, be _post hoc_ and 
>valueless in any scientific setting, regardless of its theoretical validity.

Two comments.  Firstly, "valueless in any scientific setting, regardless of
its theoretical validity" gets my vote as the best oxymoron of the year.
Secondly, given the original context of determining the theoretical
validity of functionalism, the requisite bound on input length is easily
obtained on the basis of temporal limitations on the length of human life.
Refer to the erstwhile discussion on the humongous look-up table for more
information.  

>-Anthony Francis


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