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Article 4828 of comp.ai.philosophy:
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>From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: A rock implements every FSA
Message-ID: <1992Mar30.170118.14157@oracorp.com>
Organization: ORA Corporation
Date: Mon, 30 Mar 1992 17:01:18 GMT
Lines: 34

orourke@unix1.cs.umass.edu (Joseph O'Rourke) writes:

>Was the "Zeleny machine" suggested in a posting, or are you refering
>to a private discussion?

I shouldn't have called it a Zeleny machine, since it is not identical
to the machine he was talking about. However it was inspired by his
machine.

>	M( <A,0>   ) = A
>	M( <A,y+1> ) = T( M( <A,y> ) ) = A

>Therefore, if I understand the proposal, this simple two-state
>machine includes among its corresponding Z-machine states <A,0>, <A,1>,
><A,2>, ..., all of which are mapped to A by M.  Or to reverse
>the mapping, the single state A is modeled by an infinite number
>of states in the Z-machine.
>	Is this right?

Yes. The "implements" relation is not symmetric; system S1 can
implement system S2 without S2 implementing S1, exactly because any
number of states of S1 can map to the same state of S2. I think you
want to allow many-to-one mappings; otherwise it is impossible for any
physical system (which has an infinite number of possible states) to
implement a finite state machine. For example, in the case of a single
memory bit of a computer, the voltage at that bit can be *any* value,
but all values other than 0 and 1 are unstable. To interpret the bit
as a two-state system, it is necessary to map all voltages less than
some threshold value to 0, and all voltages above the threshold to 1.

Daryl McCullough
ORA Corp.
Ithaca, NY



