From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.csri.toronto.edu!rutgers!news.cs.indiana.edu!bronze!chalmers Tue Apr  7 23:22:51 EDT 1992
Article 4780 of comp.ai.philosophy:
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>From: chalmers@bronze.ucs.indiana.edu (David Chalmers)
Newsgroups: comp.ai.philosophy
Subject: Re: A rock implements every FSA
Message-ID: <1992Mar28.204705.14884@bronze.ucs.indiana.edu>
Date: 28 Mar 92 20:47:05 GMT
References: <1992Mar27.145107.12415@oracorp.com>
Organization: Indiana University
Lines: 59

In article <1992Mar27.145107.12415@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:

>A deterministic finite state machine can be defined by the following
>functions:
>
>   T : State -> State, the internal transition function
>   I : State x Input -> State, the input transition function
>   O : State -> Output, the output function

This is more or less OK, but there's no need to split the transition
function into T and I.  T is just a special case of I, for the
case where the input is the "null input", or whatever you want
to call it.  What you have above is harmless, but more complex
than necessary (and we'll see below that you exploit this split to
do more work than it can).

>Okay, so the mapping: M(<n,0>) = State_n
>                      M(<n, y+1>) = T(M(<n,y>))
>
>We also need input function I', and the output function O':
>
>       I'(<j,y>, i) = <k,0>, where k is given by I(M(<j,y>),i) = State_k.
>
>       O'(<j,y>) = O(M(<j,y>)

This is bizarre.  This seems to me to be a perfectly good
implementation of an FSA (the issue is complicated somewhat
by the special treatment of the "T" subset of the transition
function, leading to disjunctivitis on machine states, but
let that slide).  But it certainly doesn't prove anything.
This machine is far more complex than Putnam's rocks.  Its
complex structure derives, of course, almost entirely from
the transition function I' (and from the way this relates to
the functions T and O').  You'd be hard pressed to show that
every "ordinary open system" satifies this characterization!

>Note, however, that the functions I' and O' are mathematical
>functions, not state machines. Therefore, it is sufficient to
>implement them with a table lookup, since for mathematical functions
>the only thing that matters is the input/output relation. Also, note
>that the state machine itself is entirely trivial, since it does
>nothing but count.

Any FSA can be implemented with table lookup, of course (NB this
isn't the infamous "humongous" variety).  That doesn't make it
trivial.  All the complexity is in the state-transition table.
You seem to be relying on your split of the usual FSA transition
function F:<states, inputs> -> <states> into two components T and I',
and saying "T can be implemented trivially, so the FSA can be".
But that just doesn't follow.

You might as well argue by saying "Hey, an FSA can be implemented by
an FSA.  But an FSA is trivial -- look, it just uses a table.
So anything implements an FSA."

-- 
Dave Chalmers                            (dave@cogsci.indiana.edu)      
Center for Research on Concepts and Cognition, Indiana University.
"It is not the least charm of a theory that it is refutable."


