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Article 5087 of comp.ai.philosophy:
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>From: chalmers@bronze.ucs.indiana.edu (David Chalmers)
Newsgroups: comp.ai.philosophy
Subject: Re: Functional Equivalence (Was: A rock implements every FSA)
Keywords: functionalism, behavioralism
Message-ID: <1992Apr14.064358.16495@bronze.ucs.indiana.edu>
Date: 14 Apr 92 06:43:58 GMT
Article-I.D.: bronze.1992Apr14.064358.16495
References: <1992Apr2.164457.24191@oracorp.com> <46023@dime.cs.umass.edu> <46079@dime.cs.umass.edu>
Organization: Indiana University
Lines: 56

In article <46079@dime.cs.umass.edu> orourke@sophia.smith.edu (Joseph O'Rourke) writes:

>	So here's my proposal:
>
>		Two Turing Machines are FUNCTIONALLY EQUIVALENT
>		if they make the same tape moves & writes, in the
>		same order, on every starting input tape.

That's not a bad criterion for TM equivalence -- it abstracts away
from head-states of course, but that's not an unreasonable abstraction.
Of course one gets different notions of equivalence or implementation
depending on how much one abstracts away from.

What's missing is a characterization of when a TM is actually
being implemented by a physical system.  By assuming we already
have tape squares and symbols individuated, much of the work is
already done.

>Note that there is no mention of mapping between states of the two
>machines.  So in a sense, I am defining functional equivalence in
>terms of behavioral equivalence.  But it is fine-grained behavior.
>After all, if we included state transitions as part of the behavior,
>Then behaviorism = functionalism! 

I disagree.  Changes in a TM's tape aren't behaviour, they're
internal function.  It's a mistake to think of a TM as a tape-head
that makes it's inputs and outputs onto the tape: if you did this,
then the "mechanism" itself would only be an FSA.  The tape just
allows for arbotrarily complex internal states.

(Of course, there is the question of how one then allows for
input and output to and from a TM.  The usual solution is to set
aside a certain finite portions of the tape for input and output,
or else to allow separate tapes.)

>	Consider the consequences of this definition for the two machines
>TM1 = 5-color-map-checker, and TM2 = single-state rejector, discussed
>by David Chalmers and Daryl McCullough recently.  TM2 will take a plane
>graph on its input tape, ignore it, write "no" (or 0) and halt.
>TM1 will take the graph, determine if its chromatic number is 5, and if
>not, write "no" and halt.  It should be clear that if TM1 is really
>a program that checks for chromatic # = 5, for graphs of arbitrary
>size (this is important!), then it must use the tape to write intermediate
>results.  And therefore, TM1 != TM2 under my definition of equivalence.

But this evades the question of whether an implementation of TM1
might actually implement TM2 under some mapping (and the question
of whether any old rock might).  Given a physical system, there
won't be a single canonical decomposition into tape squares and
symbols.  I think Putnam's wrong, but he can't be beaten as
easily as this.

-- 
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."


