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>From: orourke@sophia.smith.edu (Joseph O'Rourke)
Newsgroups: comp.ai.philosophy,sci.philosophy.tech
Subject: A rock implements every FSA
Keywords: Putnam, implementation
Message-ID: <44855@dime.cs.umass.edu>
Date: 14 Mar 92 03:05:10 GMT
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Reply-To: orourke@sophia.smith.edu (Joseph O'Rourke)
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Organization: Smith College, Northampton, MA, US
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A while back there was some discussion of Putnam's proof that,
roughly, a rock implements every Finite State Automaton (FSA).
I was unfamiliar with (and skeptical of) this theorem, so I studied
it a bit.  As a public service (some might say disservice!), I
herewith offer a synopsis of the argument.

[Hilary Putnam, "Representation and Reality" (1988), Appendix, 121-5]

Let S represent the state of a physical system (the rock).

Assumption 1 (Continuity): 
	S is a continuous function S(t) of time t.
Assumption 2 (Noncyclical): 
	For t1 != t2, S(t1) != S(t2)  [where "!=" means "not equal"].

Thus if S were representable as a single variable, it would be a strictly 
monotonic function with respect to time, something like this:

          S
          ^
          |                           +
          |                      +
          |                  +
          |               +
          |             +
          |            +
          |           +
          |       +
          |     +
          |  +
          | +
          +---------------------------------> t

Let the FSA go through states A,B,A,A,B in that order.
Lay out those states along the t-axis.  By the assumptions,
the physical states are partitioned as shown below:

          S
          ^
        5 |                           +
        4-|                      +
        3-|                  +
         -|               +
        2 |             +
          |            +
         -|           +
        1 |       +
         -|     +
        0 |  +
          | +   |    |    |    |    |
          +---------------------------------> t
                 A    B    A    A    B

Simply define "state A" of the rock to be the union of the physical
states s1+s3+s4 corresponding to the times when the FSA is in state
A, and define physical "state B" to be s2+s5.  Then it is immediate
that rock is in "state A" precisely when the FSA is in its state A,
and similarly for state B.  Moreover, Putnam argues that physical
"state B" is *caused by* physical "state A," in the sense that the
information that the system is in "state A" at a certain time
determines that it must be in "state B" at the next time.

	That's it.  You can see that the mathematical content of
his "theorem" is nearly trivial.  All the subtlety and complexity
resides in the I/O to and from the math:  whether his assumptions
which form the input to the mathematics are justified, and whether
his interpretation of the output of the mathematics is sound.

					:-j


