Seeing as how there is insufficient space to set out all the mathematical
niceties (they will have to wait for the full version of this paper),
let's dive in and
play an infinite game -- a game we call, in deference to some recent
investigations carried out by Robert McNaughton, ``McNaught" [McNaughton, 1993].
McNaught isn't a game like chess,
mind you: chess, as we've noted,
is after all a finite game, one handled quite
well by ordinary computation, as even Dreyfus must now admit.
We're talking about an infinite game; here's how it works. You will need
a place-marker (a dime will do nicely), and the graph shown in
Figure 3 (across which you will slide the dime). We will be black,
you will be red. Notice that the nodes in the graph of Figure 3 are
divided in half: three are red (r) nodes; three are black (b) nodes.
If the dime is on an r node, then it's your turn, as red, to move; if
the dime is on a b node, it's our turn. Here's how the game proceeds.
The dime is placed randomly on one of the nodes, and then we take turns with
you,
sliding it from node to node, making sure that a move is made in accordance
with a connecting arrow. So, if the dime is initially upon 
, you would
move, and your options are 
and 
. If you slid the dime to ,
our only
option would be 
, and so on. Now, here's the thing: you
and the two of us are going to
take turns back and forth for an infinite amount of time. Since you may
complain at this point that you are mortal, we want you to assume for the
sake of the game that the three of us, like super-machines,
can in fact take turns
forever. [Super-machines are those with more power
than Turing Machines. Super-minds are beings having, among
other things, information-processing power above
TMs. For more on super-computation in general, including an introduction
to the Arithmetic Hierarchy,
see [Bringsjord, 1997a]. For a sustained defense of the view that
human persons are indeed super-minds, see the forthcoming book
Super-Minds: [Brinsjord and Zenzen, 1997].]
Okay, now notice that nodes and are
double-circles; this
is because these two are ``winning" nodes. We win, as black, if and only
if either and are both visited only finitely many times or
are both
visited infinitely often. You, red, win if and only if one of these two
nodes is visited infinitely often and the other finitely often.
Got it?
Okay, now: What is your strategy? What is your best strategy?
What is our best strategy? If we both play our best, who will win?
And supposing we play only for a finite amount of time, how could a referee
predict a winner?
Don't read this paragraph if you intend to tackle these
questions.
Only black has an
invincible strategy, viz., from 
move to 
if has never
been visited or if has been visited since the last time
was visited; in all other circumstances move to . So there
was really no way for you to beat us! It is remarkable that
ordinary computation can find this strategy when presented
with the game in question [McNaughton, 1993]. (No ordinary
computer can literally play the game, of course.)
However, for a game utterly beyond the Turing Limit, see the
``undetermined" game featured in [Gale and Stewart, 1953]:
this is a game where a winning strategy cannot be devised by
ordinary computation (in fact, there is no mathematical function
which is a winning strategy!). It seems to us that infinite games,
perhaps especially uncomputable infinite games, provide promising
frameworks for mind-machine competition. The first step, which
we are in the process of taking, is to take a computable infinite
game and cast it in terms allowing for mind-mind competition.
[We are starting with McNaught. The task of declaring a winner
in finite time is rather challenging. See the approach indicated
in [McNaughton, 1993].] Other frameworks might involve competition
centering around the creation of infinite (and other types of)
games.