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From: Dick Menninger <Dick.Menninger@DaytonOH.ATTGIS.COM>
Subject: Re: Minesweeper
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Date: Fri, 25 Aug 1995 19:51:34 GMT
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> ==========John McNulty, 8/24/95==========
> 
> > Well, the program needs to determine the probable number of mines
> > along the frontier (possibly it needs the pobability distribution for
> > the number of mines, in particular the min/max numbers).  The
> mines along
> > the frontier are not in the unknown area and thereby affect
> the probability
> > of any of the unknown cells having a mine.  Then there is the
> assumption
> > that you can pick any of them with equal effect on winnig the game.
> > Each may have equal probability of being an immediate loss, but
> > the shape of the known area and the frontier affect the value of
> > information provided if it is not an immediate loss.
> > 
> > Even along the frontier, it is not immediately clear that the actual
> > probability
> > for a cell having a mine is readily solved in closed form.  It
> may need some
> > backtracking-based calculation subsuming the whole frontier (a
> local solution
> > that precludes a full solution for the whole frontier can happen).
> 
> Along the frontier, the probability of a cell having a mine IS readily
> solved.  You know EXACTLY how many mines are along the frontier and you
> can calculate exactly the probability of each cell having a mine.  Then

A counter-example has already been posted and there are many more
of them, as well.  This throws the later statements in doubt.

> you are correct in pointing out that this exact number of
mines along the
> frontier should be subtracted from the total number of remaining mines
> to determine the probability of non-frontier cells having a
mine.  In the
> non-frontier region, assuming a uniform distribution of mines, you can
> do no better than calculating #mines/#cells as the probability.

For determining the probability of immediate failure only (that was
already stipulated).  If you do not blow up with the guess, then
there is variability in the possible information provided that IS
affect by the shape of the frontier in the general neighborhood
of the guess.  It may even be true that there exist situations
where a slightly worse risk of immediate failure is more than
offset by the potential of better information reducing later risks.
You seem to ignore the information return of non-failing guesses.
> 
> In general, you will have to consider all possible
arrangements of mines
> along the frontier that could give the known numbers along the
frontier.
> Typically this will be a whole frontier calculation although smarter,
> local calculations could probably be made in some cases.  The point is
> that the whole frontier calculation can be done, is not that expensive
> and yields the exact probabilities so you can pick the lowest
probability
> cell along the frontier or in the non-frontier region every time.
> 
> > So there is a bit more to this than was expressed.
> 
> Granted but not enough to change the basic point that AI is
not required
> because the optimal strategy is known and easy to compute. 

Show me closed form algorithms that do solve the problem.
But even if they exist, an AI solution might be a very good solution
as the closed form may be more complex and very hard to
rigorously prove.  But so far the analysis seems leaky.

For instance, I fully expect that Backgammon can be solved
in closed form, but it is a very hard problem and an AI approach
has been best so far.

Good Day
Dick

