Go and Life game?
Charles Matthews
> It should be interesting to compare livng
> shapes in Go with living shapes in Life
I don't think so.
Charles
Xenafan
> "Jader" wrote
>
> > It should be interesting to compare livng
> > shapes in Go with living shapes in Life
>
> I don't think so.
>
> Charles
That's a bit pompous, Charles.
Denis Feldmann
Why? It *is* not very interesting. Any reasonably good (>10k) player of Go
having seriously studied Life (say, knowing some things about other ships
than gliders, or about Edens, or...) knows that the comparison makes no
sense.
-
>>> "Jader" wrote
>>>> It should be interesting to compare livng
>>>> shapes in Go with living shapes in Life
>> "Charles Matthews" <charles.r...@ntlworld.com> wrote:
>>> I don't think so.
> Xenafan wrote:
>> That's a bit pompous, Charles.
It's not pompous of Charles to express his personal opinion.
He qualified his statement by reference to the sentence subject.
"Denis Feldmann" <denis.f...@wanadoo.fr> wrote:
> Why? It *is* not very interesting. Any reasonably good (>10k) player
> of Go having seriously studied Life (say, knowing some things about
> other ships than gliders, or about Edens, or...) knows that the
> comparison makes no sense.
This, however, was pompous. There was no sentence subject
qualifying Mr.Feldmann's remarks as limited to a personal opinion.
Instead Denis Feldmann's comments were stated as generic truth.
Now the difficulty with "Jader's" question concerns a factor of many
variations to "life", concerning birth and death rules, each of which
may produce viable patterns. More generally is Stephen Wolfram's
opus, albeit overblown, portending to recast "A New Kind of Science."
The "life" game was intending to be -illustrative- of its principle, i.e.
"cellular automata." Since the game of "life" can emulate a Turing
Machine, given certain constraints, then it has potential for design
architecture on such things as field-programmable gate-arrays, and
quite possibly quantum computers built from Josephson Junctions,
and the like. Yet we also have statements from credited inventor
John H. Conway, to the effect that while he can devise powerful
theorems about abstruse results, they are generally impracticable
for mainstream mathematics. Whether the game of Go falls within
or without a domain of practicability seems also to be out to lunch.
- regards
- jb
--------------------------------------------------------------------------------
http://pi.lacim.uqam.ca/cgi-bin/lookup.pl?Submit=GO+&number=3.2437209&lookup_type=simple
Plouffe's Inverter
--------------------------------------------------------------------------------
Results of the search:
Your input of 3.2437209 was probably generated by one
the following functions or found in one of the given tables.
Answers are given from shortest to longest description
Polylogarithms type of series with the floor funtion [ ] and irrational
values.
3243720951104674 = sum([sqrt(2)*n]*(1)^n/(a*n+b)/k^n,n=0..inf),a=74,b=5,k=38
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Xenafan
That is surely true, but it was the way in which it was stated that
got to me. It seems that "jader" is a newbie in go, and so he ought to
be handled gently instead of being curtly squashed for making what he
probably thought was quite a valid and insightful observation.
Secondly, there is no current thread on Go and Life specifically, so
it seems a bit hard of Charles to start one for the purpose of
correcting jader. I have learned much from Charles's postings on go
and from his books, but, sorry, I did not appreciate this one at all.
Ralph
> "Jader" wrote
>
> > It should be interesting to compare livng
> > shapes in Go with living shapes in Life
>
> I don't think so.
>
> Charles
Where is Jadar's original post?
Charles Matthews
"Xenafan" wrote
> That's a bit pompous, Charles.
Guilty, I suppose. But it's the kind of language used from mathematician to
mathematician, if not go player to go player - signifies 'really nothing
there, don't waste your time, it's idle dreaming', that sort of thing.
Charles
Jader
news:3ebc0532...@news.cis.dfn.de...
> Now the difficulty with "Jader's" question concerns a factor of many
> variations to "life", concerning birth and death rules, each of which
Thank You, that was at last an answer. Of course I didn't mean basic
life, but I've seen an processor emulation made on cellular automaton, and
it works... I'm of course newbie in Go, but I do not think there is a chance
of made working simulation (or sth) of Go game with cellular automaton, but
I think, that sameone could (may be) made any study in any common area. A
lot of different ideas was treated by people as a waste of time, but it
didn't mean they aren't considered (or really was waste of time).
Jader
JVT
game of Go" on the CA mailing list: http://cafaq.com/threads/index.shtml
Maybe a CA could be helpful for a hardware implementation, or a mathematical
study?
--
JVT
-
Thank you for the reminder. We have previously been here,
on this newsgroup --
yet it is helpful to conduct a revisit. One essential question concerns
whether a "cellular automata" form (which re-enters the form) can be
anything other than convergent into stability, stable patterns, or death.
Yet real "life" (as with "innumerably many" games of Go), appears not
to be necessarily convergent: indeed, "life" may instead be divergent.
I've been speculating on the nature of evolutionary hypothesis: could
it not be said that, without a clearing of the deadwood, evolution might
not occur? Though it is natural for the "deadwood" to put up resistance.
And what is the rush for evolution, anyway? Where was it all going?
- regards
- jb
Ilan Vardi
> "Jader" wrote
>
> > It should be interesting to compare livng
> > shapes in Go with living shapes in Life
>
> I don't think so.
>
> Charles
It's been done already, as I recall hearing about it a long time ago,
though I forgot everything I heard.
It seems to me that arguing about the point of doing this is quite
ridiculous, when you can just think about it for a second.
First of all, there is obviously little relation between the two as
is made clear by the fact that the simplest invariant shape in Life
is the 2x2 square, which is not a living shape in Go.
This example isn't that good though, and a further thought reveals
the real problem: In Go, a living shape is determned
by only 2 factors: Connectedness on the ZxZ nearest neighbour graph,
and two eyes. In Life life, there are much more subtle factors involved,
as everyone knows the life question is turing complete. More concretely,
take a living shape in Go can consist of a completely filled NxN square
with 2 empty spaces, and this obviously leads to very strange Life
behaviour, since the interior is "overpopulated" as intended by Life rules.
There is a yet more compelling difference between the two. Go connectedness
is on the nearest neighbour graph of ZxZ, while Life is concerned with
nearest and second nearest neighbours (king moves) graph of ZxZ. This
last is the dual of the nearest neighbour graph, and it is true that
capturing pieces in Go must be connected on the second nearest neighbour
graph.
Now a word from our sponsor. It seems to me that the closest mathematical
field to Go is actually percolation, which asks what the probability is
of having an infinite group if you set pieces down randomly on an
infinitely large Go board. It has been proved that there is a certain
critical probability above which this happens. It is believed to be
about 59% (that is, if Black occupies more than 59% of the board,
then he will have an infinitely large connected group with probability one).
I refer to the above duality result and Go in a paper I wrote in Experimental
mathematics 5 years ago. It is called Prime Percolation, and is on my
website at http://www.lix.polytechnique.fr/~ilan/publications.html
And now, back to our regularly scheduled posting.
-ilan
Chris Lawrence
> I refer to the above duality result and Go in a paper I wrote in Experimental
> mathematics 5 years ago. It is called Prime Percolation, and is on my
> website at http://www.lix.polytechnique.fr/~ilan/publications.html
Where, precisely, is the paper? Last time I tried to find it I went
round the houses in a twisty little maze of redirects, all alike.
--
Chris
Ilan Vardi
> "Jader" wrote
>
> > It should be interesting to compare livng
> > shapes in Go with living shapes in Life
>
> I don't think so.
Some further thoughts:
1. There is a strong tie between Go and Life, namely John H. Conway.
He invented Life and set up many of the results which got people to
work on it by computer. Conway also invented "surreal numbers" in
an attempt to understand Go from the mathematical point of view of
games like Nim, which had a number theoretical strategy.
2. I should explain more why percolation is the natural mathematical
setting for Go. Leaving captures and eyes aside for the moment, the
basic point of Go is to encircle your opponent. The basic question of
percolation is the following: If you put down pieces randomly with
probability P (and opponent with probability 1 - P), what is the
probability that you have a connected group which is never surrounded
by your opponent? The basic result says that there is a Q such that
if P > Q then with probability one, you have such a group and if P < Q,
then with probability one, you do not have such a group. For the
standard Go case, Q is conjecture to be about .59. Inverting black and
white, this shows that if P < 1 - Q, then white will have an infinite
(not surrounded) group with probability one, and if 1 - Q < P < Q,
then both sides will have all their groups surrounded by the opponent.
In particular, if P = 1/2. Then if you set the pieces by flipping a fair
coin at each intersection of an infinite Go board, and independently
at each point, then, with probability one, you will see an infinite set
of concentric
surrounded groups, that is, an infinite sequence of sekis. That doesn't
follow immediately, but I think it has been proved, see the book
Percolation, by Grimmett.
There are other interesting results. For example, the shape of the infinite
group, when it exists, is roughly known. As P approaches Q from above,
the infnite group is a thin connected net, containing big holes (i.e.,
groups of the opponent) in a
roughly even distributed pattern all over the board. The size of the
holes as P approaches Q is roughly understood, at least in principle.
In conclusion, percolation can simply be considered as the theory
of "Random Go".
-ilan
Charles Matthews
> "Charles Matthews" wrote
> > "Jader" wrote
> >
> > > It should be interesting to compare livng
> > > shapes in Go with living shapes in Life
> >
> > I don't think so.
>
> Some further thoughts:
>
> 1. There is a strong tie between Go and Life, namely John H. Conway.
> He invented Life and set up many of the results which got people to
> work on it by computer. Conway also invented "surreal numbers" in
> an attempt to understand Go from the mathematical point of view of
> games like Nim, which had a number theoretical strategy.
Well, I know that - I believe I have the go board in question. ''Surreal
numbers'' is the Don Knuth name for numbers in what we now call
Combinatorial Game Theory. It is said (and I believe it) that Conway was
inspired to the disjunctive games part of the theory by watching go endgames
being played. But the only link here between go and cellular automata is
via Conway, AFAIK, who has never been very directly interested in go itself.
<snip>
> In conclusion, percolation can simply be considered as the theory
> of "Random Go".
Of course we go players try not to play "random go" - though no doubt we
fail from time to time.
Charles
Ilan Vardi
Sorry, I guess my site is constructed like a shopping mall. The exact address is
http://www.lix.polytechnique.fr/~ilan/prime_percolation.ps
It should also be available at the Experimental math website
http://www.expmath.com
I should also mention that there is a further Go question which has been
studied fairly extensively. This is, if you put the pieces down randomly,
what is the probability that a living shape will exist, or, more precisely,
what is the percentage of the time you will see any given shape. The
technical name in this field for "given shape" is "animal", so do a
search on the web for "percolation" and "animal". Unfortunately, I don't
think much is known, since almost all results must assume animals are convex,
and have no holes, the last part eliminating live Go shapes.
Some more about percolation and Go. One can invent a new game which I could
call Percolation Go, in which the object is to have a connected group which
connects two non-adjacent edges of a square board. Percolation is therefore
the study of this game, but in a probabilistic setting. I haven't given much
though about the deterministic (actual game) version, that is, whether
there game is trivial or not, since I just thought it up.
This is very slightly related to the "Angel Problem", in which an angel
can move anywhere within a large disk or radius X, and the Devil tries
to trap the Angel by placing Go stones on which the Angel cannot land
(angel is restricted to landing on intersections on a large Go board and
players alternate moves).
This problem and almost all results are also due to Conway. Berlekamp showed
that the devil beats an angel that can move like a chess king (X = Sqrt(2)),
and they call this game ChessGo, which may upset some people here. A survey
of results exists in the book "Games of no Chance" by Nowakowski.
-ilan
Douglas Ridgway
> "Charles Matthews" <charles.r...@ntlworld.com> wrote in message news:<C6Nua.365$hE4...@newsfep1-gui.server.ntli.net>...
> > "Jader" wrote
> >
> > > It should be interesting to compare livng
> > > shapes in Go with living shapes in Life
> >
> > I don't think so.
The real problem is that CA are deterministic finite[1] state machines,
whereas the game of go has two players who select moves. So either you need
to talk about nondeterministic finite state machines, which are rather
different beasts, or you need to include a complete description of the
players' move selection into the definition of the model.
[Ilan:]
> 2. I should explain more why percolation is the natural mathematical
> setting for Go. Leaving captures and eyes aside for the moment,
Your model has no capturing rule and no empty locations. This does allow
classical percolation results to apply, but to me seems to have little to
do with go. (For example, perfect play is rather simple without
the capturing rule -- it is the capturing rule which gives go complexity.)
More interesting would be to keep the capturing rule, define some random move
algorithm, and look for scaling results. Again, this would have little to do
with playing the game, but it would at least be inspired by it. Whether any
results would be interesting to CA theorists is a separate question.
doug.
rid...@dridgway.com
[1] Finite if considered on a finite lattice, otherwise only locally finite.
Ilan Vardi
>
> Your model has no capturing rule and no empty locations. This does allow
> classical percolation results to apply, but to me seems to have little to
> do with go. (For example, perfect play is rather simple without
> the capturing rule -- it is the capturing rule which gives go complexity.)
Such is mathematics... You gotta do what you can do, like searching
for your keys under the streetlight. For example, take billiards, which
has inspired a whole field of mathematics, and thousands of papers.
But how many consider the actual game, that is, with friction. I did
a complete search of articles from 1945 to the present, and I think
I found one, which didn't have anything original from the review.
The only known results are due to Euler (his son, actually) and
Coriolis, who wrote a book about it. How many mathematicians studying
billiards even know that their model has no bearing on the real game?
Not too many, I suspect.
-ilan
-
il...@tonyaharding.org (Ilan Vardi) wrote:
> [ ... ]
> Go connectedness is on the nearest neighbour graph of ZxZ, while
> Life is concerned with nearest and second nearest neighbours (king
> moves) graph of ZxZ. This last is the dual of the nearest neighbour
> graph, and it is true that capturing pieces in Go must be connected
> on the second nearest neighbour graph.
Well, a -very- thick "dumpling shape" could be captured if it
has only one eye, irrespective of the dual nearest neighbor graph.
il...@tonyaharding.org (Ilan Vardi) wrote:
> ... It seems to me that the closest mathematical field to Go is
> actually percolation, which asks what the probability is of having
> an infinite group if you set pieces down randomly on an infinitely
> large Go board. It has been proved that there is a certain critical
> probability above which this happens. It is believed to be about 59%
> (that is, if Black occupies more than 59% of the board, then he will
> have an infinitely large connected group with probability one). I refer
> to the above duality result and Go in a paper I wrote in Experimental
> mathematics 5 years ago. It is called Prime Percolation, and is on my
> website at http://www.lix.polytechnique.fr/~ilan/publications.html
Your result stems from speculations dealing with "Gaussian Integers
relatively prime to a given N" as an approximation for "the Poisson blob
model." Proposition 6.1 seems reasonable, however Proposition 6.3
appears to indicate departure from Go Shapes. Can you identify how
"Gaussian Integers relatively prime to a given N" are to be associated
with generalizable Go Shapes? Then your result Pc(1) = ~ 0.59 seems
to presuppose that a capturing contour would not contain one or more
"diagonal connectors" (as piecewise disconnected portions)? I am just
checking, here. Perhaps a ratio for Go is closer to 7/12 than to 0.59...
- regards
- jb