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I would guess a rough estimation of a positions "complexity" might be related to the size of the decision tree for all possible moves forward - with the trees ending only in mate or stalemate. Blueemu's post of the starting position as "complex" is a good example.
For humans, this might also invove the number of "positional changes": exchanges or "significant" pawn or piece moves.
Not exactly true. For example, an end-game position with a lot of alternate forward moves would not be "complex" if humans looking at it could tell didn't matter much what alternate paths were taken as only one or two leads to a win.
the tree can end if there is an enduring material gain I think
Not exactly. Your adjective "enduring" implies look-ahead to some outcome. If you meant "significant material gain" - then yes unless the other player can mate quickly - which also implies look-ahead.
Just the idea of putting a "complexity" rating on a position sounds too complex for me.
I think in these kind of calculations you always define the max number of moves you're going to look ahead otherwise the computation will take too much time
I don't know : for example for Deep fritz this position (white moves) is defined as having a complexity of 9.8/10 ... What do you think?
How about just adding more squares, more pieces, pieces with variable move rules, and watch chess solutions run out to a near infinity that even supercomputers won't be able to calculate.
In mathematics a solution to a problem has to be less complex than the problem itself, otherwise the problem has no solution. This is not pie-in-sky theory there are mathematical problems where the solution is more complex than the problem therefore they are theoretically unsolvable.
Fermat's last theorem?
In mathematics a solution to a problem has to be less complex than the problem itself, otherwise the problem has no solution.
For example : http://en.wikipedia.org/wiki/Fermat's_Last_Theorem
The current version of the proof is a couple of hundreds of pages, densely written, which only specialists of the domain can understand.
Does it mean the problem has no solution ?
Fermat's theorem was not a mathematical problem, it was a mathematical conjecture (a unproven statement).
example: "there are no blond girls with brown eyes." Well to prove that to be true I may have to travel around the world and look at sample all blond the girls. In math of course they had to use complex branches of geometry, ellptical functions etc.etc.
Example: Instead of looking for blond girls with blue eyes, let as look for the genes that make brown eyes and see if that gene can coexist with the gene for blond hair. If they can, then there must be a blond girl with bbrown eyes somewhere in the world.
So it was a proof of a statement not a solution to a specific problem. Capisce?
1- How to turn a conjecture into a problem : where the conjecture states "A", the problem is "is A true ?".
2- Logical error in your reasoning. The fact that there exists a gene that allows something does not mean that this something exists (in this case, some genetic characters). Other example of the same flaw : there are enough fish in the Scottish Loch to feed a dinosaur, thus the existence of the Loch Ness monster is possible, thus it exists -> hmm, no.
3- Please define a mathematical "problem" then. I don't get your subtle distinction. I think the issue arises from the word "mathematical", but not sure.
Yeah, this position is pretty complex. I would guess that the reason fritz still assigns it a very high hotness rating is because the hotness meter pays no attention to wether the game is nearly over or not. Perhaps that is what the Mate-O-Meter is for.
As a puzzle this position would have a score of less than 1000 I think
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