Author: Michael Yee
Date: 18:28:02 01/12/05
Go up one level in this thread
On January 12, 2005 at 21:07:42, chandler yergin wrote: >On January 12, 2005 at 21:03:54, Michael Yee wrote: > >>On January 12, 2005 at 20:57:40, chandler yergin wrote: >> >>>On January 12, 2005 at 20:33:25, Dann Corbit wrote: >>> >>>>On January 12, 2005 at 20:25:24, Uri Blass wrote: >>>> >>>>>On January 12, 2005 at 19:56:25, Dann Corbit wrote: >>>>> >>>>>>On January 12, 2005 at 19:37:29, Steve Maughan wrote: >>>>>> >>>>>>>Dann, >>>>>>> >>>>>>>>Things that seem impossible quickly become possible. >>>>>>> >>>>>>>I recon about 300 years before a computer will solve chess. This assumes >>>>>>> >>>>>>>1) 10^120 possible positions >>>>>> >>>>>>This is far, far too large. Chess positions have been encoded in 162 bits, >>>>>>which puts an absolute upper limit at 10^58 (and it is probably much less than >>>>>>that). >>>>>> >>>>>>>2) Alpha-beta cutting this down to 10^60 sensible positions >>>>>> >>>>>>The incorrect first assumption renders this and all following assumtions as >>>>>>moot. >>>>> >>>>>The second assumption is also not correct. >>>>> >>>>>By the same logic alphabeta can cut less than 2^30 positions in KRB vs KR to >>>>>2^15 positions but it does not happen and solving some KRB vs KR position with >>>>>no KRB vs KR tablebases is not something that you need 2^15 nodes for it. >>>> >>>>No. The second assumption would be true if the first was true. This was >>>>formally PROVEN by Donald Knuth. In a perfectly ordered alpha-beta solution >>>>tree, the number of nodes is proportional to the square root of the nodes in the >>>>full tree. >>>> >>>>If there were 10^120 in the full tree, then about 10^60 would be in the solution >>>>tree. >>>> >>>>It can be less than that. >>> >>>It "Can't be LESS than that! >>> >>> But it cannot be more. >>> >>> >>>It Certainly CAN! >>> >>>In any TREE.. the TREE ONLY represents "What HAS Been PLayed." >>>REFUTE THAT! >>>Can't HUH? >>> >>>Give it up! >> >>What you just said is correct since you're talking about the *tree* of moves. >>But Uri and Dann are talking about the *set* of unique positions (many of which >>can arise through different move orders). So you and they are talking about >>different (mathematical) objects--trees (or paths in a tree) and graphs (or >>nodes in a graph). >> >>By the way, just because some quantity is large (or infinite) doesn't mean you >>can't prove something about it mathematically. For instance, you can prove that >>a geometric series (e.g., 1/2 + 1/4 + 1/8 + ...) convergences to a number even >>though their are an infinite number of terms. >> >>Michael > > >Yeah.. ya can compute Pi to a Billion or so digits... >I round off at 3.1416... >Close enough for me.. >So What? > >Ur missing the point. Actually, I don't think I'm missing your point. What you seem to be saying is this: (1) There are approx 10^120 chess positions in the *tree* of moves (2) There aren't even that many atoms in the universe (3) Therefore, it's impossible to "mathematically prove" anything about chess (i.e., solve it) And these are my points: (1) For solving chess, you only need to consider unique positions (2) You can prove things about infinite sets of things without having to "touch" each item. For example, we can even stay with your move tree and consider a K and Q versus K ending. Ignoring the 50-move rule, there are infinitely many move-paths (in your model) starting from some root position. By your thinking (I think), it would be impossible to prove that K+Q is a win because you couldn't possibly deal with an infinite number of move paths. But I think you would agree that it's easily shown to be a win. Michael
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