Author: Robert Hyatt
Date: 13:27:16 05/16/01
Go up one level in this thread
On May 16, 2001 at 16:07:11, Robert Hyatt wrote: >On May 16, 2001 at 15:10:33, J. Wesley Cleveland wrote: > >>On May 16, 2001 at 14:07:18, Robert Hyatt wrote: >> >>>On May 16, 2001 at 13:05:13, J. Wesley Cleveland wrote: >>> >>>>On May 15, 2001 at 22:11:15, Robert Hyatt wrote: >>>> >>>>>On May 15, 2001 at 12:18:43, J. Wesley Cleveland wrote: >>>>> >>>>[snip] >>>> >>>>>>>First, how do you conclude 10^25? assuming alpha/beta and sqrt(N)? >>>>>> >>>>>>It is a classic alpha-beta search with a transposition table large enough to >>>>>>hold *all* positions found in the search. I'm guessing at the number of >>>>>>positions, but I feel that the same logic should hold, as only positions with >>>>>>one side playing perfectly would be seen. >>>>> >>>>>I don't follow. We know that within the 50 move rule, the longest game that >>>>>can be played is something over 10,000 plies. IE 50 moves, then a pawn push >>>>>or capture, then 50 more, etc. Eventually you run out of pieces and it is a >>>>>draw. But 38^10000 and 10^25 seem to have little in common. The alpha/beta >>>>>algorithm is going to search about 38^50000 nodes to search that tree to max >>>>>depth of 10,000. >>>> >>>>Look at it another way. The only positions that are visited by an alpha/beta >>>>search (with perfect move ordering) are those where one side plays perfectly. >>>>The question is what fraction of the total number of positions that is. >>>> >>> >>>The precise formula is: >>> >>> N = W^floor(D/2) + W^ceil(D/2) for all D. floor means round down in integer >>>math, ceil means round up. For the cases where D is even: >>> >>> N = 2 * W^(D/2) which is 2 * sqrt(minimax). >>> >>>If you assume that the total number of positions is roughly 2^168, then you >>>get 2 * sqrt(2^168) or 2 * 2^84. Which is fairly close to the number of atoms >>>in the universe. >> >>2^84 = 1.9*10^25 If these were atoms, it would be 32 moles, or about 1kg of >>silicon. > > >I discovered that I was having a small overflow problem on my calculator. It >told me that 2^84 was roughly 10^81, which is wrong. I should add, however, that if you searched at DB2's peak speed of one billion nodes per second, and you could search non-stop for one trillion years (longer than the expected life of the universe) you would not quite reach the 1% done stage....
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