Author: Don Dailey
Date: 03:38:00 07/29/98
Go up one level in this thread
On July 29, 1998 at 01:58:31, Steffen Jakob wrote: >On July 29, 1998 at 01:44:17, blass uri wrote: > >> >>On July 29, 1998 at 00:33:50, Steffen Jakob wrote: >> >>>On July 29, 1998 at 00:28:25, blass uri wrote: >>> >>>> >>>>On July 29, 1998 at 00:08:52, Steffen Jakob wrote: >>>> >>>>>On July 28, 1998 at 08:46:21, Komputer Korner wrote: >>>>> >>>>>[...] >>>>> >>>>>>The number of legal positions is the most important stat as far as computers are >>>>>>concerned. 10^42 has been put as a lower bound with 10^60 as an upper bound. I >>>>>>am not sure as to the mathematical accuracy of these though. >>>>> >>>>>It would be very interesting to estimate the number of positions where one side >>>>>doesn't have a "decisive advantage" which is of course not easy to define. A way >>>>>to estimate this number could be to set up positions randomly >>>> >>>>I do not understand how to set up positions randomly. >>>>If you set up a general random position then practically all the positions you >>>>set up will be illegal. >>>>For example if one side has 6 queens and 6 rooks it is illegal because at least >>>>5 queens and 4 rooks were pawns in the beginining of the game. >>>>maybe 1 out of 100000000000000000000 will be legal but you have not infinite >>>>time. >>> >>>Of course I meant to set up a random legal positions. Could be done by >making random legal moves. >> >>In this case not all the legal positions will have the same probability. > >Yes, distribution is a problem, which leads to other complex questions. E.g. >what is the probability that we have a certain amount of material in a random >position? Even if we could generate random positions easily without making >random moves we had to know this. > >>>>> and evaluate them >>>>>with a computer. Then you get the relation between balanced and >>>>>unbalanced positions which has to be multiplied with the number of legal >positions. >>>> >>>>>Greetings, >>>>>Steffen. The distribution problem is a tough one. You would like to take every possible legal position and choose one of them with equal probability. This is probably NP hard, but there may be ways that give good results. There is another factor to consider too. Are we interested in ANY legal position or just ones that can actually occur in a game? For instance a position with white pawns on b2 and d2 with a dark squared bishop not on c1 but somewhere else is illegal. (Actually this is still possible but I think you get my point.) So unless we generate random moves from the starting position we have to deal with this too, unless we don't care. - Don
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