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on board positions
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Tom Ritchford
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May 30, 2001 14:33 PDT
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Pursuing a suggestion of Cliff's, I'm entirely liberating a board
position from having to contain a sequence of moves leading
up to it.
This next part is a mathematical definition of the problem
space so it might be a little technical for some of you.
1. a set of ALL board positions.
this includes ANY number and kind of pieces on a chess board;
we don't care if it could ever have come from any possible
game or not, 32 queens, whatever.
(and this includes the castling and ep flags, everything you
need to identify the legal next moves).
we call this "all positions".
2. we define the "legal moves" or just the moves
as a relation on this set of all board positions.
if a and b are board positions, let
a->b
mean
"there's a legal move from position a to position b".
we name this move a->b as well (which uniquely determines it!
3. this induces a partial ordering(*) on these positions
so
a=>b
is read as
game b is a continuation of game a.
formally:
there's a series of zero or more legal chess moves
that will take you from position a to position b.
so if a=>b
then either a=b
or a->b
or a->a1->b for some position a1
or there are some intermediate positions
a->a1->a2...->b
4. a root position -- the starting position in chess.
The root position is named /
(you might know it as "rnbq...")
5. All legal games of chess!
This is just the set of all positions p where
/=>p
"Any continuation of the opening."
Simple, eh?
---
(*) wait a second, is this really a partial ordering?
can't you have a < b < a because of repeated positions?
no, because of the 3 repeat rule, consider the first repeat a, a', a'' and then
there can be no more.
.....all legal games of chess <http://solveChess.com/chess?refresh=0>...
........formal model of chess <http://solveChess.com/chess>.............
...programmer's documentation <http://solveChess.com/doc>...............
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