H. G. Muller wrote on Thu, Apr 21, 2022 07:54 AM UTC:
Just to make sure I fully understand this:
A Bishop on a4 could move along c5-d6-e7-f8 but not b5-c6-d7-e8. For a Bishop on 4 the reverse would be true. But that also means a Bishop on a4 can only be captured from c5-d6-e7-f8 and not from b5-c6-d7-e8.
If this is true the only thing that 'switches' are the connections of 4/a4 to the upper board half. These are either to a5 (f) and b5 (fr), or to a5 (fl), b5 (f) and c5 (fr). And the switch is only in a determined state when 4/a4 is occupied. (And that state has to be chosen as part of the move that occupies it.) When the square is empty you can use all the connections to enter it from above, or to leave it in the same move when you entered it from below.
The intepretation of this that I consider conceptualy the simplest is that 4/a4 is indeed a single square cell, but that the setting of the switch is extra game state. Which can conveniently be indicated by placing the piece that occupies it in the right or left half of the cell, as the distinction only has to be made when the square is occupied. In a computer implementation one could implement the state of the switch through making separate cells of 4 and a4, with only single exits to 5th rank, and duplicate the exits to it from a3 (f), b3 (fl) and b4 (l) so that one connects to 4, the other to a4, both available at any time. But that still would require some 'unnatural' treatment of the spaces 4/a4, to consider one blocked when the other is occupied.
The issue of teh Knight is really an independent one. If the Knight's move is defined 'subtractively', by excluding moves that a Queen can do, its mobility is reduced in the vicinity of the switch, as Q gets extra moves there. In a 'constructive' definition of the Knight move, it would benefit from the switch topology to get extra moves as well.
Just to make sure I fully understand this:
A Bishop on a4 could move along c5-d6-e7-f8 but not b5-c6-d7-e8. For a Bishop on 4 the reverse would be true. But that also means a Bishop on a4 can only be captured from c5-d6-e7-f8 and not from b5-c6-d7-e8.
If this is true the only thing that 'switches' are the connections of 4/a4 to the upper board half. These are either to a5 (f) and b5 (fr), or to a5 (fl), b5 (f) and c5 (fr). And the switch is only in a determined state when 4/a4 is occupied. (And that state has to be chosen as part of the move that occupies it.) When the square is empty you can use all the connections to enter it from above, or to leave it in the same move when you entered it from below.
The intepretation of this that I consider conceptualy the simplest is that 4/a4 is indeed a single square cell, but that the setting of the switch is extra game state. Which can conveniently be indicated by placing the piece that occupies it in the right or left half of the cell, as the distinction only has to be made when the square is occupied. In a computer implementation one could implement the state of the switch through making separate cells of 4 and a4, with only single exits to 5th rank, and duplicate the exits to it from a3 (f), b3 (fl) and b4 (l) so that one connects to 4, the other to a4, both available at any time. But that still would require some 'unnatural' treatment of the spaces 4/a4, to consider one blocked when the other is occupied.
The issue of teh Knight is really an independent one. If the Knight's move is defined 'subtractively', by excluding moves that a Queen can do, its mobility is reduced in the vicinity of the switch, as Q gets extra moves there. In a 'constructive' definition of the Knight move, it would benefit from the switch topology to get extra moves as well.