How does one know if a position is not reachable from a series of legal moves from the starting position? If one of the following is true, a position is not reachable from the starting position by a series of legal moves:


One side has two or more kings or one side has no kings;
One side has  a total of 10 or more queens and pawns, or a total of 11 or more rooks and pawns, or a total of 11 or more knights and pawns, or a total of 10 or more bishops of the same colour and pawns;
One side has a total number of more than 16 pieces;
At least one pawn is in the first rank or eighth rank;
Both kings are being checked;
The kings can capture each other in the next move;
At least one bishop is in an "impossible square" blocked by pawns in their starting positions, such as the following:





One king is being "impossibly" checked or checkmated such as the following (this should also include the case where a king is being checked by three or more pieces): (I am not sure how to define this more precisely);





One side is checking and it is his turn.


I am sure that this list is not exhaustive. Does one know a complete list? Is there an algorithm determing whether a position is reachable from the starting position by a series of legal moves?

Edit: I would also add the following:


One side has more than 8 pawns.
Here is a retro which needs deep analysis:

[Title "Thierry le Gleuher; Economy Records in Add Unit(s) Problems, 5A"]
[StartFlipped "0"]
[fen "B5bn/5PrR/5prN/6pk/7p/5P2/PPPPPP2/KR1Q2N1 w KQkq - 0 1"]


The stipulation is "Add the missing unit(s)". A solution may be found in entry P1227636 in PDB. In brief:


   To disentangle Black's kingside, White must retract most pieces back home so that White can retract g2xf3. Interleaved with these White moves are 19 Black moves. But, all this time, Black has no retrotempi with units in the diagram, so they must be provided by the added units. The only way is to add black pawns on a3, b3, c3, d3, e3. Each provides 4 retrotempi, except that the b and d pawns can't both retract home, because the bishop on g8 will need to retract to c8 eventually.


Thus the position would be illegal if


   5 black pawns were added but any of them were nearer rank 7, because Black would run out of retrotempi, but it would take deep analysis to show that this is the case.


This might be the sort of situation RemcoGerlich was thinking of, seeing that


   Some White pieces need to leave the back rank and move back in a different order.


In case you thought that there's a simpler way to add units:


   A Black piece captured White's c1 bishop on c1. White is not missing anything else, so no Black pawn ever left its file. Thus White's 3 captures g2xf3 and hxgxf were of Black's 3 missing pieces: the f8 bishop, the queen and a knight. So we may add Black pawns but not a piece.