They mean that you only need board position, you don't need the previous moves that led to that board position.

There are at least a couple of exceptions to that as far as I know.

Yes, 4 exceptions: castling rights, legal en passant captures, threefold repetition, and the 50 move rule. You actually need quite a lot of state to track all of those.

It shouldn't be too much extra state. I assume that 2 bits should be enough to cover castling rights (one for each player), whatever is necessary to store the last 3 moves should cover legal en passant captures and threefold repetition, and 12 bits to store two non-overflowing 6 bit counters (time since last capture, and time since last pawn move) should cover the 50 move rule.

So... unless I'm understanding something incorrectly, something like "the three last moves plus 17 bits of state" (plus the current board state) should be enough to treat chess as a memoryless process. Doesn't seem like too much to track.

Threefold repetition does not require the three positions to occur consecutively. So you could conceivably have a position repeat itself for first on the 1st move, second time on the 25th move, and the third time on the 50th move of a sequence and then players could claim a draw by threefold repetition or 50 move rule at the same time!

This means you do need to store the last 50 board positions in the worst case. Normally you need to store less because many moves are irreversible (pawns cannot go backwards, pieces cannot be un-captured).

Ah... gotcha. Thanks for the clarification.

The correct phrasing would be is it a Markov process?