> "If there is a mate in 1, but lots of possible moves, then complexity of the position is also very low for a player with any familiarity with the game"

This is one reason I think the concept of complexity will probably never be formalized. Because even when a position is mate in 1, the complexity varies wildly. For instance here [1] is a game between Carlsen and Nakamura where Carlsen, perhaps the strongest player of all time, failed to see a mate in 1, and for that matter Nakamura, not exactly a weaker player himself, help-mated himself by playing one of the only moves that allowed Carlsen a mate in 1! And both players still had tons of time on their clock.

But to understand how they both missed it you need to go back in the position and see that both saw black's queen was defending against the mate, and Nakamura's blunder interfered with that protection in a rather unusual way. So you can't even assess it on a positional basis, since it's dependent upon what was played before, let alone all of the poorly defined terms I'm using like 'rather unusual.'

[1] - https://www.chess.com/events/2024-titled-tuesday-blitz-may-7...

I agree that complexity can vary widely even with mate in 1. Lots of mate in 1 chess puzzles to prove that.

But in the case of this game, I wouldn’t say that they missed the mate in 1 due to the complexity of the position or the previous moves. They both just had the same blind spot, happens every so often.