Presume that a program memory only includes the current state for instance of a chess board. Does it need the variable which player, black or white, has the next turn to move or is it redundant recognizing the state?
Clearly its redundant during the first moves and then chess has so many combinations that there are combinations that can't guarantee that its deducible from just the state whether white or black has the next move, it needs that variable that wasn't needed during the first moves. How can I know the existence or non-existence of an algorithm in this case to know how the state went from deterministically knowable whether white or black has the next move to a state later in the game where the only possibility appears to keep a side variable to know whether white or black has the next move?
Is the logical consequence that I can't find a method to predict something like "making this move must activate a new boolean variable."
I'm trying to answer whether there is a method to know or just practically using the boolean variable blackHasNextMove to also include the information who is next to move for a chess game and there might be no method but in that case how do I prove that there is no method?.
I'm asking how to know if changing a state from computable to unknown has an algorithm and I'm very thankful you even brought up the idea of time bounds for a solution using ordo notation.