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This looks like a direct application of https://en.wikipedia.org/wiki/Game_semantics …
What objective function and strategy is being employed in the time-reversed game? (That page looks irrelevant, but maybe I'm not seeing the discussion of time reversal.)
I feel time reversal is immaterial here. If I understand your statement correct, state is being reversed to the original board position and that's not time dependent but turn dependent. Would that be a correct rephrasing?
May be I'm wrong here but I sense parallels between falsifier/verifier modelling propositional calculus in that page to minimax/game theory: https://en.wikipedia.org/wiki/Game_semantics#Classical_logic …
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