Our main technical contribution is a polynomial-time determinisation procedure for history-deterministic B\"uchi automata, which settles an open question of Kuperberg and Skrzypczak, 2015. A key conceptual contribution is the lookahead game, which is a variant of Bagnol and Kuperberg's token game, in which Adam is given a fixed lookahead. We prove that the lookahead game is equivalent to the 1-token game. This allows us to show that the 1-token game characterises history-determinism for semantically-deterministic B\"uchi automata, which paves the way to our polynomial-time determinisation procedure.
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Acharya et al. (2024) studied this question.
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