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For infinite Karchmer-Wigderson games, does communication complexity in the infinite setting characterize the proof complexity of the corresponding separation principle in intuitionistic logic?

Related: Karchmer-Wigderson theorem, Borel determinacy theorem, reverse mathematics of WKL0 and ATR0

The classical Karchmer-Wigderson theorem establishes a tight correspondence between the circuit depth needed to compute a Boolean function and the communication complexity of an associated two-player game. When this framework is extended to infinite combinatorics, players may exchange infinite sequences of bits and the games can run for infinite rounds, raising the fundamental question of whether the resulting infinite communication complexity still faithfully encodes logical proof complexity. Specifically, the open problem is whether there exists a precise calibration between the descriptive set-theoretic complexity of infinite KW game strategies and the strength of the intuitionistic or constructive separation principles those games represent, analogous to the classical finite case. This connection is not yet established in the infinite regime and requires reconciling tools from descriptive set theory, infinite combinatorics, and proof theory that do not align cleanly.

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