Problem 76: External Information and Amortized Expected Communication

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Suggested by Mark Braverman
Source Banff 2017
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For a function $F:\{0,1\}^n\times\{0,1\}^n\rightarrow\{0,1\}$, distribution $\mu$ on inputs $\{0,1\}^n\times\{0,1\}^n$, where Alice's and Bob's inputs are random variables $X$ and $Y$, respectively, the external information complexity for two-player zero-error protocols is defined as $$ \textrm{IC}^\text{ext}(F,0,\mu) := \inf_{\Pi \text{ that solve $F$ correctly always}} I_\mu(\Pi;XY)\,. $$ We denote by $\overline{\textrm{CC}}(F^n,0,\mu^n)$ the expected communication complexity of $F^n$ with respect to the distribution $\mu^n$ for zero-error protocols.

Either prove or disprove that $$ \textrm{IC}^\text{ext}(F,0,\mu) = \lim_{n\rightarrow\infty} \frac{\overline{\textrm{CC}}(F^n,0,\mu^n)}{n}\,. $$