We study the two-party communication complexity of functions with large outputs, and show that the communication complexity can greatly vary depending on what output model is considered. We study a variety of output models, ranging from the open model, in which an external observer can compute the outcome, to the XOR model, in which the outcome of the protocol should be the bitwise XOR of the players' local outputs. This model is inspired by XOR games, which are widely studied two-player quantum games. We focus on the question of error-reduction in these new output models. For functions of output size k, applying standard error reduction techniques in the XOR model would introduce an additional cost linear in k. We show that no dependency on k is necessary. Similarly, standard randomness removal techniques, incur a multiplicative cost of $2^k$ in the XOR model. We show how to reduce this factor to O(k). In addition, we prove analogous error reduction and randomness removal results in the other models, separate all models from each other, and show that some natural problems, including Set Intersection and Find the First Difference, separate the models when the Hamming weights of their inputs is bounded. Finally, we show how to use the rank lower bound technique for our weak output models.

翻译：我们研究具有大输出的双参与方通信复杂性，并表明根据所考虑的输出模型，通信复杂性可能存在显著差异。我们研究了多种输出模型，从开放模型（其中外部观察者可计算结果）到异或模型（其中协议输出应为参与方本地输出的按位异或）。该模型受广泛研究的双参与量子游戏——异或游戏——启发。我们聚焦于这些新型输出模型中的错误约简问题。对于输出规模为k的函数，在异或模型中应用标准错误约简技术会引入与k成线性的额外开销。我们证明无需依赖于k。类似地，标准随机性去除技术会在异或模型中引入$2^k$的乘法成本。我们展示了如何将该因子降至O(k)。此外，我们在其他模型中证明了类似的错误约简和随机性去除结果，将所有模型彼此区分开来，并表明包括集合交集和查找第一个差异在内的一些自然问题，在输入汉明权重有界时可将这些模型区分开。最后，我们展示了如何为弱输出模型应用秩下界技术。