We generalize the deterministic simulation theorem of Raz & McKenzie (Combinatorica 19(3):403–435, 1999), to any gadget which satisfies a certain hitting property. We prove that inner product and gap-Hamming satisfy this property, and as a corollary, we obtain a deterministic simulation theorem for these gadgets, where the gadget’s input size is logarithmic in the input size of the outer function. This yields the first deterministic simulation theorem with a logarithmic gadget size, answering an open question posed by Göös, Pitassi & Watson (in: Proceedings of the 56th FOCS, 2015). Our result also implies the previous results for the indexing gadget, with better parameters than was previously known. Moreover, a simulation theorem with logarithmic-sized gadget implies a quadratic separation in the deterministic communication complexity and the logarithm of the 1-partition number, no matter how high the 1-partition number is with respect to the input size—something which is not achievable by previous results of Göös, Pitassi & Watson (2015).

中文翻译：

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通过伪随机特性的模拟定理

我们将 Raz & McKenzie (Combinatorica 19(3):403–435, 1999) 的确定性模拟定理推广到满足特定命中属性的任何小工具。我们证明内积和 gap-Hamming 满足这个属性，作为推论，我们获得了这些小工具的确定性模拟定理，其中小工具的输入大小是外函数输入大小的对数。这产生了第一个具有对数小工具大小的确定性模拟定理，回答了 Göös、Pitassi 和 Watson 提出的一个悬而未决的问题（在：第 56 届 FOCS 会议录，2015 年）。我们的结果还暗示了索引小工具的先前结果，其参数比以前已知的要好。而且，