最佳短路恢复公式
Optimal Short-Circuit Resilient Formulas
摘要：
我们考虑容错布尔公式，其中一个错误的门的输出短路到门的输入之一。Kalai等人（FOCS 2012）最近的一个结果将任何布尔公式转换为多项式大小的弹性公式，如果（每个输入到输出路径上的）门的一部分$1/6$有故障，则该公式正确工作。我们改进了Kalai等人的结果，并展示了如何有效地针对每条路径的短路门的一部分$1/5$来强化任何布尔公式，而大小仅为多项式爆破。我们还表明，这是不可能获得更高的弹性和亚指数增长的大小的公式。对于我们的研究结果，我们认为互动编码方案时，无噪声反馈，这些产生弹性布尔公式通过Karchmer-Wigderson关系。我们开发了一种编码方案，抵抗高达一小部分$1/5$损坏的传输在每个方向的交互式通道。我们进一步表明，这样的噪声水平是最大的编码方案与次指数爆破通信。我们的编码方案从区块链技术中获得了令人惊讶的灵感。
英文摘要：
We consider fault-tolerant boolean formulas in which the output of a faulty gate is short-circuited to one of the gate's inputs. A recent result by Kalai et al. (FOCS 2012) converts any boolean formula into a resilient formula of polynomial size that works correctly if less than a fraction $1/6$ of the gates (on every input-to-output path) are faulty. We improve the result of Kalai et al., and show how to efficiently fortify any boolean formula against a fraction $1/5$ of short-circuit gates per path, with only a polynomial blowup in size. We additionally show that it is impossible to obtain formulas with higher resilience and sub-exponential growth in size. Towards our results, we consider interactive coding schemes when noiseless feedback is present; these produce resilient boolean formulas via a Karchmer-Wigderson relation. We develop a coding scheme that resists up to a fraction $1/5$ of corrupted transmissions in each direction of the interactive channel. We further show that such a level of noise is maximal for coding schemes with sub-exponential blowup in communication. Our coding scheme takes a surprising inspiration from Blockchain technology.
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