评列夫·戈尔杰耶夫的“论P与NP“
On Lev Gordeev's "On P Versus NP"
摘要：
在论文“On P versus NP”中，列夫·戈尔杰耶夫试图将成功证明单调电路指数下界的近似方法扩展到德摩根正规（DMN）电路的情况。正如拉兹博罗夫证明单调电路的指数下界一样，戈尔杰耶夫的工作集中在NP完全问题CLIQUE上。如果成功地证明了CLIQUE的指数DMN电路下界，Gordeev将证明P $\neq$ NP。然而，我们证明了戈尔杰耶夫在引理12中犯了一个关键性的错误。这个错误来自于仅对正电路输入进行近似运算。此外，我们认为，努力扩展DMN电路的近似方法将需要近似否定的输入。
英文摘要：
In the paper "On P versus NP," Lev Gordeev attempts to extend the method of approximation, which successfully proved exponential lower bounds for monotone circuits, to the case of De Morgan Normal (DMN) circuits. As in Razborov's proof of exponential lower bounds for monotone circuits, Gordeev's work is focused on the NP-complete problem CLIQUE. If successful in proving exponential DMN circuit lower bounds for CLIQUE, Gordeev would prove that P $\neq$ NP. However, we show that Gordeev makes a crucial mistake in Lemma 12. This mistake comes from only approximating operations over positive circuit inputs. Furthermore, we argue that efforts to extend the method of approximation to DMN circuits will need to approximate negated inputs as well.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
