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The theorem that Parity is not in $\mathsf{AC}^0$ is one of the gemstones of complexity theory. I wonder how many different proofs there are of this result? What constitutes "different" is also a part of this question. E.g., if you say there are 4 different proofs in your answer, please explain why these 4 are "different" in your opinion.

I am familiar with two proofs, one based on the switching lemma, and the other based on aprrximation by low-degree polynomials, due to Razborov-Smolensky. I think these proofs are different since they generalise in different ways. E.g., switching lemma proofs give better correlation bounds, while low-degree techniques give lower bounds for $\mathsf{AC}^0$ with mod $p$ gates.

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    $\begingroup$ Answers here give references to many proofs of this fact: cstheory.stackexchange.com/questions/3154/parity-and-ac0 $\endgroup$
    – Alex Golovnev
    Commented Oct 4, 2018 at 1:38
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    $\begingroup$ There are proofs by using Finite Model Theory due to Ajtai, and by bounding the coefficients of the Fourier transform, due to Linial et al. But both of these essentially use switching lemmas, so I don't know whether they really count as different. $\endgroup$
    – Jan Johannsen
    Commented Oct 5, 2018 at 9:33

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