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4 Karp's version makes this step only if L sa = ∅ for both a = 0 and a = 1; this note is related to footnote 1
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Next, we will show the following
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For us, it is convenient to not require it, and Karp's analysis still goes through with minor modifications that we point out
Phokion Kolaitis, Swastik Kopparty (2009)
Random graphs and the parity quantifier
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This difference is not essential for Karp's analysis but is important for us
Degree Lower Bounds of Tower-Type for Approximating Formulas with Parity Quantifiers ALBERT ATSERIAS, Universitat Polit` cnica de Catalunya e ANUJ DAWAR, University of Cambridge Kolaitis and Kopparty have shown that for any first-order formula with parity quantifiers over the language of graphs, there is a family of multivariate polynomials of constant-degree that agree with the formula on all but a 2- (n) -fraction of the graphs with n vertices. The proof bounds the degree of the polynomials by a tower of exponentials whose height is the nesting depth of parity quantifiers in the formula. We show that this tower-type dependence is necessary. We build a family of formulas of depth q whose approximating polynomials must have degree bounded from below by a tower of exponentials of height proportional to q. Our proof has two main parts. First, we adapt and extend the results by Kolaitis and Kopparty that describe the joint distribution of the parities of the numbers of copies of small subgraphs in a random graph to the setting of graphs of growing size. Second, we analyze a variant of Karp's graph canonical labeling algorithm and exploit its massive parallelism to get a formula of low depth
ACM Transactions on Computational Logic (TOCL) – Association for Computing Machinery
Published: Feb 1, 2014
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