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Refuting approaches to the log-rank conjecture for XOR functions

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Hamed Hatami, Kaave Hosseini, Shachar Lovett, Anthony Ostuni

Computational Complexity · Submitted 14 Dec 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this note, we refute two conjectures.

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arXiv ID2312.09400 (version 2)
Pith IDWD2PKD5Y, Pith's permanent record of this paper and its review
CitationAnthony Ostuni, Hamed Hatami, Kaave Hosseini, Shachar Lovett. "Refuting approaches to the log-rank conjecture for XOR functions." arXiv:2312.09400 (2023). https://pith.science/paper/WD2PKD5Y
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arXiv listing abstract v2. Results and the main claim are underlined; .

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. background 0.98 0.03 0.02 asserted 1.00
s1 Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. background 1.00 0.04 0.02 asserted 0.92
s2 In this note, we refute two conjectures. contribution 0.71 0.95 0.80 asserted 1.00
s3 The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). background 0.86 0.11 0.08 asserted 0.99
s4 These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. background 0.98 0.05 0.04 asserted 0.99
s5 Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. background 0.98 0.06 0.06 speculative 0.71
s6 We refute these conjectures by constructing two specific boolean functions tailored to each. method 0.54 0.96 0.73 asserted 1.00

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Original abstract

The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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  • In hopes of gaining further insight, many researchers have considered the special case of XOR functions, where $f_⊕(x,y) f(x + y)$ for a boolean function $f : 𝔽_2^n →{-1,1}$ [ref] .

    On the communication complexity of XOR functions · Introduction · 0.96

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