Polynomial Calculus sizes over the Boolean and Fourier bases are incomparable

Sasank Mouli

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arXiv · 2403.03933

Polynomial Calculus sizes over the Boolean and Fourier bases are incomparable

Sasank Mouli

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Abstract

For every $n >0$, we show the existence of a CNF tautology over $O(n^2)$ variables of width $O(\log n)$ such that it has a Polynomial Calculus Resolution refutation over $\{0,1\}$ variables of size $O(n^3polylog(n))$ but any Polynomial Calculus refutation over $\{+1,-1\}$ variables requires size $2^{Ω(n)}$. This shows that Polynomial Calculus sizes over the $\{0,1\}$ and $\{+1,-1\}$ bases are incomparable (since Tseitin tautologies show a separation in the other direction) and answers an open problem posed by Sokolov [Sok20] and Razborov.

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Sasank Mouli. 2024-07-01. Polynomial Calculus sizes over the Boolean and Fourier bases are incomparable. https://arxiv.org/abs/2403.03933

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