The Approximate Degree of DNF and CNF Formulas

• URL: http://arxiv.org/abs/2209.01584v1
• Date: Sun, 4 Sep 2022 10:01:39 GMT
• Title: The Approximate Degree of DNF and CNF Formulas
• Authors: Alexander A. Sherstov
• Abstract summary: For every $delta>0,$ we construct CNF and formulas of size with approximate degree $Omega(n1-delta),$ essentially matching the trivial upper bound of $n. We show that for every $delta>0$, these models require $Omega(n1-delta)$, $Omega(n/4kk2)1-delta$, and $Omega(n/4kk2)1-delta$, respectively.
• Score: 95.94432031144716
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The approximate degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that approximates $f$ pointwise: $|f(x)-p(x)|\leq1/3$ for all $x\in\{0,1\}^n.$ For every $\delta>0,$ we construct CNF and DNF formulas of polynomial size with approximate degree $\Omega(n^{1-\delta}),$ essentially matching the trivial upper bound of $n.$ This improves polynomially on previous lower bounds and fully resolves the approximate degree of constant-depth circuits ($\text{AC}^0$), a question that has seen extensive research over the past 10 years. Previously, an $\Omega(n^{1-\delta})$ lower bound was known only for $\text{AC}^0$ circuits of depth that grows with $1/\delta$ (Bun and Thaler, FOCS 2017). Moreover, our CNF and DNF formulas are the simplest possible in that they have constant width. Our result holds even for one-sided approximation, and has the following further consequences. (i) We essentially settle the communication complexity of $\text{AC}^0$ circuits in the bounded-error quantum model, $k$-party number-on-the-forehead randomized model, and $k$-party number-on-the-forehead nondeterministic model: we prove that for every $\delta>0$, these models require $\Omega(n^{1-\delta})$, $\Omega(n/4^kk^2)^{1-\delta}$, and $\Omega(n/4^kk^2)^{1-\delta}$, respectively, bits of communication even for polynomial-size constant-width CNF formulas. (ii) In particular, we show that the multiparty communication class $\text{coNP}_k$ can be separated essentially optimally from $\text{NP}_k$ and $\text{BPP}_k$ by a particularly simple function, a polynomial-size constant-width CNF. (iii) We give an essentially tight separation, of $O(1)$ versus $\Omega(n^{1-\delta})$, for the one-sided versus two-sided approximate degree of a function; and $O(1)$ versus $\Omega(n^{1-\delta})$ for the one-sided approximate degree of a function $f$ versus its negation $\neg f$.

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