The Approximate Degree of DNF and CNF Formulas
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The approximate degree of a Boolean function f{0,1}^n→{0,1} is the minimum degree of a real polynomial p that approximates f pointwise: |f(x)-p(x)|≤1/3 for all x∈{0,1}^n. For every δ>0, we construct CNF and DNF formulas of polynomial size with approximate degree Ω(n^1-δ), 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 (AC^0), a question that has seen extensive research over the past 10 years. Previously, an Ω(n^1-δ) lower bound was known only for AC^0 circuits of depth that grows with 1/δ (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 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 δ>0, these models require Ω(n^1-δ), Ω(n/4^kk^2)^1-δ, and Ω(n/4^kk^2)^1-δ, respectively, bits of communication even for polynomial-size constant-width CNF formulas. (ii) In particular, we show that the multiparty communication class coNP_k can be separated essentially optimally from NP_k and BPP_k by a particularly simple function, a polynomial-size constant-width CNF. (iii) We give an essentially tight separation, of O(1) versus Ω(n^1-δ), for the one-sided versus two-sided approximate degree of a function; and O(1) versus Ω(n^1-δ) for the one-sided approximate degree of a function f versus its negation f.
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