Nisan and Szegedy (CC 1994) showed that any Boolean function f:{0,1}ⁿ → {0,1} that depends on all its input variables, when represented as a real-valued multivariate polynomial P(x₁,…,x_n), has degree at least log n - O(log log n). This was improved to a tight (log n - O(1)) bound by Chiarelli, Hatami and Saks (Combinatorica 2020). Similar statements are also known for other Boolean function complexity measures such as Sensitivity (Simon (FCT 1983)), Quantum query complexity, and Approximate degree (Ambainis and de Wolf (CC 2014)). In this paper, we address this question for Probabilistic degree. The function f has probabilistic degree at most d if there is a random real-valued polynomial of degree at most d that agrees with f at each input with high probability. Our understanding of this complexity measure is significantly weaker than those above: for instance, we do not even know the probabilistic degree of the OR function, the best-known bounds put it between (log n)^{1/2-o(1)} and O(log n) (Beigel, Reingold, Spielman (STOC 1991); Tarui (TCS 1993); Harsha, Srinivasan (RSA 2019)). Here we can give a near-optimal understanding of the probabilistic degree of n-variate functions f, modulo our lack of understanding of the probabilistic degree of OR. We show that if the probabilistic degree of OR is (log n)^c, then the minimum possible probabilistic degree of such an f is at least (log n)^{c/(c+1)-o(1)}, and we show this is tight up to (log n)^{o(1)} factors.

In this paper, we prove the first fixed-depth size-hierarchy theorem for uniform ${\mathrm{AC}}^0[\oplus]$. In particular, we show that for any fixed $d$ and integer parameter $k$, the class ${\mathcal{{C}}}_{d,k}$ of functions that have uniform ${\mathrm{AC}}^0[\oplus]$ formulas of depth $d$ and size $n^k$ form an infinite hierarchy. We show this by exhibiting the first class of functions that have uniform ${\mathrm{AC}}^0[\oplus]$ formulas of size $n^k$ but no ${\mathrm{AC}}^0[\oplus]$ formulas of size less than $n^{\varepsilon_0 k}$ for some absolute constant $\varepsilon_0 > 0$. The uniform formulas are designed to solve the $\delta$-coin problem, which is the computational problem of distinguishing between coins that are heads with probability $(1+\delta)/2$ or $(1-\delta)/2,$ where $\delta$ is a parameter that is going to $0$. We study the complexity of this problem and make progress on both upper bound and lower bound fronts. Regarding Upper bounds, for any constant $d\geq 2$, we show that there are uniform monotone ${\mathrm{AC}}^0$ formulas (i.e., made up of AND and OR gates only) solving the $\delta$-coin problem that have depth $d$, size $\exp(O(d\cdot(1/\delta)^{1/(d-1)}))$, and sample complexity (i.e., number of inputs) ${\mathop{\mathrm{poly}}}(1/\delta).$ This matches previous upper bounds of O'Donnell and Wimmer [ICALP 2007: Automata, Languages and Programming, Lecture Notes in Comput. Sci. 4596, Springer, New York, 2007, pp. 195--206] and Amano [ICALP 2009: Automata, Languages and Programming, Lecture Notes in Comput. Sci. 5555, Springer, New York, 2009, pp. 59--70] in terms of size (which is optimal), while improving the sample complexity from $\exp(O(d\cdot(1/\delta)^{1/(d-1)}))$ to ${\mathop{\mathrm{poly}}}(1/\delta)$. The improved sample complexity is crucial for proving the size-hierarchy theorem. Regarding Lower bounds, we show that the preceding upper bounds are nearly tight (in terms of size) even for the significantly stronger model of ${\mathrm{AC}}^0[\oplus]$ formulas (which are also allowed NOT and Parity gates): formally, we show that any ${\mathrm{AC}}^0[\oplus]$ formula solving the $\delta$-coin problem must have size $\exp(\Omega(d\cdot(1/\delta)^{1/(d-1)})).$ This strengthens a result of Shaltiel and Viola [SIAM J. Comput., 39 (2010), pp. 3122--3154], who prove an $\exp(\Omega((1/\delta)^{1/(d+2)}))$ lower bound for ${\mathrm{AC}}^0[\oplus]$ circuits, and a result of Cohen, Ganor, and Raz [APPROX-RANDOM, LIPIcs. Leibniz Int. Proc. Inform. 28, Schloss Dagstuhl, Leibniz-Zentrum fuer Informatik, Wadern, 2014, pp. 618--629], who show an $\exp(\Omega((1/\delta)^{1/(d-1)}))$ lower bound for ${\mathrm{AC}}^0$ circuits. The upper bound is a derandomization involving a use of Janson's inequality and an extension of classical polynomial-based combinatorial designs. For the lower bound, we prove an optimal (up to a constant factor) degree lower bound for multivariate polynomials over ${\mathbb{F}}_2$ solving the $\delta$-coin problem, which may be of independent interest.

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial ${P}$ of degree at most $d$ that agrees with $f$ at each point with high probability. Introduced by Razborov [ Mat. Zametki, 41 (1987), pp. 598--607], upper and lower bounds on probabilistic degrees of Boolean functions---specifically symmetric Boolean functions---have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems. In this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).

In this work we prove the first Fixed-depth Size-Hierarchy Theorem for uniform AC0[⊕]. In particular, we show that for any fixed d, the class Cd,k of functions that have uniform AC0[⊕] formulas of depth d and size nk form an infinite hierarchy. We show this by exhibiting the first class of explicit functions where we have nearly (up to a polynomial factor) matching upper and lower bounds for the class of AC0[⊕] formulas.