Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let $h^d$ denote the standard $d$-fold composition of the base function $h$. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: (I) The outer function $f:\{0,1\}^n\to \{0,1\}$ is a recursive function of the form $h^d$, with $h$ being any base function and $d= Ω(\log\log n)$. (II) The inner function is a recursive function of the form $h^d$, with $h$ being any constant arity base function (other than AND and OR) and $d= Ω(\log\log n)$, where $n$ is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.
Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let hd denote the standard d-fold composition of the base function h. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: (I) The outer function f:{0,1}n→{0,1} is a recursive function of the form hd, with h being any base function and d=Ω(loglogn). (II) The inner function is a recursive function of the form hd, with h being any constant arity base function (other than AND and OR) and d=Ω(loglogn), where n is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.
A longstanding open question in Boolean function complexity asks whether approximate degree composes multiplicatively under block composition. Although a general multiplicative upper bound is known, matching lower bounds have previously been established only for restricted classes of functions. We resolve this question for all total Boolean functions by proving the matching lower bound. Together with Sherstov's upper bound, our result shows that, for every pair of total Boolean functions f:{0,1}n→{0,1} and g:{0,1}m→{0,1}, deg(f∘g)=Θ(deg(f)deg(g)), where deg denotes constant-error approximate degree.
We also consider the analogous question for exact quantum query complexity and give a counterexample showing that multiplicative composition fails in this setting.
For any Boolean functions f and g, the question whether R(f∘g)=Θ~(R(f)⋅R(g)) is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg(f∘g)=Θ~(deg(f)⋅deg(g)) . These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg compose. A recent landmark result (Ben-David and Blais, 2020) showed that R(f∘g)=Ω(noisyR(f)⋅R(g)) . This implies that composition holds iff noisyR(f)=Θ~(R(f)) . We show two results: On the other hand, we prove that deg(f∘g)=Ω(bs(f)⋅deg(g)), where bs(f) is the block sensitivity of f. This implies that deg composes when deg(f) is asymptotically equal to bs(f) . It is already known that both R and deg compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.
We describe a new hardness amplification result for point-wise approximation of Boolean functions by low-degree polynomials. Specifically, for any function f on N bits, define F(x_1,...,x_M) = OMB(f(x_1),...,f(x_M)) to be the function on M*N bits obtained by block-composing f with a function known as ODD-MAX-BIT. We show that, if f requires large degree to approximate to error 2/3 in a certain one-sided sense (captured by a complexity measure known as positive one-sided approximate degree), then F requires large degree to approximate even to error 1-2^{-M}. This generalizes a result of Beigel (Computational Complexity, 1994), who proved an identical result for the special case f=OR. Unlike related prior work, our result implies strong approximate degree lower bounds even for many functions F that have low threshold degree. Our proof is constructive: we exhibit a solution to the dual of an appropriate linear program capturing the approximate degree of any function. We describe several applications, including improved separations between the complexity classes P^{NP} and PP in both the query and communication complexity settings. Our separations improve on work of Beigel (1994) and Buhrman, Vereshchagin, and de Wolf (CCC, 2007).
The approximate degree of a Boolean function f : {-1, 1}n→ {-1, 1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n · polylog(n)) variables with approximate degree at least D = Ω(n1/3· d2/3). In particular, if d = n1-Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a constant-depth polynomial-size Boolean circuit, then so is F. By recursively applying our transformation, for any constant δ > 0 we exhibit an AC0function of approximate degree Ω(n1-δ). This improves over the best previous lower bound of Ω(n2/3) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: · For any constant δ > 0, an Ω(n1-δ) lower bound on the quantum communication complexity of a function in AC0. · A Boolean function f with approximate degree at least C(f)2-o(1), where C(f) is the certificate complexity of f. This separation is optimal up to the o(1) term in the exponent. · Improved secret sharing schemes with reconstruction procedures in AC0.
A Nearly Optimal Lower Bound on the Approximate Degree of AC0
The approximate degree of a Boolean function f:{−1,1}n→{−1,1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n⋅polylog(n)) variables with approximate degree at least D=Ω(n1/3⋅d2/3). In particular, if d=n1−Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a polynomial-size Boolean circuit of constant depth, then so is F. By recursively applying our transformation, for any constant δ>0 we exhibit an AC0 function of approximate degree Ω(n1−δ). This improves upon the best previous lower bound of Ω(n2/3) due to Aaronson and Shi [ J. ACM, 51 (2004), pp. 595--605] and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) disjunctive normal forms of polylogarithmic width. We describe several applications of these results and provide the following: (i) for any constant δ>0, an Ω(n1−δ) lower bound on the quantum communication complexity of a function in AC0; (ii) a Boolean function f with approximate degree at least C(f)2−o(1), where C(f) is the certificate complexity of f; this separation is optimal up to the o(1) term in the exponent; (iii) improved secret sharing schemes with reconstruction procedures in AC0.
The approximate non-deterministic degree of a Boolean function f, denoted ndegε(f) (written Nε(f) for brevity), is the minimum degree of a real polynomial p such that 0≤∣p(x)∣≤ε whenever f(x)=0, and ∣p(x)∣≥1 whenever f(x)=1. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on 1-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all 0-inputs and bounded away from zero on all 1-inputs.
The rational degree conjecture, open for over three decades, was recently resolved by Kothari, Kovacs-Deak, Wang, and Yang, who showed that for every total Boolean function f, deg(f)≤O(rdeg(f)3).
In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by Nε(f) and Nε(f) jointly, i.e., for every total Boolean function f and every constant 0<ε<1, deg(f)≤poly(Nε(f),Nε(f)).
This conjecture, if true, would imply a polynomial version of the rational degree result and bring us closer to resolving de Wolf's longstanding non-deterministic degree conjecture.
In this work, we make the first systematic progress on this problem, establishing the conjecture for several broad and natural function classes: monotone and unate functions, functions of bounded alternation number, symmetric functions, k-uniform hypergraph properties, and read-k Disjunctive Normal Form (DNF) formulas.