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arXiv:2310.08004

On the Rational Degree of Boolean Functions and Applications

Vishnu Iyer, Siddhartha Jain, Robin Kothari, Matt Kovacs-Deak, Vinayak M. Kumar, Luke Schaeffer, Daochen Wang, Michael Whitmeyer

Published Oct 12, 2023• 0 citations
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Abstract

We study a natural complexity measure of Boolean functions known as the rational degree. Denoted $\textrm{rdeg}(f)$, it is the minimal degree of a rational function that is equal to $f$ on the Boolean hypercube. For total functions $f$, it is conjectured that $\textrm{rdeg}(f)$ is polynomially related to the Fourier degree of $f$, $\textrm{deg}(f)$. Towards this conjecture, we show that: - Symmetric functions have rational degree at least $Ω(\textrm{deg}(f))$ and unate functions have rational degree at least $\sqrt{\textrm{deg}(f)}$. We observe that both of these lower bounds are asymptotically tight. - Read-once AC and TC formulae have rational degree at least $Ω(\sqrt{\textrm{deg}(f)})$. If these formulae contain parity gates, we show a lower bound of $Ω(\textrm{deg}(f)^{1/2d})$, where $d$ is the depth. - Almost every Boolean function on $n$ variables has rational degree at least $n/2 - O(\sqrt{n})$. In contrast, we exhibit partial functions that witness unbounded separations between rational and approximate degree, in both directions. As a consequence, we show that for quantum computers, post-selection and bounded-error are incomparable resources in the black-box model. In addition, we show AND and OR composition lemmas for the rational degree and exhibit new polynomial separations between the rational degree and other well-studied complexity measures, such as sensitivity and spectral sensitivity.

DOI
10.48550/arxiv.2310.08004

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IEEE International Symposium on Multiple-Valued Logic (ISMVL)•May 1, 2007•0 citations
We study a natural complexity measure of Boolean functions known as the rational degree. Denoted rdeg(f)\textrm{rdeg}(f)rdeg(f), it is the minimal degree of a rational function that is equal to fff on the Boolean hypercube. For total functions fff, it is conjectured that rdeg(f)\textrm{rdeg}(f)rdeg(f) is polynomially related to the Fourier degree of fff, deg(f)\textrm{deg}(f)deg(f). Towards this conjecture, we show that: - Symmetric functions have rational degree at least Ω(deg(f))Ω(\textrm{deg}(f))Ω(deg(f)) and unate functions have rational degree at least deg(f)\sqrt{\textrm{deg}(f)}deg(f)​. We observe that both of these lower bounds are asymptotically tight. - Read-once AC and TC formulae have rational degree at least Ω(deg(f))Ω(\sqrt{\textrm{deg}(f)})Ω(deg(f)​). If these formulae contain parity gates, we show a lower bound of Ω(deg(f)1/2d)Ω(\textrm{deg}(f)^{1/2d})Ω(deg(f)1/2d), where ddd is the depth. - Almost every Boolean function on nnn variables has rational degree at least n/2−O(n)n/2 - O(\sqrt{n})n/2−O(n​). In contrast, we exhibit partial functions that witness unbounded separations between rational and approximate degree, in both directions. As a consequence, we show that for quantum computers, post-selection and bounded-error are incomparable resources in the black-box model. In addition, we show AND and OR composition lemmas for the rational degree and exhibit new polynomial separations between the rational degree and other well-studied complexity measures, such as sensitivity and spectral sensitivity.
The approximate non-deterministic degree of a Boolean function fff, denoted ndegε(f)\mathsf{ndeg}_ε(f)ndegε​(f) (written Nε(f)\mathsf{N}_ε(f)Nε​(f) for brevity), is the minimum degree of a real polynomial ppp such that 0≤∣p(x)∣≤ε0 \le |p(x)| \le ε0≤∣p(x)∣≤ε whenever f(x)=0f(x) = 0f(x)=0, and ∣p(x)∣≥1|p(x)| \ge 1∣p(x)∣≥1 whenever f(x)=1f(x) = 1f(x)=1. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on 111-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all 000-inputs and bounded away from zero on all 111-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 fff, deg(f)≤O~ ⁣(rdeg⁡(f)3). deg(f) \le \widetilde O\!\left(\operatorname{rdeg}(f)^3\right). deg(f)≤O(rdeg(f)3). In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by Nε(f)\mathsf{N}_ε(f)Nε​(f) and Nε(f‾)\mathsf{N}_ε(\overline{f})Nε​(f​) jointly, i.e., for every total Boolean function fff and every constant 0<ε<10<ε<10<ε<1, deg~(f)≤poly⁡(Nε(f),Nε(f‾)). \widetilde{deg}(f) \le \operatorname{poly}(\mathsf N_ε(f), \mathsf N_ε(\overline f)). 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, kkk-uniform hypergraph properties, and read-kkk Disjunctive Normal Form (DNF) formulas.
This thesis studies three complexity measures of total Boolean functions f:{0,1}n → {0,1}: maximum sensitivity s(f), polynomial degree deg(f), and spectral sensitivity λ(f), where λ(f) is defined as the spectral norm of the adjacency matrix of the sensitivity graph. Building on the results of Aaronson et al., we examine the inequality chain √s(f) ≤ λ(f) ≤ deg(f) and investigate whether all three quantities can be simultaneously equal. The first part of the thesis reverse engineers the equality cases of the two known inequalities to isolate necessary extremal conditions on both the Fourier structure of f and the local geometry of its sensitivity graph. The second part develops Python-based exhaustive and constrained searches to test the first nontrivial candidate profiles. We compute full distributions of s(f), deg(f), and λ(f) for all Boolean functions on n=3 and n=4, showing in particular that no function realizes the candidate (s(f), λ(f), deg(f)) = (4,2,2) at n=4. We then use Möbius-based and CSP-based enumeration to study all real-degree-≤2 Boolean functions through higher dimensions, obtaining repeated evidence that s(f) ≤ 3 in this family and thereby excluding the (4,2,2) case in the tested range. Together, the theoretical and computational results support the view that the simultaneous tightness of √s(f) = λ(f) = deg(f) is obstructed by strong structural incompatibilities beyond the trivial degree-1 case.
Certificate complexity (C(f))(C(f ))(C(f)) is a fundamental measure of complexity of Boolean functions f which counts the number of bits of an input that need to be known in order for the value of the function to be determined. A certificate can be viewed as a partial assignment, or a boolean subcube where the function is constant. Certificate complexity is well understood for deterministic query (or decision tree) complexity (D)(D)(D) and other query models such as bounded-error randomized and quantum complexity (R,Q)(R, Q)(R,Q), but not as well for quantum zero-error (Q0)(Q_0)(Q0​) and exact query complexity (QE)(Q_E)(QE​), where there is no agreed-upon certificate 'object' (even for QQQ). Instead, we study an operational notion of certification and apply it to various query-based models, with a focus on zero-error and exact quantum query complexity, but also on polynomial degree measures. We give new characterizations of C,RCC, RCC,RC (randomized certificate complexity) and QCQCQC (quantum certificate complexity), in terms of various measures such as classical and quantum sabotage complexity, unambiguous certificate complexity, and variants of polynomial degree. Certification complexity also gives rise to new lower bounds on QEQ_EQE​ and Q0Q_0Q0​, the quantum analogues of DDD and R0R_0R0​, complexity measures for which few lower bound techniques are known which are not already lower bounds for two-sided error quantum query complexity. We exhibit a total Boolean function for which our certification complexity measure gives a tight lower bound for Q0Q_0Q0​, but rational degree and QQQ are asymptotically smaller.
We prove that, to compute a Boolean function fff on NNN variables with error probability εεε, any quantum black-box algorithm has to query at least 1−2ε2ρfN=1−2ε2Sˉf\frac{1 - 2\sqrtε}{2} ρ_f N = \frac{1 - 2\sqrtε}{2} \bar{S}_f21−2ε​​ρf​N=21−2ε​​Sˉf​ times, where ρfρ_fρf​ is the average influence of variables in fff, and Sˉf\bar{S}_fSˉf​ is the average sensitivity. It's interesting to contrast this result with the known lower bound of Ω(Sf)Ω(\sqrt{S_f})Ω(Sf​​), where SfS_fSf​ is the sensitivity of fff. This lower bound is tight for some functions. We also show for any polynomial f~\tilde{f}f~​ that approximates fff with error probability εεε, deg(f~)≥1/4(1−3ε1+ε)2ρfNdeg(\tilde{f}) \ge 1/4 (1 - \frac{3 ε}{1 + ε})^2 ρ_f Ndeg(f~​)≥1/4(1−1+ε3ε​)2ρf​N. This bound can be better than previous known lower bound of Ω(BSf)Ω(\sqrt{BS_f})Ω(BSf​​) for some functions. Our technique may be of intest itself: we apply Fourier analysis to functions mapping {0,1}N\{0, 1\}^N{0,1}N to unit vectors in a Hilbert space. From this viewpoint, the state of the quantum computer at step ttt can be written as ∑s∈{0,1}N,∣s∣≤tϕ^s(−1)s⋅x\sum_{s\in \{0, 1\}^N, |s| \le t} \hatϕ_s (-1)^ {s \cdot x}∑s∈{0,1}N,∣s∣≤t​ϕ^​s​(−1)s⋅x, which is handy for lower bound analysis.
A Nearly Optimal Lower Bound on the Approximate Degree of AC0^00
The approximate degree of a Boolean function f ⁣:{−1,1}n→{−1,1}f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}f:{−1,1}n→{−1,1} is the least degree of a real polynomial that approximates fff 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 fff with approximate degree ddd into a function FFF on O(n⋅polylog(n))O(n \cdot \mathrm{polylog}(n))O(n⋅polylog(n)) variables with approximate degree at least D=Ω(n1/3⋅d2/3)D = \Omega(n^{1/3} \cdot d^{2/3})D=Ω(n1/3⋅d2/3). In particular, if d=n1−Ω(1)d= n^{1-\Omega(1)}d=n1−Ω(1), then DDD is polynomially larger than ddd. Moreover, if fff is computed by a polynomial-size Boolean circuit of constant depth, then so is FFF. By recursively applying our transformation, for any constant δ>0\delta > 0δ>0 we exhibit an AC0^00 function of approximate degree Ω(n1−δ)\Omega(n^{1-\delta})Ω(n1−δ). This improves upon the best previous lower bound of Ω(n2/3)\Omega(n^{2/3})Ω(n2/3) due to Aaronson and Shi [ J. ACM, 51 (2004), pp. 595--605] and nearly matches the trivial upper bound of nnn 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\delta > 0δ>0, an Ω(n1−δ)\Omega(n^{1-\delta})Ω(n1−δ) lower bound on the quantum communication complexity of a function in AC0^00; (ii) a Boolean function fff with approximate degree at least C(f)2−o(1)C(f)^{2-o(1)}C(f)2−o(1), where C(f)C(f)C(f) is the certificate complexity of fff; this separation is optimal up to the o(1)o(1)o(1) term in the exponent; (iii) improved secret sharing schemes with reconstruction procedures in AC0^00.
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. This is why Boolean functions are needed with a high number of essential variables and a low polynomial degree. Unfortunately, it is a well-known problem to construct such functions. The best separation between these two complexity measures of a Boolean function was exhibited by Ambai- nis [5]. He constructed functions with polynomial degree M and number of variables Omega(M2). We improve such a separation to become exponential. On the other hand, we use a computerized exhaustive search to prove tightness of this bound.