On the Rational Degree of Boolean Functions and Applications

TL;DR

This work investigates the rational degree $\mathrm{rdeg}(f)$, the minimal degree of a rational representation matching a Boolean function on the Boolean cube, and its relation to other complexity measures. It establishes tight lower bounds for broad function classes (e.g., symmetric: $\mathrm{rdeg}(f) \ge (\deg(f)+1)/3$, unate/monotone: $\mathrm{rdeg}(f)=\mathrm{s}(f) \ge \sqrt{\deg(f)}$), and shows $\mathrm{rdeg}$ remains large for read-once AC/TC formulas, with parity gates yielding stronger bounds. The authors also prove that almost all functions satisfy $\mathrm{rdeg}(f) \ge n/2 - O(\sqrt{n})$ and provide AND/OR composition lemmas for $\mathrm{rdeg}$, together with explicit separations between $\mathrm{rdeg}$ and approximate degree via partial functions. On the complexity-theory side, they demonstrate both weak and strong oracle separations between zero-error post-selection and bounded-error quantum models, and show that NP ∩ coNP sits inside $\mathsf{PostEQP}$, illustrating powerful implications of post-selection. Overall, the paper advances the understanding of how rational representations interact with standard Boolean-function measures and quantum resources, offering new tools for separating complexity classes in the black-box setting.

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.

Figures (3)

• Figure 1: A table summarizing our lower bounds on rational degree for total functions. The third column gives an example of a function that demonstrates the asymptotic tightness of our lower bound, where applicable.
• Figure 2: Relevant complexity classes. We are able to obtain the strongest possible oracle separations in this picture. An arrow $\cA\rightarrow\cB$ means $\cA\subseteq \cB$ relative to all oracles. A dashed arrow $\cA\dashrightarrow\cB$ means $\cA\not\subseteq \cB$ relative to some oracle.
• Figure 3: A depth-$d$$\mathop{\mathrm{\mathsf{AND}}}\nolimits$-$\mathop{\mathrm{\mathsf{OR}}}\nolimits$ tree with certificate complexity $\sqrt{n}$, and thus rational degree at most $\sqrt{n}$. Indeed, setting all input wires to $1$ for $\mathop{\mathrm{\mathsf{AND}}}\nolimits$ functions and a single input wire to $1$ for $\mathop{\mathrm{\mathsf{OR}}}\nolimits$ functions along a single path to root gives a $1$ certificate, and setting all input wires to $0$ for $\mathop{\mathrm{\mathsf{OR}}}\nolimits$ functions and a single input wire to $0$ for $\mathop{\mathrm{\mathsf{AND}}}\nolimits$ functions gives a $0$-certificate. One can easily verify that both of these certificates are of size $\sqrt{n}$.

Theorems & Definitions (72)

• Conjecture 1: de Wolf dewolf2000
• Corollary 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 5
• proof
• Proposition 6
• proof
• Proposition 7