On the Rational Degree of Boolean Functions and Applications

2023-10-12

Electron. Colloquium Comput. Complex.

AI Summary

This paper investigates the rational degree (rdeg) of Boolean functions—the minimal degree of a rational function that exactly represents the function on the hypercube—and aims to establish polynomial relationships between rdeg and classical complexity measures such as Fourier degree. The authors combine combinatorial and Fourier-analytic techniques, algebraic complexity theory, and quantum query methods. They establish tight lower bounds on rdeg for symmetric functions, unate functions, and read-once AC⁰/TC⁰ formulas—the first such results. They prove that almost all $n$-variable Boolean functions satisfy $ ext{rdeg} = n/2 - O(sqrt{n})$. They demonstrate a bidirectional unbounded separation between rdeg and approximate degree, uncover new polynomial separations between rdeg and sensitivity as well as spectral sensitivity, and refute the equivalence of postselected and bounded-error quantum query models. Collectively, these results provide critical criteria for resolving the Boolean function complexity hierarchy.

Abstract

We study a natural complexity measure of Boolean functions known as the rational degree. Denoted $ extrm{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 $ extrm{rdeg}(f)$ is polynomially related to the Fourier degree of $f$, $ extrm{deg}(f)$. Towards this conjecture, we show that: - Symmetric functions have rational degree at least $Omega( extrm{deg}(f))$ and unate functions have rational degree at least $sqrt{ extrm{deg}(f)}$. We observe that both of these lower bounds are asymptotically tight. - Read-once AC and TC formulae have rational degree at least $Omega(sqrt{ extrm{deg}(f)})$. If these formulae contain parity gates, we show a lower bound of $Omega( extrm{deg}(f)^{1/2d})$, 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.

Problem

• Studies rational degree of Boolean functions.
• Explores relations between rational and Fourier degrees.
• Examines rational degree in quantum computing contexts.

Innovation

• Symmetric functions have tight rational degree bounds
• Read-once formulae show depth-dependent rational degree
• Partial functions separate rational and approximate degrees