On the Rational Degree of Boolean Functions and Applications

• URL: http://arxiv.org/abs/2310.08004v1
• Date: Thu, 12 Oct 2023 03:14:44 GMT
• Title: On the Rational Degree of Boolean Functions and Applications
• Authors: Vishnu Iyer, Siddhartha Jain, Matt Kovacs-Deak, Vinayak M. Kumar, Luke Schaeffer, Daochen Wang, Michael Whitmeyer
• Abstract summary: It is conjectured that $mathrmrdeg(f)$ is unboundedly related to $mathrmdeg(f)$. symmetric functions have rational degree $mathrmdeg(f)/2$ and monotone functions have rational degree at least $sqrtmathrmdeg(f)$.
• Score: 1.8125795677957914
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: We study a natural complexity measure of Boolean functions known as the (exact) rational degree. For total functions $f$, it is conjectured that $\mathrm{rdeg}(f)$ is polynomially related to $\mathrm{deg}(f)$, where $\mathrm{deg}(f)$ is the Fourier degree. Towards this conjecture, we show that symmetric functions have rational degree at least $\mathrm{deg}(f)/2$ and monotone functions have rational degree at least $\sqrt{\mathrm{deg}(f)}$. We observe that both of these lower bounds are tight. In addition, we show that all read-once depth-$d$ Boolean formulae have rational degree at least $\Omega(\mathrm{deg}(f)^{1/d})$. Furthermore, we show that almost every Boolean function on $n$ variables has rational degree at least $n/2 - O(\sqrt{n})$. In contrast to total functions, 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.

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