On the Rational Degree of Boolean Functions and Applications

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 $\Omega(\textrm{deg}(f))$ 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 $\Omega(\sqrt{\textrm{deg}(f)})$. If these formulae contain parity gates, we show a lower bound of $\Omega(\textrm{deg}(f)^{1/2d})$, 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.