VC-Dimension vs Degree: An Uncertainty Principle for Boolean Functions
Fan Chang, Yijia Fang
TL;DR
The paper establishes a sharp uncertainty-type principle for Boolean functions by proving that for any nonzero function $f:\{0,1 ight ext{^}n o\\{0,1 ight ext}$, one has ${\rm VC}(f)+{\rm deg}(f)\ge n$ and ${\rm VC}(f)+{\rm deg}_{\mathbb{F}_2}(f)\ge n$, tying combinatorial and algebraic complexity. The proofs connect Fourier-analytic properties on the hypercube with combinatorial design theory, notably via null $d$-designs and parity arguments, and show how low degree forces combinatorial richness. These results recover the classical uncertainty principle on the discrete cube and yield the Sziklai–Weiner bound in the $\mathbb{F}_2$ setting, while also implying trade-offs with other notions such as decision-tree complexity and sensitivity. The work further discusses extensions to slices of the cube and general finite abelian groups, and addresses equality cases, with computational exploration hinting at intricate extremal structures. Overall, the findings reveal a fundamental tension between a Boolean function’s combinatorial and algebraic aspects, with potential implications for extremal combinatorics and complexity theory.
Abstract
In this paper, we uncover a new uncertainty principle that governs the complexity of Boolean functions. This principle manifests as a fundamental trade-off between two central measures of complexity: a combinatorial complexity of its supported set, captured by its Vapnik-Chervonenkis dimension ($\mathrm{VC}(f)$), and its algebraic structure, captured by its polynomial degree over various fields. We establish two primary inequalities that formalize this trade-off: $\mathrm{VC}(f)+\mathrm{deg}(f)\ge n,$ and $\mathrm{VC}(f)+\mathrm{deg}_{\mathbb{F}_2}(f)\ge n$. In particular, these results recover the classical uncertainty principle on the discrete hypercube, as well as the Sziklai--Weiner's bound in the case of $\mathbb{F}_2$.