Subcube Stifling
For researchers in Boolean function analysis and complexity theory, this work provides a new measure that may lead to tighter composition theorems for approximate degree, though the main question remains open.
The paper introduces the subcube stifling number, a new combinatorial measure for Boolean functions, and shows it yields an approximate-degree composition theorem. They prove that random functions have subcube stifling number Θ(log n), and construct functions from linear codes with high stifling number but approximate degree Ω(μ(f)), leaving open whether any function achieves approximate degree Θ(√μ(f)).
We introduce the subcube stifling number, a new combinatorial measure of total Boolean functions. This measure is the largest integer $k$ such that, for every set $S$ of at most $k$ input variables and every assignment $b \in \{0,1\}^S$, there is a fixing of the variables outside $S$ under which the resulting function on the free variables $S$ is the point indicator $\mathbb{I}[x_S=b]$. Equivalently, for every small set of coordinates, the function can isolate any prescribed point of the corresponding Boolean cube by suitably fixing all remaining coordinates. This measure is inspired by the stifling number of Chattopadhyay et al.~(ITCS'23); whereas their measure asks for restrictions realizing every constant function, ours asks for restrictions realizing every point indicator. Our results are as follows. 1) We show that the subcube stifling number gives rise to an approximate-degree composition theorem. In particular, if a Boolean function $f$ has approximate degree $O(\sqrt{μ(f)})$, then for every Boolean function $g$, approximate degree composes tightly. This motivates the study of the subcube stifling number, and in particular the search for functions whose approximate degree is $O(\sqrt{μ(f)})$. 2) We show that a random Boolean function on $n$ input bits has subcube stifling number $Θ(\log(n))$ with high probability. 3) We show that indicators of linear codes over $\mathbb{F}_2$ whose minimum distance and dual distance are both linear have high subcube stifling number. 4) We prove that the functions arising from this linear-code construction do not have approximate degree $O(\sqrt{μ(f)})$; in fact, they have approximate degree $Ω(μ(f))$. The main question left open is whether there exists a Boolean function $f$ with approximate degree $Θ(\sqrt{μ(f)})$. A positive answer would yield new instances of tight approximate-degree composition.