Lifting for Arbitrary Gadgets
Siddharth Iyer
TL;DR
The paper addresses how hard it is to compute the composition $f∘g$ in deterministic communication when $g$ is an arbitrary gadget, and proves a lifting theorem that translates sensitivity of $f$ into a lower bound for $D(f∘g)$. The authors develop a density-rectangle lemma, leverage entropy-based arguments, and iteratively decompose the problem to obtain the main bound $D(f∘g) \ge s(f) \left( \frac{\Omega(D(g))}{\log rk(g)} - \log rk(g) \right)$, with corollaries showing $D(f∘g) = \Omega(\min\{s,d\}\cdot\sqrt{D(g)})$ when $D(g)$ is large and a specific OR-case lower bound of $\Omega(n\sqrt{D(g)})$. The approach connects to direct-sum/XOR-lemma techniques and situates the result among lifting theorems that relate inner-gadget complexity to outer-function measures such as sensitivity and degree. This work broadens lifting theory beyond parity-type inner functions and has implications for understanding how outer Boolean function complexity translates into communication overhead under arbitrary gadgets.
Abstract
We prove a sensitivity-to-communication lifting theorem for arbitrary gadgets. Given functions $f: \{0,1\}^n\to \{0,1\}$ and $g : \mathcal X\times \mathcal Y\to \{0,1\}$, denote $f\circ g(x,y) := f(g(x_1,y_1),\ldots,g(x_n,y_n))$. We show that for any $f$ with sensitivity $s$ and any $g$, \[D(f\circ g) \geq s\cdot \bigg(\frac{Ω(D(g))}{\log\mathsf{rk}(g)} - \log\mathsf{rk}(g)\bigg),\] where $D(\cdot)$ denotes the deterministic communication complexity and $\mathsf{rk}(g)$ is the rank of the matrix associated with $g$. As a corollary, we get that if $D(g)$ is a sufficiently large constant, $D(f\circ g) = Ω(\min\{s,d\}\cdot \sqrt{D(g)})$, where $s$ and $d$ denote the sensitivity and degree of $f$. In particular, computing the OR of $n$ copies of $g$ requires $Ω(n\cdot\sqrt{D(g)})$ bits.