Toward Better Depth Lower Bounds: Strong Composition of XOR and a Random Function
Nikolai Chukhin, Alexander S. Kulikov, Ivan Mihajlin
TL;DR
This work advances depth lower bounds for formula computation by proving a strong composition result for KW relations: for the strong composition $\KW_{\XOR_m} \circledast \KW_f$ with a random inner function $f$, any protocol requires $n^{3 - o(1)}$ leaves (where $n = m \log m$), matching the $(3 - o(1))\log n$ bound known for XOR outer functions. The authors extend KRW-type arguments beyond difficult outer functions by combining Khrapchenko-style graph measures, heavy/light edge analysis, and a two-stage strategy that first establishes a quadratic-in-$n$ bound modulated by $\L_{\frac{3}{4}}(f)$ and then leverages formula-balancing to generalize to wider protocol classes. They show that for 0.49-balanced inner functions, the leaves are at least $n^{2 - o(1)} \cdot \L_{\frac{3}{4}}(f)$, and for a random $f$, $\L_{\frac{3}{4}}(f)=\Omega(m/\log\log m)$ with high probability, yielding $n^{3 - o(1)}$ bounds. These results broaden the regime where strong lower bounds apply and provide tools that help bridge toward resolving the KRW conjecture in its full generality, potentially informing formula-depth separations.
Abstract
Proving formula depth lower bounds is a fundamental challenge in complexity theory, with the strongest known bound of $(3 - o(1))\log n$ established by Hastad over 25 years ago. The Karchmer-Raz-Wigderson (KRW) conjecture offers a promising approach to advance these bounds and separate P from NC$^{1}$. It suggests that the depth complexity of a function composition $f \diamond g$ approximates the sum of the depth complexities of $f$ and $g$. The Karchmer-Wigderson (KW) relation framework translates formula depth into communication complexity, restating the KRW conjecture as $\mathsf{CC}(\mathsf{KW}_f \diamond \mathsf{KW}_g) \approx \mathsf{CC}(\mathsf{KW}_f) + \mathsf{CC}(\mathsf{KW}_g)$. Prior work has confirmed the conjecture under various relaxations, often replacing one or both KW relations with the universal relation or constraining the communication game through strong composition. In this paper, we examine the strong composition $\mathsf{KW}_{\mathsf{XOR}} \circledast \mathsf{KW}_f$ of the parity function and a random Boolean function $f$. We prove that with probability $1-o(1)$, any protocol solving this composition requires at least $n^{3 - o(1)}$ leaves. This result establishes a depth lower bound of $(3 - o(1))\log n$, matching Hastad's bound, but is applicable to a broader class of inner functions, even when the outer function is simple. Though bounds for the strong composition do not translate directly to formula depth bounds, they usually help to analyze the standard composition (of the corresponding two functions) which is directly related to formula depth.