New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String
Lijie Chen, Yang Hu, Hanlin Ren
TL;DR
This paper establishes new algebrization barriers to circuit lower bounds by introducing XOR-Missing-String, a duality-based communication problem whose multilinear-extension behavior captures nonrelativizing, arithmetization-based techniques. Through a trio of oracle constructions A1, A2, and A3, the authors derive barriers for pr-PostBPE, BPE, and MA_E, respectively, showing that certain lower bounds cannot be achieved via algebrizing methods alone. The core technical engine is a rectangle-based analysis of XOR-Missing-String under various communication models, translated into algebrization barriers via approximate majority covers and robust MA_E frameworks. The results collectively demonstrate that surpassing half-exponential and even almost-everywhere circuit lower bounds for these classes would require fundamentally non-algebrizing techniques, shaping our understanding of the limits of arithmetization-based approaches in circuit lower bounds.
Abstract
The *algebrization barrier*, proposed by Aaronson and Wigderson (STOC '08, ToCT '09), captures the limitations of many complexity-theoretic techniques based on arithmetization. Notably, several circuit lower bounds that overcome the relativization barrier (Buhrman--Fortnow--Thierauf, CCC '98; Vinodchandran, TCS '05; Santhanam, STOC '07, SICOMP '09) remain subject to the algebrization barrier. In this work, we establish several new algebrization barriers to circuit lower bounds by studying the communication complexity of the following problem, called XOR-Missing-String: For $m < 2^{n/2}$, Alice gets a list of $m$ strings $x_1, \dots, x_m\in\{0, 1\}^n$, Bob gets a list of $m$ strings $y_1, \dots, y_m\in\{0, 1\}^n$, and the goal is to output a string $s\in\{0, 1\}^n$ that is not equal to $x_i\oplus y_j$ for any $i, j\in [m]$. 1. We construct an oracle $A_1$ and its multilinear extension $\widetilde{A_1}$ such that ${\sf PostBPE}^{\widetilde{A_1}}$ has linear-size $A_1$-oracle circuits on infinitely many input lengths. 2. We construct an oracle $A_2$ and its multilinear extension $\widetilde{A_2}$ such that ${\sf BPE}^{\widetilde{A_2}}$ has linear-size $A_2$-oracle circuits on all input lengths. 3. Finally, we study algebrization barriers to circuit lower bounds for $\sf MA_E$. Buhrman, Fortnow, and Thierauf proved a *sub-half-exponential* circuit lower bound for $\sf MA_E$ via algebrizing techniques. Toward understanding whether the half-exponential bound can be improved, we define a natural subclass of $\sf MA_E$ that includes their hard $\sf MA_E$ language, and prove the following result: For every *super-half-exponential* function $h(n)$, we construct an oracle $A_3$ and its multilinear extension $\widetilde{A_3}$ such that this natural subclass of ${\sf MA}_{\sf E}^{\widetilde{A_3}}$ has $h(n)$-size $A_3$-oracle circuits on all input lengths.