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.

翻译：*代数化障碍*由Aaronson和Wigderson（STOC '08, ToCT '09）提出，刻画了基于算术化的许多复杂性理论技术的局限性。值得注意的是，多个突破相对化障碍的电路下界结果（Buhrman--Fortnow--Thierauf, CCC '98；Vinodchandran, TCS '05；Santhanam, STOC '07, SICOMP '09）仍然受限于代数化障碍。本文通过研究以下问题（称为XOR-缺失字符串）的通信复杂度，建立了若干电路下界的新代数化障碍：当$m < 2^{n/2}$时，Alice获得$m$个字符串列表$x_1, \dots, x_m\\in\\{0, 1\\}^n$，Bob获得$m$个字符串列表$y_1, \dots, y_m\\in\\{0, 1\\}^n$，目标是输出一个字符串$s\\in\\{0, 1\\}^n$，使得对于任意$i, j\\in [m]$均满足$s \\neq x_i\\oplus y_j$。1. 我们构造了一个预言机$A_1$及其多重线性扩展$\\widetilde{A_1}$，使得${\\sf PostBPE}^{\\widetilde{A_1}}$在无限多个输入长度上具有线性规模的$A_1$-预言机电路。2. 我们构造了一个预言机$A_2$及其多重线性扩展$\\widetilde{A_2}$，使得${\\sf BPE}^{\\widetilde{A_2}}$在所有输入长度上具有线性规模的$A_2$-预言机电路。3. 最后，我们研究了$\\sf MA_E$电路下界的代数化障碍。Buhrman、Fortnow和Thierauf通过代数化技术证明了$\\sf MA_E$的*次半指数*电路下界。为探究半指数界限是否可改进，我们定义了一个包含其困难$\\sf MA_E$语言的自然子类，并证明以下结果：对于每个*超半指数*函数$h(n)$，我们构造了一个预言机$A_3$及其多重线性扩展$\\widetilde{A_3}$，使得${\\sf MA}_{\\sf E}^{\\widetilde{A_3}}$的这个自然子类在所有输入长度上具有$h(n)$规模的$A_3$-预言机电路。