关于扩展器随机游走对只读ACC0电路伪随机性的研究
Towards the Pseudorandomness of Expander Random Walks for Read-Once ACC0 circuits
AI总结：
本研究探讨扩展器随机游走对非对称复杂性类ACC0的伪随机性，证明其能欺骗特定MOD[k]门两层电路，并构造TC0阈值电路作为反例，提供欺骗函数集上界。
摘要：
扩展图是理论计算机科学中最有用的组合对象之一。一系列工作研究了扩展器上的随机游走，针对各种测试函数类（包括对称函数、只读分支程序、置换分支程序和$\mathrm{AC}^0$电路）的伪随机性。扩展器随机游走对$\mathrm{AC}^0$电路的伪随机性研究结果显示出其超越对称函数的鲁棒性，这促使我们思考扩展器随机游走是否能够欺骗更鲁棒的\emph{非对称}复杂性类，例如$\mathrm{ACC}^0$。在本工作中，我们通过考虑$\mathrm{MOD}[k]$门的特定两层电路组合来推进这一问题，并证明这些电路族被扩展器随机游走以总变差距离误差$O(λ)$欺骗，其中$λ$是底层扩展器图的第二大特征值。对于$k\geq 3$，这些电路可能高度非对称且具有复杂的傅里叶特征。在此背景下，我们的工作朝着欺骗更复杂的非对称电路迈出了一步。另外，借鉴学习理论文献，我们在电路族$\mathrm{TC}^0$中构造了一个显式阈值电路，并证明它\emph{不被}扩展器随机游走欺骗，从而为扩展器随机游走所欺骗的函数集提供了上界。
英文摘要：
Expander graphs are among the most useful combinatorial objects in theoretical computer science. A line of work studies random walks on expander graphs for their pseudorandomness against various classes of test functions, including symmetric functions, read-only branching programs, permutation branching programs, and $\mathrm{AC}^0$ circuits. The promising results of pseudorandomness of expander random walks against $\mathrm{AC}^0$ circuits indicate a robustness of expander random walks beyond symmetric functions, motivating the question of whether expander random walks can fool more robust \emph{asymmetric} complexity classes, such as $\mathrm{ACC}^0$. In this work, we make progress towards this question by considering certain two-layered circuit compositions of $\mathrm{MOD}[k]$ gates, where we show that these family of circuits are fooled by expander random walks with total variation distance error $O(λ)$, where $λ$ is the second largest eigenvalue of the underlying expander graph. For $k\geq 3$, these circuits can be highly asymmetric with complicated Fourier characters. In this context, our work takes a step in the direction of fooling more complex asymmetric circuits. Separately, drawing from the learning-theory literature, we construct an explicit threshold circuit in the circuit family $\mathrm{TC}^0$, and show that it is \emph{not} fooled by expander random walk, providing an upper bound on the set of functions fooled by expander random walks.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
