AI 中文总结
研究多数函数是否属于或非门电路类的问题，借助环面多项式逼近，将其相关猜想简化为线性规划问题，通过受对偶多项式方法启发的方式，为一大类线性规划找到可行解并证明多个关键结果。
AI 中文摘要
或非门电路类由可以用具有与门、非门和模m门的多项式规模的常数深度电路计算的布尔函数组成，其中m是自然数。一个广泛相信的猜想是多数函数不属于或非门电路类。几年前，Bhrushundi等人引入环面多项式逼近作为研究该猜想的一种方法。他们将多数函数不属于或非门电路类的猜想简化为关于不存在逼近多数函数的低次环面多项式的猜想。我们进一步将不存在问题简化为关于为一个无限线性规划族找到可行解的陈述。这一陈述的主要优点是允许逐步推进，即依次为这些规划的更大集合找到可行解。作为第一步，我们为一大类这些线性规划找到了可行解，只留下有限集以供进一步考虑。我们的方法受用于研究布尔函数逼近度的对偶多项式方法的启发。使用我们的方法，我们还提出了进一步推进的方法。我们用相同方法证明了几个额外的关键结果，包括与函数逼近的下界、逼近多项式对称时的下界，展示了我们方法的威力。
英文摘要
The class $ACC^0$ consists of Boolean functions that can be computed by constant-depth circuits of polynomial size with $AND, NOT$ and $MOD_m$ gates, where $m$ is a natural number. At the frontier of our understanding lies a widely believed conjecture asserting that $MAJORITY$ does not belong to $ACC^0$. A few years ago, Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach towards this conjecture. Torus polynomials approximate Boolean functions when the fractional part of their value on Boolean points is close to half the value of the function. They reduced the conjecture that $MAJORITY \notin ACC^0$ to a conjecture concerning the non-existence of low degree torus polynomials that approximate $MAJORITY$. We reduce the non-existence problem further, to a statement about finding feasible solutions for an infinite family of linear programs. The main advantage of this statement is that it allows for incremental progress, which means finding feasible solutions for successively larger collections of these programs. As an immediate first step, we find feasible solutions for a large class of these linear programs, leaving only a finite set for further consideration. Our method is inspired by the method of dual polynomials, which is used to study the approximate degree of Boolean functions. Using our method, we also propose a way to progress further. We prove several additional key results with the same method, including lower bounds for approximating the $AND$ function, lower bounds when the approximating polynomial is symmetric, showcasing the power of our machinery.
