We consider a basic question on the expressiveness of $k$-CNF formulas: How well can $k$-CNF formulas capture threshold functions? Specifically, what is the largest number of assignments (of Hamming weight $t$) accepted by a $k$-CNF formula that only accepts assignments of weight at least $t$? Among others, we provide the following results: - While an optimal solution is known for $t \leq n/k$, the problem remains open for $t > n/k$. We formulate a (monotone) version of the problem as an extremal hypergraph problem and show that for $t = n-k$, the problem is exactly the Tur\'{a}n problem. - For $t = \alpha n$ with constant $\alpha$, we provide a construction and show its optimality for $2$-CNF. Optimality of the construction for $k>2$ would give improved lower bounds for depth-$3$ circuits.

翻译：我们探讨k-CNF公式表达能力的一个基本问题：k-CNF公式能以何种程度捕获阈值函数？具体而言，在仅接受汉明权重至少为t的赋值时，k-CNF公式最多能接受多少个（汉明权重为t的）赋值？除其他成果外，我们提供了以下结论：- 虽然当t ≤ n/k时最优解已知，但t > n/k时该问题仍未解决。我们将该问题的（单调）版本表述为极值超图问题，并证明当t = n-k时，该问题等价于Turán问题。- 对于t = αn（其中α为常数），我们给出了一个构造方法，并证明该构造对2-CNF是最优的。若该构造对k>2的情况也能达到最优，将为深度-3电路提供改进的下界。