We investigate the randomized decision tree complexity of a specific class of read-once threshold functions. A read-once threshold formula can be defined by a rooted tree, every internal node of which is labeled by a threshold function $T_k^n$ (with output 1 only when at least $k$ out of $n$ input bits are 1) and each leaf by a distinct variable. Such a tree defines a Boolean function in a natural way. We focus on the randomized decision tree complexity of such functions, when the underlying tree is a uniform tree with all its internal nodes labeled by the same threshold function. We prove lower bounds of the form $c(k,n)^d$, where $d$ is the depth of the tree. We also treat trees with alternating levels of AND and OR gates separately and show asymptotically optimal bounds, extending the known bounds for the binary case.

翻译：我们研究了特定类别的只读阈值函数的随机决策树复杂度。由一棵有根树可定义只读阈值公式，其中每个内部节点标记为阈值函数 $T_k^n$（仅当 $n$ 个输入比特中至少 $k$ 个为 1 时输出 1），每个叶子节点标记为不同变量。此类树以自然方式定义布尔函数。我们关注此类函数的随机决策树复杂度，尤其当底层树为均匀树且所有内部节点标记相同阈值函数时。我们证明了形如 $c(k,n)^d$ 的下界，其中 $d$ 为树的深度。此外，我们分别处理了由 AND 和 OR 门交替层级构成的树，并给出了渐近最优界，从而推广了二元情形的已知结果。