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：我们研究了特定一类只读阈值函数的随机决策树复杂度。只读阈值公式可由一棵有根树定义，其中每个内部节点标记为阈值函数 $T_k^n$（仅当 $n$ 个输入比特中至少有 $k$ 个为1时输出1），每个叶子节点标记为不同的变量。此类树以自然方式定义布尔函数。我们聚焦于此类函数在底层树为均匀树（所有内部节点标记相同阈值函数）时的随机决策树复杂度。我们证明了形如 $c(k,n)^d$ 的下界，其中 $d$ 为树的深度。此外，我们单独处理了AND与OR门交替层级的树结构，并给出了渐近最优界，将已知二元情形下的结论进行了推广。