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：我们研究了一类特定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，其中每个内部节点被标记为阈值函数 $T_k^n$（仅当至少 $k$ 个输入位为1时输出1），每个叶节点被标记为一个不同的变量。这类树以自然方式定义了一个布尔函数。我们重点关注当底层树是均匀树（所有内部节点被标记为同一阈值函数）时此类函数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，我们单独处理了交替包含AND和OR门层的树，并证明了渐近最优的界，从而推广了已知的二元情况下的界。