Lower bounds for uniform read-once threshold formulae in the randomized decision tree model

TL;DR

This paper establishes lower bounds on the randomized decision tree complexity for uniform read-once threshold functions, revealing how complexity scales with tree depth and structure.

Contribution

It provides new lower bounds for the complexity of uniform read-once threshold formulae, including cases with alternating AND and OR levels, extending previous binary case results.

Findings

Lower bounds of the form c(k,n)^d for decision tree complexity

Asymptotically optimal bounds for trees with alternating AND/OR levels

Extension of known bounds from binary to more general threshold functions

Abstract

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.