# Tree tribes and lower bounds for switching lemmas

## TL;DR

This paper establishes tight bounds on the effect of random p-restrictions on t-clipped decision trees, providing new insights into the structure and limitations of such boolean functions.

## Contribution

It introduces tight upper and lower bounds for switching lemmas applied to t-clipped decision trees under random p-restrictions, advancing understanding of decision tree complexity.

## Key findings

- Upper bound: probability of large decision tree depth after restriction is at most (4p2^t)^d.
- Lower bound: existence of functions with probability at least (c_0 p 2^t)^d of large depth after restriction.
- Results are tight bounds for the behavior of t-clipped decision trees under random restrictions.

## Abstract

We show tight upper and lower bounds for switching lemmas obtained by the action of random $p$-restrictions on boolean functions that can be expressed as decision trees in which every vertex is at a distance of at most $t$ from some leaf, also called $t$-clipped decision trees. More specifically, we show the following: $\bullet$ If a boolean function $f$ can be expressed as a $t$-clipped decision tree, then under the action of a random $p$-restriction $\rho$, the probability that the smallest depth decision tree for $f|_{\rho}$ has depth greater than $d$ is upper bounded by $(4p2^{t})^{d}$. $\bullet$ For every $t$, there exists a function $g_{t}$ that can be expressed as a $t$-clipped decision tree, such that under the action of a random $p$-restriction $\rho$, the probability that the smallest depth decision tree for $g_{t}|_{\rho}$ has depth greater than $d$ is lower bounded by $(c_{0}p2^{t})^{d}$, for $0\leq p\leq c_{p}2^{-t}$ and $0\leq d\leq c_{d}\frac{\log n}{2^{t}\log t}$, where $c_{0},c_{p},c_{d}$ are universal constants.