Shrinkage of Decision Lists and DNF Formulas

TL;DR

This paper provides nearly tight bounds on how decision lists and DNF formulas shrink on average under random restrictions, revealing how their sizes decrease depending on the restriction parameter p.

Contribution

It establishes new bounds on the expected size reduction of decision lists and DNF formulas under p-random restrictions, extending understanding of their structural properties.

Findings

Expected decision list size after restriction is bounded by a power of original size.

The bounds are nearly tight for all p in [0,1].

Special case: size reduces to roughly the square root at p ≈ 0.24.

Abstract

We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the $p$-random restriction $\mathbf R_p$ for all values of $p \in [0,1]$. For a function $f$ with domain $\{0,1\}^n$, let $\mathrm{DL}(f)$ denote the minimum size of a decision list that computes $f$. We show that \[ \mathbb E[\ \mathrm{DL}(f{\upharpoonright}\mathbf R_p)\ ] \le \mathrm{DL}(f)^{\log_{2/(1-p)}(\frac{1+p}{1-p})}. \] For example, this bound is $\sqrt{\mathrm{DL}(f)}$ when $p = \sqrt{5}-2 \approx 0.24$. For Boolean functions $f$, we obtain the same shrinkage bound with respect to DNF formula size plus $1$ (i.e., replacing $\mathrm{DL}(\cdot)$ with $\mathrm{DNF}(\cdot)+1$ on both sides of the inequality).