A Distance Amplification Lemma for Monotonicity

TL;DR

This paper introduces a procedure that amplifies the distance from monotonicity of Boolean functions, enabling stronger lower bounds on property testing algorithms by transforming functions while preserving monotonicity properties.

Contribution

The paper presents a distance amplification lemma for monotonicity, linking the distance from monotonicity to query complexity lower bounds in property testing.

Findings

Amplifies the distance from monotonicity while controlling function size.

Enables deriving near-optimal lower bounds for monotonicity testing.

Connects the amplification technique with existing lower bound results.

Abstract

We show a procedure that, given oracle access to a function $f\colon \{0,1\}^n\to\{0,1\}$, produces oracle access to a function $f'\colon \{0,1\}^{n'}\to\{0,1\}$ such that if $f$ is monotone, then $f'$ is monotone, and if $f$ is $\varepsilon$-far from monotone, then $f'$ is $\Omega(1)$-far from monotone. Moreover, $n' \leq n 2^{O(1/\varepsilon)}$ and each oracle query to $f'$ can be answered by making $2^{O(1/\varepsilon)}$ oracle queries to $f$. Our lemma is motivated by a recent result of [Chen, Chen, Cui, Pires, Stockwell, arXiv:2511.04558], who showed that for all $c>0$ there exists $\varepsilon_c>0$, such that any (even two-sided, adaptive) algorithm distinguishing between monotone functions and $\varepsilon_c$-far from monotone functions, requires $\Omega(n^{1/2-c})$ queries. Combining our lemma with their result implies a similar result, except that the distance from monotonicity is an absolute constant $\varepsilon>0$, and the lower bound is $\Omega(n^{1/2-o(1)})$ queries.