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A basic lower bound for property testing

Eldar Fischer

TL;DR

The paper proves a universal lower bound on the query complexity of $\\epsilon$-testing for non-trivial properties, showing that any such test must use at least a constant multiple of $1/\\epsilon$ queries. The main technique constructs a pair of inputs $S$ and $U$ with distance $d(U,\\mathcal{P}) \ge \alpha$, then builds a family of perturbed inputs $U_A$ on a distinguished set $D$ to form a $(U,V,l)$-distinguisher and analyzes it under non-adaptive querying via a union-bound argument to obtain a lower bound $|D|/(3l) = \\\Omega(k/l)$. By relating $k$ to $\alpha n$ and $l$ to $\\epsilon n$, the result yields $\\Omega(\\alpha/\\epsilon)$ queries for infinitely many $n$ and $0<\\epsilon<\\alpha$. Importantly, the proof avoids Yao's principle and applies to general non-trivial properties, not just dense or graph-based settings. This formalizes the intuitive difficulty of property testing and provides a foundational, model-free lower bound with broad applicability.

Abstract

An $ε$-test for any non-trivial property (one for which there are both satisfying inputs and inputs of large distance from the property) should use a number of queries that is at least inversely proportional in $ε$. However, to the best of our knowledge there is no reference proof for this intuition. Such a proof is provided here. It is written so as to not require any prior knowledge of the related literature, and in particular does not use Yao's method.

A basic lower bound for property testing

TL;DR

The paper proves a universal lower bound on the query complexity of -testing for non-trivial properties, showing that any such test must use at least a constant multiple of queries. The main technique constructs a pair of inputs and with distance , then builds a family of perturbed inputs on a distinguished set to form a -distinguisher and analyzes it under non-adaptive querying via a union-bound argument to obtain a lower bound . By relating to and to , the result yields queries for infinitely many and . Importantly, the proof avoids Yao's principle and applies to general non-trivial properties, not just dense or graph-based settings. This formalizes the intuitive difficulty of property testing and provides a foundational, model-free lower bound with broad applicability.

Abstract

An -test for any non-trivial property (one for which there are both satisfying inputs and inputs of large distance from the property) should use a number of queries that is at least inversely proportional in . However, to the best of our knowledge there is no reference proof for this intuition. Such a proof is provided here. It is written so as to not require any prior knowledge of the related literature, and in particular does not use Yao's method.
Paper Structure (2 sections, 6 theorems)

This paper contains 2 sections, 6 theorems.

Table of Contents

  1. Introduction and statement
  2. The main proof

Key Result

Theorem 1.5

If a property $\mathcal{P}\subset\Sigma^n$ (for any alphabet $\Sigma$) admits both a satisfying input $S\in\Sigma^n\cap\mathcal{P}$ and an input $U\in\Sigma^n$ for which $d(U,\mathcal{P})\geq k/n$, then for any $1\leq l<k$, an $(l/n)$-test for $\mathcal{P}$ requires at least $\Omega(k/l)$ many queri

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5: Meticulous statement
  • Corollary 1.6: Straightforward statement
  • proof
  • Lemma 2.1
  • proof
  • Definition 2.2
  • ...and 6 more