Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Relative-error unateness testing

Xi Chen, Diptaksho Palit, Kabir Peshawaria, William Pires, Rocco A. Servedio, Yiding Zhang

TL;DR

This work studies relative-error unateness testing for Boolean functions f: {0,1}^n -> {0,1}, where distance is measured relative to the number of 1s in f, |f^{-1}(1)|=N. It develops two testers: a known-N one-sided non-adaptive tester with query complexity ilde{O}( abla abla log N / abla epsilon) and an unknown-N one-sided adaptive tester achieving the same order without knowledge of N. The paper also establishes lower bounds: a non-adaptive two-sided lower bound of ilde{ Omega}( log N) and a two-sided adaptive lower bound of ilde{ Omega}(( log N)^{2/3}), which together delineate the landscape in this relative-error model. The results reveal a nuanced role for sparsity N in relative-error unateness testing, including a gap between adaptive and non-adaptive regimes and between relative-error and standard-model bounds, driven by the fact that the closure under truncation concentrates action on adjacent layers rather than long hypercube paths. These findings connect to prior relative-error monotonicity results and open avenues for exploring other semantic properties under the relative-error framework.

Abstract

The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [CDH+24][CPPS25a][CPPS25b] In this paper we consider the problem of relative-error testing an unknown and arbitrary $f: \{0,1\}^n \to \{0,1\}$ for the property of being a unate function, i.e. a function that is either monotone non-increasing or monotone non-decreasing in each of the $n$ input variables. Our first result is a one-sided non-adaptive algorithm for this problem that makes $\tilde{O}(\log(N)/ε)$ samples and queries, where $N=|f^{-1}(1)|$ is the number of satisfying assignments of the function that is being tested and the value of $N$ is given as an input parameter to the algorithm. Building on this algorithm, we next give a one-sided adaptive algorithm for this problem that does not need to be given the value of $N$ and with high probability makes $\tilde{O}(\log(N)/ε)$ samples and queries. We also give lower bounds for both adaptive and non-adaptive two-sided algorithms that are given the value of $N$ up to a constant multiplicative factor. In the non-adaptive case, our lower bounds essentially match the complexity of the algorithm that we provide.

Relative-error unateness testing

TL;DR

This work studies relative-error unateness testing for Boolean functions f: {0,1}^n -> {0,1}, where distance is measured relative to the number of 1s in f, |f^{-1}(1)|=N. It develops two testers: a known-N one-sided non-adaptive tester with query complexity ilde{O}( abla abla log N / abla epsilon) and an unknown-N one-sided adaptive tester achieving the same order without knowledge of N. The paper also establishes lower bounds: a non-adaptive two-sided lower bound of ilde{ Omega}( log N) and a two-sided adaptive lower bound of ilde{ Omega}(( log N)^{2/3}), which together delineate the landscape in this relative-error model. The results reveal a nuanced role for sparsity N in relative-error unateness testing, including a gap between adaptive and non-adaptive regimes and between relative-error and standard-model bounds, driven by the fact that the closure under truncation concentrates action on adjacent layers rather than long hypercube paths. These findings connect to prior relative-error monotonicity results and open avenues for exploring other semantic properties under the relative-error framework.

Abstract

The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [CDH+24][CPPS25a][CPPS25b] In this paper we consider the problem of relative-error testing an unknown and arbitrary for the property of being a unate function, i.e. a function that is either monotone non-increasing or monotone non-decreasing in each of the input variables. Our first result is a one-sided non-adaptive algorithm for this problem that makes samples and queries, where is the number of satisfying assignments of the function that is being tested and the value of is given as an input parameter to the algorithm. Building on this algorithm, we next give a one-sided adaptive algorithm for this problem that does not need to be given the value of and with high probability makes samples and queries. We also give lower bounds for both adaptive and non-adaptive two-sided algorithms that are given the value of up to a constant multiplicative factor. In the non-adaptive case, our lower bounds essentially match the complexity of the algorithm that we provide.
Paper Structure (50 sections, 37 theorems, 85 equations, 8 algorithms)

This paper contains 50 sections, 37 theorems, 85 equations, 8 algorithms.

Table of Contents

  1. Introduction
  2. Our results
  3. Discussion
  4. Techniques
  5. Upper bounds
  6. The known-$N$ algorithm.
  7. The unknown-$N$ algorithm.
  8. Overview of both algorithms.
  9. Lower bounds
  10. Preliminaries
  11. Notation
  12. Strings.
  13. Edges.
  14. Distances.
  15. Relative-error testing of Boolean functions
  16. ...and 35 more sections

Key Result

Theorem 1

There is a one-sided, non-adaptive algorithm, algo:Known-N, for $\varepsilon$-relative-error testing of unateness of an unknown and arbitrary function $f: \{0,1\}^n \to \{0,1\}$. The algo:Known-N algorithm is given as input the value of $N$, and makes $\tilde{O}(\log(N)/\varepsilon)$ samples and que

Theorems & Definitions (89)

  • Theorem 1: One-sided non-adaptive algorithm, known $N$, informal version of \ref{['thm:known-N']}
  • Theorem 2: One-sided adaptive algorithm, unknown $N$, informal version of \ref{['thm:unknown-N']}
  • Theorem 3: Non-adaptive lower bound
  • Theorem 4: Adaptive lower bound
  • Lemma 5: Approximate triangle inequality for relative distance, Lemma 5 of rel-error-conj-DL
  • Definition 6
  • Lemma 8
  • proof
  • Theorem 10: Theorem 1.3 of CS16
  • Corollary 11
  • ...and 79 more