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Sensitivity of $m$-ary functions and low degree partitions of Hamming graphs

Sara Asensio, Ignacio García-Marco, Kolja Knauer

Abstract

The study of complexity measures of Boolean functions led Nisan and Szegedy to state the sensitivity conjecture in 1994, claiming a polynomial relation between degree and sensitivity. This problem remained unsolved until 2019, when Huang proved the conjecture via an equivalent graph theoretical reformulation due to Gotsman and Linial. We study $m$-ary functions, i.e., functions $f: T^n \rightarrow T$ where $T\subseteq \mathbb{C}$ is a finite alphabet of cardinality $|T| = m $ and extend the notions of degree $\mathrm{deg}(f)$ and sensitivity $s(f)$ to $m$-ary functions and show $s(f)\in O(\mathrm{deg}(f)^2)$. This generalizes results of Nisan and Szegedy. Conversely, we introduce the $m$-ary sensitivity conjecture, claiming a polynomial upper bound for $\mathrm{deg}(f)$ in terms of $s(f)$. Analogously to results of Gotsman and Linial, we provide a formulation of the conjecture in terms of imbalanced partitions of Hamming graphs into low degree subgraphs. Combining this with ideas of Chung, Füredi, Graham and Seymour, we show that for any prime $p$ the bound in the $p$-ary sensitivity conjecture has to be at least quadratic: there exist $p$-ary functions $f$ of arbitrarily large degree and $\mathrm{deg}(f)\in Ω(s(f)^2)$.

Sensitivity of $m$-ary functions and low degree partitions of Hamming graphs

Abstract

The study of complexity measures of Boolean functions led Nisan and Szegedy to state the sensitivity conjecture in 1994, claiming a polynomial relation between degree and sensitivity. This problem remained unsolved until 2019, when Huang proved the conjecture via an equivalent graph theoretical reformulation due to Gotsman and Linial. We study -ary functions, i.e., functions where is a finite alphabet of cardinality and extend the notions of degree and sensitivity to -ary functions and show . This generalizes results of Nisan and Szegedy. Conversely, we introduce the -ary sensitivity conjecture, claiming a polynomial upper bound for in terms of . Analogously to results of Gotsman and Linial, we provide a formulation of the conjecture in terms of imbalanced partitions of Hamming graphs into low degree subgraphs. Combining this with ideas of Chung, Füredi, Graham and Seymour, we show that for any prime the bound in the -ary sensitivity conjecture has to be at least quadratic: there exist -ary functions of arbitrarily large degree and .
Paper Structure (6 sections, 19 theorems, 48 equations, 7 figures)

This paper contains 6 sections, 19 theorems, 48 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Some complexity measures of $m$-ary functions
  3. An upper bound for the sensitivity in terms of the degree
  4. The equivalence theorem for $m$-ary functions
  5. Quadratic separation between the sensitivity and the degree
  6. Conclusions

Key Result

Theorem 1.1

For every $m$-ary function $f$, we have $s(f)\leq 2\ (m-1)^3\ {\rm deg}(f)^2.$

Figures (7)

  • Figure 1: Some Hamming graphs, the upper two with their proper $3$ and $4$-coloring, respectively
  • Figure 2: Three induced subgraphs of $H(3,3)$ whose vertices partition $\mathcal{U}_{3}^3$
  • Figure 3: Vertices of $\rho(V_{\varepsilon^k})$ as a certain cyclic rotation of the vertices of $V_{\varepsilon^k}$ through the partition given by the $C_{\varepsilon^j}$'s
  • Figure 4: Partition associated to $f_\varepsilon$ for the $3$-ary function $f$ of \ref{['ex:running']}, where $H_i'$ is the induced subgraph of $H(3,3)$ on $\rho(V(H_i))$
  • Figure 5: \ref{['thm:equivalence']} generalizes Gotsman-Linial's Boolean equivalence theorem
  • ...and 2 more figures

Theorems & Definitions (51)

  • Theorem 1.1
  • Conjecture 1.1: $m$-ary Sensitivity Conjecture
  • Theorem 1.2
  • Conjecture 1.2
  • Definition 2.1: Degree
  • Proposition 2.2
  • proof
  • Definition 2.3: Sensitivity
  • Definition 2.4: Block sensitivity
  • Remark 2.5
  • ...and 41 more