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Approximate Degree Composition for Recursive Functions

Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh

TL;DR

The paper advances the long-standing question of how approximate degree behaves under function composition by focusing on recursive bases $h^d$. It develops a primal–dual framework centered on a majority amplification gadget and shows that dominant lower bounds survive when either the outer or inner function is recursive, with $d=\Omega(\log\log n)$ and a polylogarithmic factor loss. The results are established first for recursive Maj or alternating $\mathsf{AND}$-$\mathsf{OR}$ trees, then extended to general recursive functions by simulating $\mathsf{AND}_2$ and $\mathsf{OR}_2$ within $h^3$, and further broadened to cases with full sign-degree amplifiers. This work sharpens our understanding of approximate-degree composition and informs related domains such as quantum query and learning theory, while identifying central open cases such as composition when the inner function is $\mathsf{OR}$.

Abstract

Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let $h^d$ denote the standard $d$-fold composition of the base function $h$. The main result of this work is to show that the approximate degree composes if either of the following conditions holds: (I) The outer function $f:\{0,1\}^n\to \{0,1\}$ is a recursive function of the form $h^d$, with $h$ being any base function and $d= Ω(\log\log n)$. (II) The inner function is a recursive function of the form $h^d$, with $h$ being any constant arity base function (other than AND and OR) and $d= Ω(\log\log n)$, where $n$ is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.

Approximate Degree Composition for Recursive Functions

TL;DR

The paper advances the long-standing question of how approximate degree behaves under function composition by focusing on recursive bases . It develops a primal–dual framework centered on a majority amplification gadget and shows that dominant lower bounds survive when either the outer or inner function is recursive, with and a polylogarithmic factor loss. The results are established first for recursive Maj or alternating - trees, then extended to general recursive functions by simulating and within , and further broadened to cases with full sign-degree amplifiers. This work sharpens our understanding of approximate-degree composition and informs related domains such as quantum query and learning theory, while identifying central open cases such as composition when the inner function is .

Abstract

Determining the approximate degree composition for Boolean functions remains a significant unsolved problem in Boolean function complexity. In recent decades, researchers have concentrated on proving that approximate degree composes for special types of inner and outer functions. An important and extensively studied class of functions are the recursive functions, i.e.~functions obtained by composing a base function with itself a number of times. Let denote the standard -fold composition of the base function . The main result of this work is to show that the approximate degree composes if either of the following conditions holds: (I) The outer function is a recursive function of the form , with being any base function and . (II) The inner function is a recursive function of the form , with being any constant arity base function (other than AND and OR) and , where is the arity of the outer function. In terms of proof techniques, we first observe that the lower bound for composition can be obtained by introducing majority in between the inner and the outer functions. We then show that majority can be \emph{efficiently eliminated} if the inner or outer function is a recursive function.
Paper Structure (14 sections, 35 theorems, 57 equations, 2 figures)

This paper contains 14 sections, 35 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Results
  3. Proof Ideas
  4. Notations and Preliminaries
  5. Composition theorem for recursive Majority and alternating $\mathsf{AND}$-$\mathsf{OR}$ trees
  6. Approximate Degree of Recursive Majority
  7. Proof of Lemma \ref{['lem: Maj as amplification function']}
  8. Proof of Theorem \ref{['th: RecMaj composition']}
  9. Composition theorem for recursive functions
  10. Proof of the main technical lemma (Lemma \ref{['thm:simulate_AND-OR']})
  11. Functions with full sign degree as hardness amplifier
  12. Composition theorem for recursive functions with full sign degree
  13. Conclusion
  14. One-sided approximate degree

Key Result

Theorem 1.1

Let $f\colon {\left\{ 0,1 \right\}}^{n} \to \left\{ 0,1 \right\}$ and $g\colon {\left\{ 0,1 \right\}}^{m} \to \left\{ 0,1 \right\}$ be two Boolean functions and $d=\Omega(\log\log n)$. Then, if either of the following conditions hold:

Figures (2)

  • Figure 1: A function on $3$ bits with a shifted $\mathsf{OR}$ marked with red edges.
  • Figure 2: An example for constructing $\mathsf{AND}_2$ using a non-monotone function. Let $\mathsf{h: \left\{ 0,1 \right\}^3 \to \left\{ 0,1 \right\}}$ be $0$ at $x= 001$ and $1$ otherwise. We use the shifted $\mathsf{OR}_2$/minimal sensitive block at $001$ with indices $\{2,3\}$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.2
  • Definition 2.1: Generalized Composition of functions
  • Definition 2.2: Recursive functions
  • Definition 2.3: Approximate degree $(\widetilde{\deg})$
  • Lemma 2.4: Error reduction Sherstov11
  • Lemma 2.5: Sherstovpattern11Sherstov12
  • Theorem 2.6: robustSherstov13
  • Definition 2.7: $(\delta,\varepsilon)$-robust approximating polynomial
  • ...and 50 more