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Top-Down Lower Bounds for Depth-Four Circuits

Mika Göös, Artur Riazanov, Anastasia Sofronova, Dmitry Sokolov

TL;DR

This work develops a complete top-down framework for proving lower bounds against constant-depth circuits, focusing on depth-$4$ circuits computing the parity function. The authors extend unpredictability methods from single bits to blocks of coordinates and couple them with robust sunflower techniques to build mirror sets, enabling a depth-$4$ lower bound of size $2^{n^{1/3-o(1)}}$. The core contributions are the block unpredictability lemma with near-optimal parameters, a spreadify step that centers local density around a near-parity point, and a detailed downwards argument that yields a contradiction for any too-small depth-$4$ circuit computing parity. This approach illustrates the potential completeness of top-down methods for constant-depth circuits and may inform broader strategies for related circuit classes and complexity separations.

Abstract

We present a top-down lower-bound method for depth-$4$ boolean circuits. In particular, we give a new proof of the well-known result that the parity function requires depth-$4$ circuits of size exponential in $n^{1/3}$. Our proof is an application of robust sunflowers and block unpredictability.

Top-Down Lower Bounds for Depth-Four Circuits

TL;DR

This work develops a complete top-down framework for proving lower bounds against constant-depth circuits, focusing on depth- circuits computing the parity function. The authors extend unpredictability methods from single bits to blocks of coordinates and couple them with robust sunflower techniques to build mirror sets, enabling a depth- lower bound of size . The core contributions are the block unpredictability lemma with near-optimal parameters, a spreadify step that centers local density around a near-parity point, and a detailed downwards argument that yields a contradiction for any too-small depth- circuit computing parity. This approach illustrates the potential completeness of top-down methods for constant-depth circuits and may inform broader strategies for related circuit classes and complexity separations.

Abstract

We present a top-down lower-bound method for depth- boolean circuits. In particular, we give a new proof of the well-known result that the parity function requires depth- circuits of size exponential in . Our proof is an application of robust sunflowers and block unpredictability.
Paper Structure (13 sections, 11 theorems, 23 equations, 2 figures)

This paper contains 13 sections, 11 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof overview
  3. Depth-3 via bit unpredictability
  4. Depth-4 via block unpredictability
  5. Challenge 1: Which subcircuit to pick?
  6. Challenge 2: Block unpredictability.
  7. Challenge 3: Flipping into target set.
  8. Proof of \ref{['thm:main']}
  9. Proof of \ref{['lem:spreadify']}
  10. Block unpredictability lemma
  11. Entropy
  12. Shattered sets
  13. Proof of \ref{['thm:unpred']}

Key Result

Theorem 1

Every depth-$4$ circuit computing the $n$-bit parity requires $2^{n^{1/3-o(1)}}$ gates.

Figures (2)

  • Figure 1: Flipping one bit yields a local limit
  • Figure 2: Choosing a mirror set

Theorems & Definitions (30)

  • Theorem 1
  • Lemma 1: Bit unpredictability Meir2019
  • Claim 1: Local limit
  • proof
  • Remark 1: Completeness of the top-down method
  • Lemma 2: Block unpredictability
  • Remark 2: Optimality of \ref{['thm:unpred']}
  • Lemma 3: Rao2020
  • Claim 2
  • proof
  • ...and 20 more