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Symmetric Arithmetic Circuits

Anuj Dawar, Gregory Wilsenach

TL;DR

This work studies arithmetic circuits under natural symmetry constraints when computing polynomials on $n\times n$ matrices, distinguishing the determinant from the permanent. By formalizing matrix, square, and transpose symmetry and connecting circuit automorphisms to counting-width via the Weisfeiler-Leman framework, the authors reduce lower-bound problems to graph-counting parameters such as the number of perfect matchings. They prove exponential lower bounds for square-symmetric circuits computing the permanent over char $0$, while giving polynomial-size transpose-symmetric circuits for the determinant, using Le Verrier’s method and a symmetry-preserving Boolean translation. The results reveal a fundamental barrier that symmetry imposes on permanent computations and provide a robust methodology—via counting width and CFI-style constructions—for proving lower bounds in restricted circuit classes. They also discuss extensions to positive characteristics and relate their findings to equivariant determinantal representations, suggesting broad implications for algebraic complexity and symmetry-aware computation.

Abstract

We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under simultaneous row and column permutations of the matrix. We establish unconditional exponential lower bounds on the size of any symmetric circuit for computing the permanent. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.

Symmetric Arithmetic Circuits

TL;DR

This work studies arithmetic circuits under natural symmetry constraints when computing polynomials on matrices, distinguishing the determinant from the permanent. By formalizing matrix, square, and transpose symmetry and connecting circuit automorphisms to counting-width via the Weisfeiler-Leman framework, the authors reduce lower-bound problems to graph-counting parameters such as the number of perfect matchings. They prove exponential lower bounds for square-symmetric circuits computing the permanent over char , while giving polynomial-size transpose-symmetric circuits for the determinant, using Le Verrier’s method and a symmetry-preserving Boolean translation. The results reveal a fundamental barrier that symmetry imposes on permanent computations and provide a robust methodology—via counting width and CFI-style constructions—for proving lower bounds in restricted circuit classes. They also discuss extensions to positive characteristics and relate their findings to equivariant determinantal representations, suggesting broad implications for algebraic complexity and symmetry-aware computation.

Abstract

We introduce symmetric arithmetic circuits, i.e. arithmetic circuits with a natural symmetry restriction. In the context of circuits computing polynomials defined on a matrix of variables, such as the determinant or the permanent, the restriction amounts to requiring that the shape of the circuit is invariant under simultaneous row and column permutations of the matrix. We establish unconditional exponential lower bounds on the size of any symmetric circuit for computing the permanent. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.

Paper Structure

This paper contains 26 sections, 19 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Groups
  4. Fields and Linear Algebra
  5. Graphs, Matrices and Matchings
  6. Counting Width
  7. Circuits
  8. Symmetric Circuits
  9. Symmetric Functions
  10. Symmetric Circuits
  11. Polynomials
  12. An Upper-Bound for the Determinant
  13. From Arithmetic To Boolean Circuits
  14. Supports and Counting-Width
  15. A Lower-Bound for the Permanent
  16. ...and 11 more sections

Key Result

Theorem 10

For $\mathbb{F}$ be a field of characteristic $0$, there exists a family of transpose symmetric arithmetic circuits $(\Phi_n)_{n \in \mathbb{N}}$ over $\mathbb{F}$ computing $\{{\normalfont\textsc{DET}}_n\}$ for which the function $n \mapsto \Phi_n$ is computable in time $\mathcal{O}(n^4)$.

Figures (1)

  • Figure 1: A gadget in $X(\Gamma)$ corresponding to vertex $v$ with incident edges $f,g,h$

Theorems & Definitions (41)

  • Definition 1
  • Definition 2: Circuit
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8: Circuit Automorphism
  • Definition 9: Symmetric Circuit
  • Theorem 10
  • ...and 31 more