Symmetric Arithmetic Circuits

Authors Anuj Dawar , Gregory Wilsenach

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Author Details

Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK
Gregory Wilsenach
  • Department of Computer Science and Technology, University of Cambridge, UK

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Anuj Dawar and Gregory Wilsenach. Symmetric Arithmetic Circuits. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 row and column permutations of the matrix. We establish unconditional, nearly exponential, lower bounds on the size of any symmetric circuit for computing the permanent over any field of characteristic other than 2. In contrast, we show that there are polynomial-size symmetric circuits for computing the determinant over fields of characteristic zero.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Algebraic complexity theory
  • arithmetic circuits
  • symmetric arithmetic circuits
  • Boolean circuits
  • symmetric circuits
  • permanent
  • determinant
  • counting width
  • Weisfeiler-Leman dimension
  • Cai-Fürer-Immerman constructions


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