Lower Bounds for Symmetric Circuits for the Determinant

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


We are grateful to Albert Atserias for useful discussions on the construction in Section 5.2

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Anuj Dawar and Gregory Wilsenach. Lower Bounds for Symmetric Circuits for the Determinant. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 52:1-52:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is that the circuits which take a matrix input are unchanged by a permutation applied simultaneously to the rows and columns of the matrix. Under such restrictions we have polynomial-size circuits for computing the determinant but no subexponential size circuits for the permanent. Here, we consider a more stringent symmetry requirement, namely that the circuits are unchanged by arbitrary even permutations applied separately to rows and columns, and prove an exponential lower bound even for circuits computing the determinant. The result requires substantial new machinery. We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games. These are applied to the determinant problem with a novel construction of matrices that are bi-adjacency matrices of graphs based on the CFI construction. Our general framework opens the way to exploring a variety of symmetry restrictions and studying trade-offs between symmetry and other resources used by arithmetic circuits.

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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