License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2023.68
URN: urn:nbn:de:0030-drops-175717
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He, William ; Rossman, Benjamin

Symmetric Formulas for Products of Permutations

LIPIcs-ITCS-2023-68.pdf (0.8 MB)


We study the formula complexity of the word problem Word_{S_n,k} : {0,1}^{kn²} → {0,1}: given n-by-n permutation matrices M₁,… ,M_k, compute the (1,1)-entry of the matrix product M₁⋯ M_k. An important feature of this function is that it is invariant under action of S_n^{k-1} given by (π₁,… ,π_{k-1})(M₁,… ,M_k) = (M₁π₁^{-1},π₁M₂π₂^{-1},… ,π_{k-2}M_{k-1}π_{k-1}^{-1},π_{k-1}M_k).
This symmetry is also exhibited in the smallest known unbounded fan-in {and,or,not}-formulas for Word_{S_n,k}, which have size n^O(log k).
In this paper we prove a matching n^{Ω(log k)} lower bound for S_n^{k-1}-invariant formulas computing Word_{S_n,k}. This result is motivated by the fact that a similar lower bound for unrestricted (non-invariant) formulas would separate complexity classes NC¹ and Logspace.
Our more general main theorem gives a nearly tight n^d(k^{1/d}-1) lower bound on the G^{k-1}-invariant depth-d {maj,and,or,not}-formula size of Word_{G,k} for any finite simple group G whose minimum permutation representation has degree n. We also give nearly tight lower bounds on the G^{k-1}-invariant depth-d {and,or,not}-formula size in the case where G is an abelian group.

BibTeX - Entry

  author =	{He, William and Rossman, Benjamin},
  title =	{{Symmetric Formulas for Products of Permutations}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{68:1--68:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-175717},
  doi =		{10.4230/LIPIcs.ITCS.2023.68},
  annote =	{Keywords: circuit complexity, group-invariant formulas}

Keywords: circuit complexity, group-invariant formulas
Collection: 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)
Issue Date: 2023
Date of publication: 01.02.2023

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