A Variant of the VC-Dimension with Applications to Depth-3 Circuits

Authors Peter Frankl, Svyatoslav Gryaznov , Navid Talebanfard

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Author Details

Peter Frankl
  • Rényi Institute, Budapest, Hungary
Svyatoslav Gryaznov
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic
  • St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, Russia
Navid Talebanfard
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic


We are grateful to Vojtěch Rödl for discussions and to ITCS'22 reviewers for useful comments.

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Peter Frankl, Svyatoslav Gryaznov, and Navid Talebanfard. A Variant of the VC-Dimension with Applications to Depth-3 Circuits. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 72:1-72:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We introduce the following variant of the VC-dimension. Given S ⊆ {0,1}ⁿ and a positive integer d, we define 𝕌_d(S) to be the size of the largest subset I ⊆ [n] such that the projection of S on every subset of I of size d is the d-dimensional cube. We show that determining the largest cardinality of a set with a given 𝕌_d dimension is equivalent to a Turán-type problem related to the total number of cliques in a d-uniform hypergraph. This allows us to beat the Sauer-Shelah lemma for this notion of dimension. We use this to obtain several results on Σ₃^k-circuits, i.e., depth-3 circuits with top gate OR and bottom fan-in at most k: - Tight relationship between the number of satisfying assignments of a 2-CNF and the dimension of the largest projection accepted by it, thus improving Paturi, Saks, and Zane (Comput. Complex. '00). - Improved Σ₃³-circuit lower bounds for affine dispersers for sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture under which we get further improvement. - We make progress towards settling the Σ₃² complexity of the inner product function and all degree-2 polynomials over 𝔽₂ in general. The question of determining the Σ₃³ complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21).

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Hypergraphs
  • VC-dimension
  • Hypergraph
  • Clique
  • Affine Disperser
  • Circuit


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