Document Open Access Logo

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

Authors Peter Frankl, Svyatoslav Gryaznov , Navid Talebanfard

Thumbnail PDF


  • Filesize: 0.74 MB
  • 19 pages

Document Identifiers

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.

Cite AsGet BibTex

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. V. E. Alekseev. An upper bound for the number of maximal independent sets in a graph. Discrete Math. Appl., 17(4):355-359, 2007. URL:
  2. Noga Alon, Guy Moshkovitz, and Noam Solomon. Traces of hypergraphs. J. Lond. Math. Soc., 100(2):498-517, 2019. URL:
  3. Eli Ben-Sasson and Swastik Kopparty. Affine dispersers from subspace polynomials. SIAM J. Comput., 41(4):880-914, 2012. URL:
  4. Béla Bollobás and A. J. Radcliffe. Defect sauer results. J. Comb. Theory, Ser. A, 72(2):189-208, 1995. Google Scholar
  5. J. Adrian Bondy. Induced subsets. J. Combinatorial Theory Ser. B, 12:201-202, 1972. URL:
  6. Vasek Chvátal and Colin McDiarmid. Small transversals in hypergraphs. Comb., 12(1):19-26, 1992. URL:
  7. Gil Cohen and Igor Shinkar. The complexity of DNF of parities. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, pages 47-58. ACM, 2016. URL:
  8. Irit Dinur and Or Meir. Toward the KRW composition conjecture: Cubic formula lower bounds via communication complexity. Comput. Complex., 27(3):375-462, 2018. URL:
  9. Jeff Edmonds, Russell Impagliazzo, Steven Rudich, and Jirí Sgall. Communication complexity towards lower bounds on circuit depth. Comput. Complex., 10(3):210-246, 2001. URL:
  10. Magnus Gausdal Find, Alexander Golovnev, Edward A. Hirsch, and Alexander S. Kulikov. A better-than-3n lower bound for the circuit complexity of an explicit function. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 89-98. IEEE Computer Society, 2016. URL:
  11. Peter Frankl. On the trace of finite sets. J. Comb. Theory, Ser. A, 34(1):41-45, 1983. URL:
  12. Zoltán Füredi. A proof of the stability of extremal graphs, simonovits' stability from szemerédi’s regularity. J. Comb. Theory, Ser. B, 115:66-71, 2015. URL:
  13. Robert G. Gallager. Low-density parity-check codes. IRE Trans. Inf. Theory, 8(1):21-28, 1962. URL:
  14. Alexander Golovnev, Edward A. Hirsch, Alexander Knop, and Alexander S. Kulikov. On the limits of gate elimination. J. Comput. Syst. Sci., 96:107-119, 2018. URL:
  15. Alexander Golovnev, Alexander S. Kulikov, and R. Ryan Williams. Circuit depth reductions. In 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 24:1-24:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL:
  16. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL:
  17. Stasys Jukna. Boolean function complexity, volume 27 of Algorithms and Combinatorics. Springer, Heidelberg, 2012. Advances and frontiers. URL:
  18. Peter Keevash. Shadows and intersections: stability and new proofs. Adv. Math., 218(5):1685-1703, 2008. URL:
  19. A. V. Kostochka. A class of constructions for Turán’s (3, 4)-problem. Combinatorica, 2(2):187-192, 1982. URL:
  20. Or Meir. Toward better depth lower bounds: Two results on the multiplexor relation. Comput. Complex., 29(1):4, 2020. URL:
  21. Peter Bro Miltersen, Jaikumar Radhakrishnan, and Ingo Wegener. On converting CNF to DNF. Theor. Comput. Sci., 347(1-2):325-335, 2005. URL:
  22. Ramamohan Paturi, Pavel Pudlák, Michael E. Saks, and Francis Zane. An improved exponential-time algorithm for k-SAT. J. ACM, 52(3):337-364, 2005. URL:
  23. Ramamohan Paturi, Pavel Pudlák, and Francis Zane. Satisfiability coding lemma. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 566-574. IEEE Computer Society, 1997. URL:
  24. Ramamohan Paturi, Michael E. Saks, and Francis Zane. Exponential lower bounds for depth three boolean circuits. Comput. Complex., 9(1):1-15, 2000. URL:
  25. Benjamin Rossman. Criticality of regular formulas. In 34th Computational Complexity Conference, CCC 2019, July 18-20, 2019, New Brunswick, NJ, USA, volume 137 of LIPIcs, pages 1:1-1:28. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL:
  26. Norbert Sauer. A generalization of a theorem of Turán. Journal of Combinatorial Theory, Series B, 10(2):109-112, 1971. URL:
  27. Norbert Sauer. On the density of families of sets. J. Comb. Theory, Ser. A, 13(1):145-147, 1972. URL:
  28. Saharon Shelah. A combinatorial problem; stability and order for models and theories in infinitary languages. Pacific J. Math., 41:247-261, 1972. URL:
  29. Alexander Sidorenko. Exact values of Turán numbers. Mat. Zametki, 42(5):751-760, 764, 1987. Google Scholar
  30. Stéphan Thomassé and Anders Yeo. Total domination of graphs and small transversals of hypergraphs. Comb., 27(4):473-487, 2007. URL:
  31. Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In Mathematical foundations of computer science (Proc. Sixth Sympos., Tatranská Lomnica, 1977), pages 162-176. Lecture Notes in Comput. Sci., Vol. 53, 1977. Google Scholar
  32. Vladimir. N. Vapnik and Alexey. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probab. and its Applications, 16(2):264-280, 1971. Google Scholar
  33. Alexander Aleksandrovich Zykov. On some properties of linear complexes. Matematicheskii sbornik, 66(2):163-188, 1949. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail