Document Open Access Logo

Cubic Formula Size Lower Bounds Based on Compositions with Majority

Authors Anna Gál, Avishay Tal, Adrian Trejo Nuñez

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2019.35.pdf
  • Filesize: 470 kB
  • 13 pages

Document Identifiers

Author Details

Anna Gál
  • The University of Texas at Austin, Austin, TX, USA
Avishay Tal
  • Stanford University, Palo Alto, CA, USA
Adrian Trejo Nuñez
  • The University of Texas at Austin, Austin, TX, USA

Funding

  • Gál, Anna: Part of this work was done while visiting the Simons Institute for the Theory of Computing.
  • Tal, Avishay: Supported by a Motwani Postdoctoral Fellowship and by NSF grant CCF-1763299. Part of this work was done while visiting the Simons Institute for the Theory of Computing.

Cite As Get BibTex

Anna Gál, Avishay Tal, and Adrian Trejo Nuñez. Cubic Formula Size Lower Bounds Based on Compositions with Majority. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ITCS.2019.35

Abstract

We define new functions based on the Andreev function and prove that they require n^{3}/polylog(n) formula size to compute. The functions we consider are generalizations of the Andreev function using compositions with the majority function. Our arguments apply to composing a hard function with any function that agrees with the majority function (or its negation) on the middle slices of the Boolean cube, as well as iterated compositions of such functions. As a consequence, we obtain n^{3}/polylog(n) lower bounds on the (non-monotone) formula size of an explicit monotone function by combining the monotone address function with the majority function.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Circuit complexity
Keywords
  • formula lower bounds
  • random restrictions
  • KRW conjecture
  • composition

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. A. E. Andreev. On a method for obtaining more than quadratic effective lower bounds for the complexity of π-scheme. Moscow University Mathematics Bulletin, 42(1):63-66, 1987. Google Scholar
  2. Andrej Bogdanov. Small Bias Requires Large Formulas. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 22:1-22:12, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.22.
  3. Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel, and David Zuckerman. Mining Circuit Lower Bound Proofs for Meta-Algorithms. Computational Complexity, 24(2):333-392, June 2015. URL: http://dx.doi.org/10.1007/s00037-015-0100-0.
  4. Irit Dinur and Or Meir. Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 3:1-3:51, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.3.
  5. Johan Håstad. The Shrinkage Exponent of de Morgan Formulas is 2. SIAM Journal on Computing, 27(1):48-64, 1998. URL: http://dx.doi.org/10.1137/S0097539794261556.
  6. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American statistical association, 58(301):13-30, 1963. Google Scholar
  7. Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from Shrinkage. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 111-119, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.78.
  8. Russell Impagliazzo and Noam Nisan. The effect of random restrictions on formula size. Random Structures &Algorithms, 4(2):121-133, 1993. Google Scholar
  9. Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5(3-4):191-204, 1995. Google Scholar
  10. V. M. Khrapchenko. Complexity of the realization of a linear function in the class of Π-circuits. Mathematical notes of the Academy of Sciences of the USSR, 9(1):21-23, January 1971. URL: http://dx.doi.org/10.1007/BF01405045.
  11. Ilan Komargodski and Ran Raz. Average-case lower bounds for formula size. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 171-180, 2013. URL: http://dx.doi.org/10.1145/2488608.2488630.
  12. Ilan Komargodski, Ran Raz, and Avishay Tal. Improved Average-Case Lower Bounds for De Morgan Formula Size: Matching Worst-Case Lower Bound. SIAM Journal on Computing, 46(1):37-57, 2017. URL: http://dx.doi.org/10.1137/15M1048045.
  13. Michael S Paterson and Uri Zwick. Shrinkage of de Morgan formulae under restriction. Random Structures &Algorithms, 4(2):135-150, 1993. Google Scholar
  14. P. Pudlak. Personal communication, 2018. Google Scholar
  15. John Riordan and Claude E Shannon. The Number of Two-Terminal Series-Parallel Networks. Studies in Applied Mathematics, 21(1-4):83-93, 1942. Google Scholar
  16. I. S. Sergeev. Complexity and depth of formulas for symmetric Boolean functions. Moscow University Mathematics Bulletin, 71(3):127-130, 2016. Google Scholar
  17. Bella Abramovna Subbotovskaya. Realizations of linear functions by formulas using +, ∗, and -. Doklady Akademii Nauk SSSR, 136(3):553-555, 1961. Google Scholar
  18. Avishay Tal. Shrinkage of De Morgan Formulae by Spectral Techniques. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 551-560, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.65.
  19. Avishay Tal. Formula lower bounds via the quantum method. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 1256-1268, 2017. URL: http://dx.doi.org/10.1145/3055399.3055472.
  20. Leslie G. Valiant. Short monotone formulae for the majority function. Journal of Algorithms, 5(3):363-366, 1984. Google Scholar
  21. I. Wegener. The critical complexity of all (monotone) Boolean functions and monotone graph properties. Information and Control, 67:212-222, 1985. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact