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Small Bias Requires Large Formulas

Author Andrej Bogdanov

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Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • formula lower bounds
  • natural proofs
  • pseudorandomness

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Abstract

A small-biased function is a randomized function whose distribution of truth-tables is small-biased. We demonstrate that known explicit lower bounds on (1) the size of general Boolean formulas, (2) the size of De Morgan formulas, and (3) correlation against small De Morgan formulas apply to small-biased functions. As a consequence, any strongly explicit small-biased generator is subject to the best-known explicit formula lower bounds in all these models.
On the other hand, we give a construction of a small-biased function that is tight with respect to lower bound (1) for the relevant range of parameters. We interpret this construction as a natural-type barrier against substantially stronger lower bounds for general formulas.

Cite As Get BibTex

Andrej Bogdanov. Small Bias Requires Large Formulas. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 22:1-22:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.22

Author Details

Andrej Bogdanov
  • Department of Computer Science and Engineering and , Institute of Theoretical Computer Science and Communications, Chinese University of Hong Kong.

Funding

  • Bogdanov, Andrej: Work supported by HK RGC GRF grant no. CUHK14208215.

References

  1. Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ron M. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. Information Theory, 38(2):509-516, 1992. URL: http://dx.doi.org/10.1109/18.119713.
  2. Noga Alon, Oded Goldreich, Johan Hastad, and René Peralta. Simple construction of almost k-wise independent random variables. Random Struct. Algorithms, 3(3):289-304, 1992. URL: http://dx.doi.org/10.1002/rsa.3240030308.
  3. A. E. Andreev. On a method for obtaining more than quadratic effective lower bounds for the complexity of π-schemes. Moscow Univ. Math. Bull., 42(1):63-66, 1987. Google Scholar
  4. Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, New York, NY, USA, 1st edition, 2009. Google Scholar
  5. Abhishek Banerjee, Chris Peikert, and Alon Rosen. Pseudorandom functions and lattices. In EUROCRYPT, pages 719-737, 2012. URL: http://dx.doi.org/10.1007/978-3-642-29011-4_42.
  6. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778-797, jul 2001. URL: http://dx.doi.org/10.1145/502090.502097.
  7. Avraham Ben-Aroya and Amnon Ta-Shma. Constructing small-bias sets from algebraic-geometric codes. Theory of Computing, 9:253-272, 2013. URL: http://dx.doi.org/10.4086/toc.2013.v009a005.
  8. Ravi B. Boppana and Michael Sipser. The complexity of finite functions. In Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity (A), pages 757-804. Elsevier and MIT Press, 1990. Google Scholar
  9. 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.
  10. Johan Håstad. The shrinkage exponent of de Morgan formulas is 2. SIAM J. Comput., 27(1):48-64, 1998. URL: http://dx.doi.org/10.1137/S0097539794261556.
  11. Russell Impagliazzo and Noam Nisan. The effect of random restrictions on formula size. Random Struct. Algorithms, 4(2):121-134, 1993. URL: http://dx.doi.org/10.1002/rsa.3240040202.
  12. Stasys Jukna. Extremal Combinatorics: With Applications in Computer Science. Springer Publishing Company, Incorporated, 1st edition, 2010. Google Scholar
  13. 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.
  14. Ilan Komargodski, Ran Raz, and Avishay Tal. Improved average-case lower bounds for de Morgan formula size. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 588-597, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.69.
  15. Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput, 22:838-856, 1993. Google Scholar
  16. Moni Naor and Omer Reingold. On the construction of pseudorandom permutations: Luby-Rackoff revisited. J. Cryptology, 12(1):29-66, 1999. URL: http://dx.doi.org/10.1007/PL00003817.
  17. Moni Naor, Omer Reingold, and Alon Rosen. Pseudorandom functions and factoring. SIAM J. Comput., 31(5):1383-1404, 2002. URL: http://dx.doi.org/10.1137/S0097539701389257.
  18. E. I. Nechiporuk. On a Boolean function. Soviet Math. Dokl., 7(4):999-1000, 1966. Google Scholar
  19. Mike Paterson and Uri Zwick. Shrinkage of de Morgan formulae under restriction. Random Struct. Algorithms, 4(2):135-150, 1993. URL: http://dx.doi.org/10.1002/rsa.3240040203.
  20. Alexander A. Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Matematicheskie Zametki, 41(4):598-607, 1987. URL: http://dx.doi.org/10.1007-BF01137685.
  21. Alexander A. Razborov and Steven Rudich. Natural proofs. In STOC, pages 204-213, 1994. URL: http://dx.doi.org/10.1145/195058.195134.
  22. Ben W. Reichardt. Reflections for quantum query algorithms. In Proceedings of the Twenty-second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '11, pages 560-569, Philadelphia, PA, USA, 2011. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2133036.2133080.
  23. Rahul Santhanam. Fighting perebor: New and improved algorithms for formula and QBF satisfiability. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 183-192, 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.25.
  24. Roman Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In STOC, pages 77-82, 1987. URL: http://dx.doi.org/10.1145/28395.28404.
  25. B. A. Subbotovskaya. Realizations of linear functions by formulas using +, ⋅, -. Soviet Math. Dokl., 2:110-112, 1961. Google Scholar
  26. Amnon Ta-Shma. Explicit, almost optimal, epsilon-balanced codes. Electronic Colloquium on Computational Complexity (ECCC), 24:41, 2017. URL: https://eccc.weizmann.ac.il/report/2017/041.
  27. 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.
  28. Avishay Tal. Computing requires larger formulas than approximating. Electronic Colloquium on Computational Complexity (ECCC), 23:179, 2016. URL: http://eccc.hpi-web.de/report/2016/179.
  29. Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theor. Comput. Sci., 47(3):85-93, 1986. URL: http://dx.doi.org/10.1016/0304-3975(86)90135-0.
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