Document Open Access Logo

Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates

Authors Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, Igor C. Oliveira

PDF
Thumbnail PDF

File

LIPIcs.CCC.2020.15.pdf
  • Filesize: 0.74 MB
  • 41 pages

Document Identifiers

Author Details

Valentine Kabanets
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Sajin Koroth
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
Zhenjian Lu
  • School of Computing Science, Simon Fraser University, Burnaby, BC, Canada
  • Department of Computer Science, University of Warwick, UK
Dimitrios Myrisiotis
  • Department of Computing, Imperial College London, UK
Igor C. Oliveira
  • Department of Computer Science, University of Warwick, UK

Funding

  • Oliveira, Igor C.: This work was funded in part by a Royal Society University Research Fellowship (URFโงตR1โงต191059).

Acknowledgements

We would like to thank Rocco Servedio for bringing to our attention the work by Kalai, Mansour, and Verbin [Adam Tauman Kalai et al., 2008], which is a central ingredient in the proof of Theorem 7. We also thank Mahdi Cheraghchi for several discussions on the analysis of Boolean circuits with a bottom layer of parity gates.

Cite AsGet BibTex

Valentine Kabanets, Sajin Koroth, Zhenjian Lu, Dimitrios Myrisiotis, and Igor C. Oliveira. Algorithms and Lower Bounds for De Morgan Formulas of Low-Communication Leaf Gates. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 15:1-15:41, Schloss Dagstuhl โ€“ Leibniz-Zentrum fรผr Informatik (2020)
https://doi.org/10.4230/LIPIcs.CCC.2020.15

Abstract

The class ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [s]โˆ˜๐’ข consists of Boolean functions computable by size-s de Morgan formulas whose leaves are any Boolean functions from a class ๐’ข. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n^{1.99}]โˆ˜๐’ข, for classes ๐’ข of functions with low communication complexity. Let R^(k)(๐’ข) be the maximum k-party number-on-forehead randomized communication complexity of a function in ๐’ข. Among other results, we show that: - The Generalized Inner Product function ๐–ฆ๐–จ๐–ฏ^k_n cannot be computed in ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [s]โˆ˜๐’ข on more than 1/2+ฮต fraction of inputs for s = o(nยฒ/{(kโ‹…4^kโ‹…R^(k)(๐’ข)โ‹…log (n/ฮต)โ‹…log(1/ฮต))ยฒ}). This significantly extends the lower bounds against bipartite formulas obtained by [Avishay Tal, 2017]. As a corollary, we get an average-case lower bound for ๐–ฆ๐–จ๐–ฏ^k_n against ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [n^{1.99}]โˆ˜๐–ฏ๐–ณ๐–ฅ^{k-1}, i.e., sub-quadratic-size de Morgan formulas with degree-(k-1) PTF (polynomial threshold function) gates at the bottom. - There is a PRG of seed length n/2 + O(โˆšsโ‹…R^(2)(๐’ข)โ‹…log(s/ฮต)โ‹…log(1/ฮต)) that ฮต-fools FORMULA[s]โˆ˜๐’ข. For the special case of FORMULA[s]โˆ˜๐–ซ๐–ณ๐–ฅ, i.e., size-s formulas with LTF (linear threshold function) gates at the bottom, we get the better seed length O(n^{1/2}โ‹…s^{1/4}โ‹…log(n)โ‹…log(n/ฮต)). In particular, this provides the first non-trivial PRG (with seed length o(n)) for intersections of n half-spaces in the regime where ฮต โ‰ค 1/n, complementing a recent result of [Ryan O'Donnell et al., 2019]. - There exists a randomized 2^{n-t}-time #SAT algorithm for ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [s]โˆ˜๐’ข, where t = ฮฉ(n/{โˆšsโ‹…logยฒ(s)โ‹…R^(2)(๐’ข)})^{1/2}. In particular, this implies a nontrivial #SAT algorithm for ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [n^1.99]โˆ˜๐–ซ๐–ณ๐–ฅ. - The Minimum Circuit Size Problem is not in ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [n^1.99]โˆ˜๐–ท๐–ฎ๐–ฑ; thereby making progress on hardness magnification, in connection with results from [Igor Carboni Oliveira et al., 2019; Lijie Chen et al., 2019]. On the algorithmic side, we show that the concept class ๐–ฅ๐–ฎ๐–ฑ๐–ฌ๐–ด๐–ซ๐– [n^1.99]โˆ˜๐–ท๐–ฎ๐–ฑ can be PAC-learned in time 2^O(n/log n).

Subject Classification

ACM Subject Classification
  • Theory of computation โ†’ Pseudorandomness and derandomization
  • Theory of computation โ†’ Circuit complexity
  • Theory of computation โ†’ Communication complexity
  • Theory of computation โ†’ Complexity classes
Keywords
  • de Morgan formulas
  • circuit lower bounds
  • satisfiability (SAT)
  • pseudorandom generators (PRGs)
  • learning
  • communication complexity
  • polynomial threshold functions (PTFs)
  • parities

Metrics

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

Related Versions

References

  1. Amir Abboud and Karl Bringmann. Tighter connections between formula-SAT and shaving logs. In ICALP, pages 8:1-8:18, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.8.
  2. Josh Alman, Timothy M. Chan, and R. Ryan Williams. Polynomial representations of threshold functions and algorithmic applications. In FOCS, pages 467-476, 2016. URL: https://doi.org/10.1109/FOCS.2016.57.
  3. Noga Alon, Oded Goldreich, Johan Hรฅstad, and Renรฉ Peralta. Simple construction of almost k-wise independent random variables. Random Struct. Algorithms, 3(3):289-304, 1992. URL: https://doi.org/10.1002/rsa.3240030308.
  4. Andris Ambainis, Andrew M. Childs, Ben Reichardt, Robert Spalek, and Shengyu Zhang. Any AND-OR formula of size N can be evaluated in time N^1/2+o(1) on a quantum computer. SIAM J. Comput., 39(6):2513-2530, 2010. URL: https://doi.org/10.1137/080712167.
  5. Alexander E Andreev. On a method for obtaining more than quadratic effective lower bounds for the complexity of ฯ€-schemes. Moscow Univ. Math. Bull., 42(1):63-66, 1987. Google Scholar
  6. Roy Armoni, Michael E. Saks, Avi Wigderson, and Shiyu Zhou. Discrepancy sets and pseudorandom generators for combinatorial rectangles. In FOCS, pages 412-421, 1996. URL: https://doi.org/10.4230/10.1109/SFCS.1996.548500.
  7. Lรกszlรณ Babai, Noam Nisan, and Mario Szegedy. Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. Syst. Sci., 45(2):204-232, 1992. URL: https://doi.org/10.1016/0022-0000(92)90047-M.
  8. Swapnam Bajpai, Vaibhav Krishan, Deepanshu Kush, Nutan Limaye, and Srikanth Srinivasan. A #sat algorithm for small constant-depth circuits with PTF gates. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, volume 124 of LIPIcs, pages 8:1-8:20. Schloss Dagstuhl - Leibniz-Zentrum fรผr Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.8.
  9. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  10. Dan Boneh, Yuval Ishai, Alain Passelรจgue, Amit Sahai, and David J. Wu. Exploring crypto dark matter: New simple PRF candidates and their applications. In TCC, pages 699-729, 2018. URL: https://doi.org/10.1007/978-3-030-03810-6_25.
  11. Harry Buhrman, Ilan Newman, Hein Rรถhrig, and Ronald de Wolf. Robust polynomials and quantum algorithms. Theory Comput. Syst., 40(4):379-395, 2007. URL: https://doi.org/10.1007/s00224-006-1313-z.
  12. Lijie Chen, Ce Jin, and Ryan Williams. Hardness magnification for all sparse NP languages. In FOCS, 2019. URL: https://eccc.weizmann.ac.il/report/2019/118/.
  13. Lijie Chen and Ruosong Wang. Classical algorithms from quantum and Arthur-Merlin communication protocols. In ITCS, pages 23:1-23:20, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.23.
  14. 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, 2015. URL: https://doi.org/10.1007/s00037-015-0100-0.
  15. Ruiwen Chen, Rahul Santhanam, and Srikanth Srinivasan. Average-case lower bounds and satisfiability algorithms for small threshold circuits. Theory of Computing, 14(1):1-55, 2018. URL: https://doi.org/10.4086/toc.2018.v014a009.
  16. Mahdi Cheraghchi, Valentine Kabanets, Zhenjian Lu, and Dimitrios Myrisiotis. Circuit lower bounds for MCSP from local pseudorandom generators. In ICALP, pages 39:1-39:14, 2019. URL: http://eccc.hpi-web.de/report/2019/022.
  17. Don Coppersmith. Rapid multiplication of rectangular matrices. SIAM J. Comput., 11(3):467-471, 1982. URL: https://doi.org/10.1137/0211037.
  18. Evgeny Dantsin and Edward A Hirsch. Worst-case upper bounds. Handbook of Satisfiability, 185:403-424, 2009. Google Scholar
  19. Irit Dinur and Or Meir. Toward the KRW composition conjecture: Cubic formula lower bounds via communication complexity. Computational Complexity, 27(3):375-462, 2018. URL: https://doi.org/10.1007/s00037-017-0159-x.
  20. Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A quantum algorithm for the hamiltonian NAND tree. Theory of Computing, 4(1):169-190, 2008. URL: https://doi.org/10.4086/toc.2008.v004a008.
  21. Bill Fefferman, Ronen Shaltiel, Christopher Umans, and Emanuele Viola. On beating the hybrid argument. Theory of Computing, 9:809-843, 2013. URL: https://doi.org/10.4086/toc.2013.v009a026.
  22. Yoav Freund. Boosting a weak learning algorithm by majority. In COLT, pages 202-216, 1990. URL: http://dl.acm.org/citation.cfm?id=92640.
  23. Mika Gรถรถs, Toniann Pitassi, and Thomas Watson. The landscape of communication complexity classes. Computational Complexity, 27(2):245-304, 2018. URL: https://doi.org/10.1007/s00037-018-0166-6.
  24. Parikshit Gopalan, Ryan O'Donnell, Yi Wu, and David Zuckerman. Fooling functions of halfspaces under product distributions. In CCC, pages 223-234, 2010. URL: http://arxiv.org/abs/1001.1593.
  25. Lov K. Grover. A fast quantum mechanical algorithm for database search. In STOC, pages 212-219, 1996. URL: https://doi.org/10.1145/237814.237866.
  26. Prahladh Harsha and Srikanth Srinivasan. On polynomial approximations to AC. Random Struct. Algorithms, 54(2):289-303, 2019. URL: https://doi.org/10.1002/rsa.20786.
  27. Johan Hรฅstad. The shrinkage exponent of de Morgan formulas is 2. SIAM J. Comput., 27(1):48-64, 1998. URL: https://doi.org/10.1137/S0097539794261556.
  28. Peter Hรธyer, Troy Lee, and Robert Spalek. Negative weights make adversaries stronger. In STOC, pages 526-535, 2007. URL: https://doi.org/10.1145/1250790.1250867.
  29. Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from shrinkage. In FOCS, pages 111-119, 2012. URL: https://eccc.weizmann.ac.il/report/2012/057/.
  30. Russell Impagliazzo and Noam Nisan. The effect of random restrictions on formula size. Random Struct. Algorithms, 4(2):121-134, 1993. URL: https://doi.org/10.1002/rsa.3240040202.
  31. Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In STOC, pages 356-364, 1994. URL: https://doi.org/10.1145/195058.195190.
  32. Stasys Jukna. Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and combinatorics. Springer, 2012. Google Scholar
  33. Valentine Kabanets. Derandomization: A brief overview. Current Trends in Theoretical Computer Science, 1:165-188, 2002. Google Scholar
  34. Valentine Kabanets and Zhenjian Lu. Satisfiability and derandomization for small polynomial threshold circuits. In APPROX/RANDOM, pages 46:1-46:19, 2018. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.46.
  35. Adam Tauman Kalai, Adam R. Klivans, Yishay Mansour, and Rocco A. Servedio. Agnostically learning halfspaces. SIAM J. Comput., 37(6):1777-1805, 2008. URL: https://doi.org/10.1137/060649057.
  36. Adam Tauman Kalai, Yishay Mansour, and Elad Verbin. On agnostic boosting and parity learning. In STOC, pages 629-638, 2008. URL: https://doi.org/10.1145/1374376.1374466.
  37. Michael J. Kearns and Umesh V. Vazirani. An Introduction to Computational Learning Theory. MIT Press, 1994. URL: https://mitpress.mit.edu/books/introduction-computational-learning-theory.
  38. Valeriy M Khrapchenko. Method of determining lower bounds for the complexity of ฯ€-schemes. Mathematical Notes, 10(1):474-479, 1971. Google Scholar
  39. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. Google Scholar
  40. Sophie Laplante, Troy Lee, and Mario Szegedy. The quantum adversary method and classical formula size lower bounds. Computational Complexity, 15(2):163-196, 2006. URL: https://doi.org/10.1007/s00037-006-0212-7.
  41. Saburo Muroga, Iwao Toda, and Satoru Takasu. Theory of majority decision elements. Journal of the Franklin Institute, 271:376-418, 1961. URL: https://doi.org/10.1016/0016-0032(61)90702-5.
  42. E.I. Nechiporuk. On a Boolean function. Doklady Akademii Nauk SSSR, 169(4):765-766, 1966. English translation in Soviet Mathematics Doklady. Google Scholar
  43. Noam Nisan. The communication complexity of threshold gates. In Proceedings of "Combinatorics, Paul Erdos is Eighty", pages 301-315, 1994. Google Scholar
  44. Ryan O'Donnell, Rocco A. Servedio, and Li-Yang Tan. Fooling polytopes. In STOC, pages 614-625, 2019. URL: http://arxiv.org/abs/1808.04035.
  45. Igor Carboni Oliveira, Jรกn Pich, and Rahul Santhanam. Hardness magnification near state-of-the-art lower bounds. In CCC, pages 27:1-27:29, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.27.
  46. Igor Carboni Oliveira and Rahul Santhanam. Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness. In CCC, pages 18:1-18:49, 2017. URL: https://eccc.weizmann.ac.il/report/2016/197/.
  47. Mike Paterson and Uri Zwick. Shrinkage of de Morgan formulae under restriction. Random Struct. Algorithms, 4(2):135-150, 1993. URL: https://doi.org/10.1002/rsa.3240040203.
  48. Mihai Patrascu and Ryan Williams. On the possibility of faster SAT algorithms. In SODA, pages 1065-1075, 2010. URL: https://doi.org/10.1137/1.9781611973075.86.
  49. Pavel Pudlรกk, Vojtech Rรถdl, and Petr Savickรฝ. Graph complexity. Acta Inf., 25(5):515-535, 1988. URL: https://doi.org/10.1007/BF00279952.
  50. Ben Reichardt. Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function. In FOCS, pages 544-551, 2009. URL: https://doi.org/10.1109/FOCS.2009.55.
  51. Ben Reichardt. Faster quantum algorithm for evaluating game trees. In SODA, pages 546-559, 2011. URL: http://arxiv.org/abs/0907.1623.
  52. Ben Reichardt. Reflections for quantum query algorithms. In SODA, pages 560-569, 2011. URL: http://arxiv.org/abs/1005.1601.
  53. Ben Reichardt and Robert Spalek. Span-program-based quantum algorithm for evaluating formulas. Theory of Computing, 8(1):291-319, 2012. URL: https://doi.org/10.4086/toc.2012.v008a013.
  54. Igor S Sergeev. Upper bounds for the size and the depth of formulae for MOD-functions. Discrete Mathematics and Applications, 27(1):15-22, 2017. Google Scholar
  55. Rocco A. Servedio and Li-Yang Tan. What circuit classes can be learned with non-trivial savings? In ITCS, pages 30:1-30:21, 2017. URL: https://doi.org/10.4230/LIPIcs.ITCS.2017.30.
  56. Rocco A. Servedio and Li-Yang Tan. Improved pseudorandom generators from pseudorandom multi-switching lemmas. In APPROX/RANDOM, pages 45:1-45:23, 2019. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.45.
  57. Bella Abramovna Subbotovskaya. Realization of linear functions by formulas using โˆจ, &, -. In Doklady Akademii Nauk, volume 136, pages 553-555. Russian Academy of Sciences, 1961. Google Scholar
  58. Avishay Tal. Shrinkage of de Morgan formulae by spectral techniques. In FOCS, pages 551-560, 2014. URL: https://eccc.weizmann.ac.il/report/2014/048/.
  59. Avishay Tal. #SAT algorithms from shrinkage. Electronic Colloquium on Computational Complexity (ECCC), 22:114, 2015. URL: http://eccc.hpi-web.de/report/2015/114.
  60. Avishay Tal. The bipartite formula complexity of inner-product is quadratic. Electronic Colloquium on Computational Complexity (ECCC), 23:181, 2016. URL: http://eccc.hpi-web.de/report/2016/181.
  61. Avishay Tal. Formula lower bounds via the quantum method. In STOC, pages 1256-1268, 2017. URL: https://doi.org/10.1145/3055399.3055472.
  62. Salil P. Vadhan. Pseudorandomness. Foundations and Trends in Theoretical Computer Science, 7(1-3):1-336, 2012. URL: https://doi.org/10.1561/0400000010.
  63. Leslie G. Valiant. A theory of the learnable. In STOC, pages 436-445, 1984. URL: https://doi.org/10.1145/800057.808710.
  64. Emanuele Viola. The communication complexity of addition. Combinatorica, 35(6):703-747, 2015. URL: https://doi.org/10.1145/10.1007/s00493-014-3078-3.
  65. Ryan Williams. Nonuniform ACC circuit lower bounds. J. ACM, 61(1):2:1-2:32, 2014. URL: https://doi.org/10.1109/10.1145/2559903.
  66. Andrew Chi-Chih Yao. Theory and applications of trapdoor functions. In FOCS, pages 80-91, 1982. URL: https://doi.org/10.1109/SFCS.1982.45.
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 โ€“ LZI GmbH Imprint Privacy Contact