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Circuit Depth Reductions

Authors Alexander Golovnev, Alexander S. Kulikov, R. Ryan Williams

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Author Details

Alexander Golovnev
  • Georgetown University, Washington, DC, USA
Alexander S. Kulikov
  • Steklov Institute of Mathematics at St. Petersburg, Russia
  • St. Petersburg State University, Russia
R. Ryan Williams
  • CSAIL & EECS, MIT, Cambridge, MA, USA

Funding

  • Kulikov, Alexander S.: Research presented in Section 4 is supported by the RNF grant 18-71-10042.
  • Williams, R. Ryan: Supported by NSF CCF-1909429 and CCF-1741615.

Cite AsGet BibTex

Alexander Golovnev, Alexander S. Kulikov, and R. Ryan Williams. Circuit Depth Reductions. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.24

Abstract

The best known size lower bounds against unrestricted circuits have remained around 3n for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than 5n. In this work, we propose a non-gate-elimination approach for obtaining circuit lower bounds, via certain depth-three lower bounds. We prove that every (unbounded-depth) circuit of size s can be expressed as an OR of 2^{s/3.9} 16-CNFs. For DeMorgan formulas, the best known size lower bounds have been stuck at around n^{3-o(1)} for decades. Under a plausible hypothesis about probabilistic polynomials, we show that n^{4-ε}-size DeMorgan formulas have 2^{n^{1-Ω(ε)}}-size depth-3 circuits which are approximate sums of n^{1-Ω(ε)}-degree polynomials over F₂. While these structural results do not immediately lead to new lower bounds, they do suggest new avenues of attack on these longstanding lower bound problems. Our results complement the classical depth-3 reduction results of Valiant, which show that logarithmic-depth circuits of linear size can be computed by an OR of 2^{ε n} n^δ-CNFs, and slightly stronger results for series-parallel circuits. It is known that no purely graph-theoretic reduction could yield interesting depth-3 circuits from circuits of super-logarithmic depth. We overcome this limitation (for small-size circuits) by taking into account both the graph-theoretic and functional properties of circuits and formulas. We show that improvements of the following pseudorandom constructions imply super-linear circuit lower bounds for log-depth circuits via Valiant’s reduction: dispersers for varieties, correlation with constant degree polynomials, matrix rigidity, and hardness for depth-3 circuits with constant bottom fan-in. On the other hand, our depth reductions show that even modest improvements of the known constructions give elementary proofs of improved (but still linear) circuit lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Circuit complexity
  • formula complexity
  • pseudorandomness
  • matrix rigidity

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References

  1. Josh Alman and Ryan Williams. Probabilistic polynomials and hamming nearest neighbors. In FOCS 2015, pages 136-150. IEEE, 2015. Google Scholar
  2. Alexander 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
  3. Ravi B. Boppana. The average sensitivity of bounded-depth circuits. Inf. Process. Lett., 63(5):257-261, 1997. Google Scholar
  4. Chris Calabro. A lower bound on the size of series-parallel graphs dense in long paths. In ECCC, volume 15, 2008. Google Scholar
  5. Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi. A duality between clause width and clause density for SAT. In CCC 2006, pages 252-260, 2006. Google Scholar
  6. Aleksandr V. Chashkin. On the complexity of Boolean matrices, graphs and their corresponding Boolean functions. Discrete Math. and Appl., 4(3):229-257, 1994. Google Scholar
  7. Lijie Chen and Ryan Williams. Circuit lower bounds from PCP of proximity. Unpublished manuscript, 2019. Google Scholar
  8. Ruiwen Chen and Valentine Kabanets. Correlation bounds and #SAT algorithms for small linear-size circuits. In COCOON 2015, pages 211-222. Springer, 2015. Google Scholar
  9. Gil Cohen and Avishay Tal. Two structural results for low degree polynomials and applications. In RANDOM 2015, pages 680-709, 2015. Google Scholar
  10. Vlado Dančík. Complexity of Boolean functions over bases with unbounded fan-in gates. Inf. Process. Lett., 57(1):31-34, 1996. Google Scholar
  11. Evgeny Demenkov, Arist Kojevnikov, Alexander S. Kulikov, and Grigory Yaroslavtsev. New upper bounds on the boolean circuit complexity of symmetric functions. Inf. Process. Lett., 110(7):264-267, 2010. Google Scholar
  12. Irit Dinur and Or Meir. Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity. In CCC 2016, pages 3:1-3:51, 2016. Google Scholar
  13. Paul Erdös, Ronald L. Graham, and Endre Szemerédi. On sparse graphs with dense long paths. Comp. and Math. with Appl., 1:145-161, 1975. Google Scholar
  14. Magnus G. 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 FOCS 2016, pages 89-98, 2016. Google Scholar
  15. Joel Friedman. A note on matrix rigidity. Combinatorica, 13(2):235-239, 1993. Google Scholar
  16. Dmitry Gavinsky, Or Meir, Omri Weinstein, and Avi Wigderson. Toward better formula lower bounds: An information complexity approach to the KRW composition conjecture. In STOC 2014, pages 213-222, 2014. Google Scholar
  17. 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. Google Scholar
  18. Alexander Golovnev, Alexander S. Kulikov, and R. Ryan Williams. Circuit depth reductions. arXiv, 2018. URL: http://arxiv.org/abs/1811.04828.
  19. Dmitrii Yu. Grigoriev. Application of separability and independence notions for proving lower bounds of circuit complexity. Zap. Nauch. Sem. POMI, 60:38-48, 1976. Google Scholar
  20. Johan Håstad. Almost optimal lower bounds for small depth circuits. In STOC 1986, pages 6-20, 1986. Google Scholar
  21. Johan Håstad. The shrinkage exponent of de Morgan formulas is 2. SIAM J. Comput., 27(1):48-64, 1998. Google Scholar
  22. Johan Håstad, Stasys Jukna, and Pavel Pudlák. Top-down lower bounds for depth 3 circuits. In FOCS 1993, pages 124-129, 1993. Google Scholar
  23. Russell Impagliazzo and Valentine Kabanets. Fourier concentration from shrinkage. Comput. Complex., 26(1):275-321, 2017. Google Scholar
  24. Russell Impagliazzo and Noam Nisan. The effect of random restrictions on formula size. Random Struct. Algorithms, 4(2):121-134, 1993. Google Scholar
  25. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. Google Scholar
  26. Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Comput. Complex., 5(3/4):191-204, 1995. Google Scholar
  27. Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM J. Discrete Math., 3(2):255-265, 1990. Google Scholar
  28. Tali Kaufman, Shachar Lovett, and Ely Porat. Weight distribution and list-decoding size of reed-muller codes. IEEE Trans. Inf. Theory, 58(5):2689-2696, 2012. Google Scholar
  29. Valeriy M. Khrapchenko. A method of determining lower bounds for the complexity of π-schemes. Math. Notes of the Acad. of Sci. of the USSR, 10(1):474-479, 1971. Google Scholar
  30. Maria M. Klawe. Shallow grates. Theor. Comput. Sci., 123(2):389-395, 1994. Google Scholar
  31. Ilan Komargodski, Ran Raz, and Avishay Tal. Improved average-case lower bounds for demorgan formula size. In FOCS 2013, pages 588-597, 2013. Google Scholar
  32. Satyanarayana V. Lokam. Complexity lower bounds using linear algebra. Found. Trends Theor. Comput. Sci., 4(1-2):1-155, 2009. Google Scholar
  33. Oleg B. Lupanov. On rectifier and switching-and-rectifier schemes. Dokl. Akad. Nauk SSSR, 111(6):1171-1174, 1956. In Russian. Google Scholar
  34. Or Meir and Avi Wigderson. Prediction from partial information and hindsight, with application to circuit lower bounds. In ECCC, volume 24, 2017. Google Scholar
  35. Edward I. Nechiporuk. On a Boolean function. Dokl. Akad. Nauk SSSR, 169(4):765-766, 1966. Google Scholar
  36. Mike Paterson and Uri Zwick. Shrinkage of de Morgan formulae under restriction. Random Struct. Algorithms, 4(2):135-150, 1993. Google Scholar
  37. Ramamohan Paturi and Pavel Pudlák. Circuit lower bounds and linear codes. J. Math. Sci., 134(5):2425-2434, 2006. Google Scholar
  38. 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. Google Scholar
  39. Ramamohan Paturi, Pavel Pudlák, and Francis Zane. Satisfiability coding lemma. In FOCS 1997, pages 566-574, 1997. Google Scholar
  40. Ramamohan Paturi, Michael E. Saks, and Francis Zane. Exponential lower bounds for depth 3 Boolean circuits. In STOC 1997, pages 86-91, 1997. Google Scholar
  41. Pavel Pudlák and Zdeněk Vavřín. Computation of rigidity of order n²/r for one simple matrix. Comment. Math. Univ. Carolinae, 32(2):213-218, 1991. Google Scholar
  42. Alexander A. Razborov. Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Mat. Zametki, 41(4):598-607, 1987. Google Scholar
  43. Ben W. Reichardt. Reflections for quantum query algorithms. In SODA 2011, pages 560-569. SIAM, 2011. Google Scholar
  44. Zachary Remscrim. The Hilbert function, algebraic extractors, and recursive fourier sampling. In FOCS 2016, pages 197-208, 2016. Google Scholar
  45. Rahul Santhanam. Fighting perebor: New and improved algorithms for formula and QBF satisfiability. In FOCS 2010, pages 183-192, 2010. Google Scholar
  46. Rahul Santhanam and Srikanth Srinivasan. On the limits of sparsification. In ICALP 2012, pages 774-785, 2012. Google Scholar
  47. Georg Schnitger. A family of graphs with expensive depth-reduction. Theor. Comput. Sci., 18(1):89-93, 1982. Google Scholar
  48. Georg Schnitger. On depth-reduction and grates. In FOCS 1983, pages 323-328, 1983. Google Scholar
  49. Rainer Schuler. An algorithm for the satisfiability problem of formulas in conjunctive normal form. J. Algorithms, 54(1):40-44, 2005. Google Scholar
  50. Igor S. Sergeev. On complexity of circuits and formulas of bounded depth over unbounded fan-in bases. Disc. Math. Appl., 30(2):120-137, 2018. In Russian. Google Scholar
  51. Kazuhisa Seto and Suguru Tamaki. A satisfiability algorithm and average-case hardness for formulas over the full binary basis. Comput. Complex., 22(2):245-274, 2013. Google Scholar
  52. Claude E. Shannon. The synthesis of two-terminal switching circuits. Bell Syst. Tech. J., 28:59-98, 1949. Google Scholar
  53. Alexander A Sherstov. Making polynomials robust to noise. In STOC 2012, pages 747-758. ACM, 2012. Google Scholar
  54. Bella A. Subbotovskaya. Realizations of linear functions by formulas using +,⋅,-. Dokl. Akad. Nauk SSSR, 136(3):553-555, 1961. Google Scholar
  55. Avishay Tal. Shrinkage of De Morgan formulae by spectral techniques. In FOCS 2014, pages 551-560. IEEE, 2014. Google Scholar
  56. Leslie G. Valiant. Graph-theoretic arguments in low-level complexity. In MFCS 1977, pages 162-176, 1977. Google Scholar
  57. Emanuele Viola. On the power of small-depth computation. Found. Trends Theor. Comput. Sci., 5(1):1-72, 2009. Google Scholar
  58. Emanuele Viola and Avi Wigderson. Norms, XOR lemmas, and lower bounds for polynomials and protocols. Theory Comput., 4(1):137-168, 2008. Google Scholar
  59. Richard Ryan Williams. Limits on representing Boolean functions by linear combinations of simple functions: Thresholds, ReLUs, and low-degree polynomials. In CCC 2018, pages 6:1-6:24, 2018. Google Scholar
  60. Guy Wolfovitz. The complexity of depth-3 circuits computing symmetric Boolean functions. Inf. Process. Lett., 100(2):41-46, 2006. Google Scholar
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