Abstract
While it can be easily shown by counting that almost all Boolean predicates of n variables have circuit size \(\Omega(2^n/n)\), we have no example of an NP function requiring even a superlinear number of gates. Moreover, only modest linear lower bounds are known. Until recently, the strongest known lower bound was \(3n - o(n)\) presented by Blum in 1984. In 2011, Demenkov and Kulikov presented a much simpler proof of the same lower bound, but for a more complicated function —an affine disperser for sublinear dimension. Informally, this is a function that is resistant to any \(n - o(n)\) affine substitutions. In 2011, Ben-Sasson and Kopparty gave an explicit construction of such a function. The proof of the lower bound basically goes by showing that for any circuit there exists an affine hyperplane where the function complexity decreases at least by three gates. In this paper, we prove the following two extensions.
1. A \((3+\frac {1}{86}\, n - o(n)\) lower bound for the circuit size of an affine disperser for sublinear dimension. The proof is based on the gate elimination technique extended with the following three ideas: (i) generalizing the computational model by allowing circuits to contain cycles, this in turn allows us to perform affine substitutions, (ii) a carefully chosen circuit complexity measure to track the progress of the gate elimination process, and (iii) quadratic substitutions that may be viewed as delayed affine substitutions.
2. A much simpler proof of a stronger lower bound of \(3.11n\) for a quadratic disperser. Informally, a quadratic disperser is resistant to sufficiently many substitutions of the form \(x \leftarrow p\), where p is a polynomial of degree at most two. Currently, there are no constructions of quadratic dispersers in NP (although there are constructions over large fields, and constructions with weaker parameters over GF(2)). The key ingredient of this proof is the induction on the size of the underlying quadratic variety instead of the number of variables as in the previously known proofs.
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References
Kazuyuki Amano & Jun Tarui (2011). A well-mixed function with circuit complexity 5n: Tightness of the Lachish-Raz-type bounds. Theor. Comput. Sci. 412(18), 1646-1651. URL http://dx.doi.org/10.1016/j.tcs.2010.12.040.
Alexander E. Andreev (1987). On a method for obtaining more than quadratic effective lower bounds for the complexity of \(\prod\)-schemes. Moscow Univ. Math. Bull. 42(1), 63-66.
Eli Ben-Sasson & Ariel Gabizon (2012). Extractors for polynomials sources over constant-size fields of small characteristic. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 399-410. Springer.
Eli Ben-Sasson & Swastik Kopparty (2009). Affine dispersers from subspace polynomials. In Proceedings of the Annual Symposium on Theory of Computing (STOC), volume 679, 65-74. ACM Press. ISBN 9781605585062. URL http://portal.acm.org/citation.cfm?doid=1536414.1536426.
Eli Ben-Sasson & Emanuele Viola (2014). Short PCPs with Projection Queries. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, Javier Esparza, Pierre Fraigniaud, Thore Husfeldt & Elias Koutsoupias, editors, volume 8572 of Lecture Notes in Computer Science, 163-173. Springer. ISBN 978-3-662-43947-0. URL http://dx.doi.org/10.1007/978-3-662-43948-7.
Norbert Blum (1984). A Boolean Function Requiring 3n Network Size. Theor. Comput. Sci. 28, 337-345. URL http://dx.doi.org/10.1016/0304-3975(83)90029-4.
Harry Buhrman, Lance Fortnow & Thomas Thierauf (1998). Nonrelativizing Separations. In CCC-98.
Jin-Yi Cai (2001). \(S_{2} \subseteq ZPP^{NP}\). In Proceedings 2001 IEEE International Conference on Cluster Computing, 620-628.
Eshan Chattopadhyay, Jesse Goodman & Jyun-Jie Liao (2022). Affine extractors for almost logarithmic entropy. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), 622-633. IEEE.
Ruiwen Chen & Valentine Kabanets (2015). Correlation Bounds and #SAT Algorithms for Small Linear-Size Circuits. In Computing and Combinatorics - 21st International Conference, COCOON 2015, Beijing, China, August 4-6, 2015, Proceedings, Dachuan Xu, Donglei Du & Dingzhu Du, editors, volume 9198 of Lecture Notes in Computer Science, 211-222. Springer. ISBN 978-3-319-21397-2. URL http://dx.doi.org/10.1007/978-3-319-21398-9.
Ruiwen Chen, Valentine Kabanets, Antonina Kolokolova, Ronen Shaltiel & David Zuckerman (2015). Mining Circuit Lower Bound Proofs for Meta-Algorithms. Computational Complexity 24(2), 333-392. URL http://dx.doi.org/10.1007/s00037-015-0100-0.
Gil Cohen & Igor Shinkar (2014). The Complexity of DNF of Parities. Technical Report 99, Electronic Colloquium on Computational Complexity.
Gil Cohen & Avishay Tal (2015). Two Structural Results for Low Degree Polynomials and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2015, August 24-26, 2015, Princeton, NJ, USA, Naveen Garg, Klaus Jansen, Anup Rao & JOSÉ D. P. Rolim, editors, volume 40 of LIPIcs, 680-709. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. ISBN 978-3-939897-89-7. URL http://dx.doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.680.
Evgeny Demenkov & Alexander S. Kulikov (2011). An Elementary Proof of a 3n - o(n) Lower Bound on the Circuit Complexity of Affine Dispersers. In Proceedings of 36th International Symposium on Mathematical Foundations of Computer Science (MFCS), volume 6907 of Lecture Notes in Computer Science, 256-265. Springer.
Evgeny Demenkov, Alexander S. Kulikov, Olga Melanich & Ivan Mihajlin (2015). New Lower Bounds on Circuit Size of Multioutput Functions. Theory Comput. Syst. 56(4), 630-642. URL http://dx.doi.org/10.1007/s00224-014-9590-4.
Zeev Dvir (2012). Extractors for varieties. Computational complexity 21(4), 515-572.
Patrick W. Dymond & Stephen A. Cook (1989). Complexity Theory of Parallel Time and Hardware. Inf. Comput. 80(3), 205-226. URL http://dx.doi.org/10.1016/0890-5401(89)90009-6.
Magnus G. Find, Alexander Golovnev, Edward A. Hirsch & Alexander S. Kulikov (2016). A better-than-3n lower bound for the circuit complexity of an explicit function. In 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October, 2016, New Brunswick, NJ, USA, 89-98. IEEE Computer Society.
Fedor V. Fomin, Fabrizio Grandoni & Dieter Kratsch (2009). A measure & conquer approach for the analysis of exact algorithms. J. ACM 56(5). URL http://doi.acm.org/10.1145/1552285.1552286.
Alexander Golovnev, Edward A. Hirsch, Alexander Knop & Alexander S. Kulikov (2016). On the Limits of Gate Elimination. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), volume 58 of Leibniz International Proceedings in Informatics (LIPIcs), 46:1-46:13.
Alexander Golovnev & Alexander S. Kulikov (2016). Weighted Gate Elimination: Boolean Dispersers for Quadratic Varieties Imply Improved Circuit Lower Bounds. In Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science, Cambridge, MA, USA, January 14-16, 2016, Madhu Sudan, editor, 405-411. ACM. ISBN 978-1-4503-4057-1. URL http://doi.acm.org/10.1145/2840728.2840755.
Johan Håstad (1986). Almost Optimal Lower Bounds for Small Depth Circuits. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, Juris Hartmanis, editor, 6-20. ACM. ISBN 0-89791-193-8. URL http://doi.acm.org/10.1145/12130.12132.
Johan Håstad (1998). The Shrinkage Exponent of de Morgan Formulas is 2. SIAM J. Comput. 27(1), 48-64. URL http://dx.doi.org/10.1137/S0097539794261556.
Erhard Heinz (1951). Beiträge zur Störungstheorie der Spektralzerleung. Mathematische Annalen 123(1), 415-438.
Russell Impagliazzo, Valentine Kabanets & Ilya Volkovich (2018). The power of natural properties as oracles. In LIPIcs-Leibniz International Proceedings in Informatics, volume 102. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Russell Impagliazzo & Noam Nisan (1993). The Effect of Random Restrictions on Formula Size. Random Struct. Algorithms 4(2), 121-134. URL http://dx.doi.org/10.1002/rsa.3240040202.
Kazuo Iwama & Hiroki Morizumi (2002). An Explicit Lower Bound of 5n - o(n) for Boolean Circuits. In Mathematical Foundations of Computer Science 2002, 27th International Symposium, MFCS 2002, Warsaw, Poland, August 26-30, 2002, Proceedings, Krzysztof Diks & Wojciech Rytter, editors, volume 2420 of Lecture Notes in Computer Science, 353-364. Springer. ISBN 3-540-44040-2.
Valeriy M. Khrapchenko (1971). A method of determining lower bounds for the complexity of \(\prod\)-schemes. Math. Notes of the Acad. of Sci. of the USSR 10(1), 474-479.
Boris M. Kloss & Vadim A. Malyshev (1965). Estimates of the complexity of certain classes of functions. Vestn.Moskov.Univ.Ser.1 4, 44-51. In Russian.
Donald E. Knuth (2015). The Art of Computer Programming, volume 4, pre-fascicle 6a. Addison-Wesley. Section 7.2.2.2. Satisfiability. Draft available at http://www-cs-faculty.stanford.edu/~uno/fasc6a.ps.gz.
Arist Kojevnikov & Alexander S. Kulikov (2006). A new approach to proving upper bounds for MAX-2-SAT. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, 11-17. ACM Press. ISBN 0-89871-605-5. URL http://dl.acm.org/citation.cfm?id=1109557.1109559.
Arist Kojevnikov & Alexander S. Kulikov (2010). Circuit Complexity and Multiplicative Complexity of Boolean Functions. In Programs, Proofs, Processes, 6th Conference on Computability in Europe, CiE 2010, Fernando Ferreira, Benedikt L"owe, Elvira Mayordomo & Lu'is Mendes Gomes, editors, volume 6158 of Lecture Notes in Computer Science, 239-245. Springer. ISBN 978-3-642-13961-1. URL http://dx.doi.org/10.1007/978-3-642-13962-8.
Ilan Komargodski, Ran Raz & Avishay Tal (2013). Improved Average-Case Lower Bounds for DeMorgan Formula Size. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, 588-597. IEEE Computer Society. ISBN 978-0-7695-5135-7. URL http://dx.doi.org/10.1109/FOCS.2013.69.
Alexander S. Kulikov & Nikita Slezkin (2021). SAT-based Circuit Local Improvement. CoRR abs/2102.12579. URL https://arxiv.org/abs/2102.12579.
Oliver Kullmann (1999). New Methods for 3-SAT Decision and Worst-case Analysis. Theor. Comput. Sci. 223(1-2), 1-72. URL http://dx.doi.org/10.1016/S0304-3975(98)00017-6.
Oded Lachish & Ran Raz (2001). Explicit lower bound of \(4.5n-o(n)\) for Boolean circuits. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, Jeffrey Scott Vitter, Paul G. Spirakis & Mihalis Yannakakis, editors, 399-408. ACM. ISBN 1-58113-349-9. URL ahttp://doi.acm.org/10.1145/380752.380832.
Jiatu Li & Tianqi Yang (2022). \(3.1n - o(n)\) Circuit Lower Bounds for Explicit Functions. In Proceedings on 54th Annual ACM Symposium on Theory of Computing. To appear.
Xin Li (2011). A New Approach to Affine Extractors and Dispersers. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC 2011, San Jose, California, June 8-10, 2011, 137-147. IEEE Computer Society. URL http://dx.doi.org/10.1109/CCC.2011.27.
Xin Li (2016). Improved two-source extractors, and affine extractors for polylogarithmic entropy. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), 168-177. IEEE.
A. A. Lialina (2018). On the complexity of unique circuit SAT. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 475(Kombinatorika i Teoriya Grafov. X), 122-136. ISSN 0373-2703.
Edward I. Nechiporuk (1966). On a Boolean function. Doklady Akademii Nauk. SSSR 169(4), 765-766.
Arfst Nickelsen, Till Tantau & Lorenz Weizs"acker (2004). Aggregates with Component Size One Characterize Polynomial Space. Electronic Colloquium on Computational Complexity (ECCC) 028. URL http://eccc.hpi-web.de/eccc-reports/2004/TR04-028/index.html.
Sergey Nurk (2009). An \(2^{0.4058m}\) upper bound for Circuit SAT. Technical Report 10, Steklov Institute of Mathematics at St.Petersburg. PDMI Preprint.
Mike Paterson & Uri Zwick (1993). Shrinkage of de Morgan Formulae under Restriction. Random Struct. Algorithms 4(2), 135-150. URL http://dx.doi.org/10.1002/rsa.3240040203.
Wolfgang J. Paul (1977). A 2.5n-Lower Bound on the Combinational Complexity of Boolean Functions. SIAM J. Comput. 6(3), 427-443. URL http://dx.doi.org/10.1137/0206030.
Alexander A. Razborov (1985). Lower bound on monotone complexity of some Boolean functions. Doklady Akademii Nauk. SSSR 281(4), 798-801.
Marc D. Riedel & Jehoshua Bruck (2012). Cyclic Boolean circuits. Discrete Applied Mathematics 160(13-14), 1877-1900. URL http://dx.doi.org/10.1016/j.dam.2012.03.039.
Ronald L. Rivest (1977). The Necessity of Feedback in Minimal Monotone Combinational Circuits. IEEE Trans. Computers 26(6), 606-607. URL http://doi.ieeecomputersociety.org/10.1109/TC.1977.1674886.
Rahul Santhanam (2009). Circuit lower bounds for Merlin-Arthur classes. SIAM J. Comput. 39(3), 1038-1061.
Rahul Santhanam (2010). 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, 183-192. IEEE Computer Society. ISBN 978-0-7695-4244-7. URL http://dx.doi.org/10.1109/FOCS.2010.25.
Sergey Savinov (2014). Upper bounds for the Boolean circuit satisfiability problem. Master Thesis defended at St.Petersburg Academic University of Russian Academy of Sciences. In Russian.
Claus-Peter Schnorr (1974). Zwei lineare untere Schranken f"ur die Komplexit"at Boolescher Funktionen. Computing 13(2), 155-171. URL http://dx.doi.org/10.1007/BF02246615.
Claus-Peter Schnorr (1976). The Combinational Complexity of Equivalence. Theor. Comput. Sci. 1(4), 289-295. URL http://dx.doi.org/10.1016/0304-3975(76)90073-6.
Kazuhisa Seto & Suguru Tamaki (2013). A satisfiability algorithm and average-case hardness for formulas over the full binary basis. Computational Complexity 22(2), 245-274. URL http://dx.doi.org/10.1007/s00037-013-0067-7.
Ronen Shaltiel (2011a). Dispersers for Affine Sources with Subpolynomial Entropy. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, Rafail Ostrovsky, editor, 247-256. IEEE Computer Society. ISBN 978-1-4577-1843-4. URL http://dx.doi.org/10.1109/FOCS.2011.37.
Ronen Shaltiel (2011b). An introduction to randomness extractors. In Proceedings of 38th International Colloquium on Automata, Languages and Programming (ICALP), volume 6756 of Lecture Notes in Computer Science, 21-41. Springer.
Claude E. Shannon (1949). The synthesis of two-terminal switching circuits. Bell Systems Technical Journal 28, 59-98.
Victor Shoup & Roman Smolensky (1991). Lower Bounds for Polynomial Evaluation and Interpolation Problems. In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1-4 October 1991, 378-383. IEEE Computer Society. ISBN 0-8186-2445-0. URL http://dx.doi.org/10.1109/SFCS.1991.185394.
Larry J. Stockmeyer (1977). On the Combinational Complexity of Certain Symmetric Boolean Functions. Mathematical Systems Theory 10, 323-336. URL http://dx.doi.org/10.1007/BF01683282.
Bella A. Subbotovskaya (1961). Realizations of linear functions by formulas using +, Doklady Akademii Nauk. SSSR 136(3), 553-555.
Avishay Tal (2014). Shrinkage of De Morgan Formulae by Spectral Techniques. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, 551-560. IEEE.
Salil Vadhan & Ryan Williams (2013). Personal communication.
E. Warning (1935). Bemerkung zur vorstehenden Arbeit von Herrn Chevalley. Abh. Math. Sem. Univ. Hamburg 11(1), 76-83.
Ryan Williams (2013). Improving exhaustive search implies superpolynomial lower bounds. SIAM J. Comput. 42(3), 1218-1244. Extended abstract appeared in Proc. STOC-2010.
Ryan Williams (2014). Nonuniform ACC circuit lower bounds. JACM 61(1). Extended abstract appears in Proc. CCC-2011.
Andrew C. Yao (1985). Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version). In FOCS, 1-10. IEEE Computer Society. ISBN 0-8186-0644-4. URL http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4568115.
Amir Yehudayoff (2011). Affine extractors over prime fields. Combinatorica 31(2), 245-256. URL http://dx.doi.org/10.1007/s00493-011-2604-9.
Uri Zwick (1991). A 4n Lower Bound on the Combinational Complexity of Certain Symmetric Boolean Functions over the Basis of Unate Dyadic Boolean Functions. SIAM J. Comput. 20(3), 499-505. URL http://dx.doi.org/10.1137/0220032.
Acknowledgements
We would like to thank Dmitry Itsykson and Alexander Knop for their valuable comments, and Olga Melanich for proofreading an earlier version of this paper. We are thankful to Tianqi Yang and Jiatu Li for pointing out a gap (and proposing a solution) in the Case 7.2.4.2 in Section 3.5. We are very grateful to the anonymous reviewers whose suggestions and comments significantly helped to improve the presentation in this paper. Preliminary versions of the results presented in this work appeared in (Find et al. 2016) and (Golovnev & Kulikov 2016).
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Find , M.G., Golovnev , A., Hirsch , E. et al. Improving \(3N\) Circuit Complexity Lower Bounds. comput. complex. 32, 13 (2023). https://doi.org/10.1007/s00037-023-00246-9
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DOI: https://doi.org/10.1007/s00037-023-00246-9