Document Open Access Logo

Constant-Depth Circuits vs. Monotone Circuits

Authors Bruno P. Cavalar , Igor C. Oliveira

PDF
Thumbnail PDF

File

LIPIcs.CCC.2023.29.pdf
  • Filesize: 1.09 MB
  • 37 pages

Document Identifiers

Author Details

Bruno P. Cavalar
  • University of Warwick, Coventry, UK
Igor C. Oliveira
  • University of Warwick, Coventry, UK

Funding

This work received support from the Royal Society University Research Fellowship URF⧵R1⧵191059, the EPSRC New Horizons Grant EP/V048201/1, and the Centre for Discrete Mathematics and its Applications (DIMAP) at the University of Warwick.

Acknowledgements

We thank Arkadev Chattopadhyay for several conversations about the AC⁰ versus mSIZE[poly] problem and related questions. We are also grateful to Denis Kuperberg for explaining to us the results from [Denis Kuperberg, 2021; Denis Kuperberg, 2022]. The first author thanks Ninad Rajgopal for helpful discussions about depth reduction. Finally, we thank Gernot Salzer for the code used to generate Figures 1, 2, and 3.

Cite AsGet BibTex

Bruno P. Cavalar and Igor C. Oliveira. Constant-Depth Circuits vs. Monotone Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 29:1-29:37, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.29

Abstract

We establish new separations between the power of monotone and general (non-monotone) Boolean circuits: - For every k ≥ 1, there is a monotone function in AC⁰ (constant-depth poly-size circuits) that requires monotone circuits of depth Ω(log^k n). This significantly extends a classical result of Okol'nishnikova [Okol'nishnikova, 1982] and Ajtai and Gurevich [Ajtai and Gurevich, 1987]. In addition, our separation holds for a monotone graph property, which was unknown even in the context of AC⁰ versus mAC⁰. - For every k ≥ 1, there is a monotone function in AC⁰[⊕] (constant-depth poly-size circuits extended with parity gates) that requires monotone circuits of size exp(Ω(log^k n)). This makes progress towards a question posed by Grigni and Sipser [Grigni and Sipser, 1992]. These results show that constant-depth circuits can be more efficient than monotone formulas and monotone circuits when computing monotone functions. In the opposite direction, we observe that non-trivial simulations are possible in the absence of parity gates: every monotone function computed by an AC⁰ circuit of size s and depth d can be computed by a monotone circuit of size 2^{n - n/O(log s)^{d-1}}. We show that the existence of significantly faster monotone simulations would lead to breakthrough circuit lower bounds. In particular, if every monotone function in AC⁰ admits a polynomial size monotone circuit, then NC² is not contained in NC¹. Finally, we revisit our separation result against monotone circuit size and investigate the limits of our approach, which is based on a monotone lower bound for constraint satisfaction problems (CSPs) established by Göös, Kamath, Robere and Sokolov [Göös et al., 2019] via lifting techniques. Adapting results of Schaefer [Thomas J. Schaefer, 1978] and Allender, Bauland, Immerman, Schnoor and Vollmer [Eric Allender et al., 2009], we obtain an unconditional classification of the monotone circuit complexity of Boolean-valued CSPs via their polymorphisms. This result and the consequences we derive from it might be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • circuit complexity
  • monotone circuit complexity
  • bounded-depth circuis
  • constraint-satisfaction problems

Metrics

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

Related Versions

References

  1. Miklós Ajtai and Yuri Gurevich. Monotone versus positive. J. Assoc. Comput. Mach., 34(4):1004-1015, 1987. URL: https://doi.org/10.1145/31846.31852.
  2. Miklós Ajtai, János Komlós, and Endre Szemerédi. An O(n log n) sorting network. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 1-9. ACM, 1983. URL: https://doi.org/10.1145/800061.808726.
  3. Jin Akiyama and Mikio Kano. Factors and Factorizations of Graphs, volume 2031 of Lecture Notes in Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011. URL: https://doi.org/10.1007/978-3-642-21919-1.
  4. Eric Allender, Michael Bauland, Neil Immerman, Henning Schnoor, and Heribert Vollmer. The complexity of satisfiability problems: Refining Schaefer’s theorem. J. Comput. Syst. Sci., 75(4):245-254, 2009. URL: https://doi.org/10.1016/j.jcss.2008.11.001.
  5. Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael E. Saks. Minimizing disjunctive normal form formulas and AC⁰ circuits given a truth table. SIAM J. Comput., 38(1):63-84, 2008. URL: https://doi.org/10.1137/060664537.
  6. Eric Allender, Michal Koucký, Detlef Ronneburger, Sambuddha Roy, and V. Vinay. Time-space tradeoffs in the counting hierarchy. In Proceedings of the 16th Annual IEEE Conference on Computational Complexity, Chicago, Illinois, USA, June 18-21, 2001, pages 295-302. IEEE Computer Society, 2001. URL: https://doi.org/10.1109/CCC.2001.933896.
  7. Noga Alon and Ravi B. Boppana. The monotone circuit complexity of boolean functions. Combinatorica, 7(1):1-22, 1987. Google Scholar
  8. Alexander E Andreev. On a method for obtaining lower bounds for the complexity of individual monotone functions. Doklady Akademii Nauk SSSR, 282:1033-1037, 1985. Google Scholar
  9. László Babai, Anna Gál, and Avi Wigderson. Superpolynomial lower bounds for monotone span programs. Combinatorica, 19(3):301-319, 1999. URL: https://doi.org/10.1007/s004930050058.
  10. Josh Cohen Benaloh and Jerry Leichter. Generalized secret sharing and monotone functions. In Advances in Cryptology (CRYPTO), pages 27-35, 1988. Google Scholar
  11. S. J. Berkowitz. On some relationships between monotone and non-monotone circuit complexity. Technical report, University of Toronto, 1982. Google Scholar
  12. Eric Blais, Johan Håstad, Rocco A. Servedio, and Li-Yang Tan. On DNF approximators for monotone boolean functions. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 235-246, 2014. Google Scholar
  13. Eric Blais, Dominik Scheder, and Li-Yang Tan. Ajtai-gurevich redux. Manuscript, 2013. Google Scholar
  14. Nader H. Bshouty and Christino Tamon. On the Fourier spectrum of monotone functions. J. ACM, 43(4):747-770, 1996. URL: https://doi.org/10.1145/234533.234564.
  15. Andrei A. Bulatov. A dichotomy theorem for nonuniform csps. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 319-330. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.37.
  16. Andrei A. Bulatov. Constraint satisfaction problems: complexity and algorithms. ACM SIGLOG News, 5(4):4-24, 2018. URL: https://doi.org/10.1145/3292048.3292050.
  17. Gerhard Buntrock, Carsten Damm, Ulrich Hertrampf, and Christoph Meinel. Structure and importance of logspace-mod class. Math. Syst. Theory, 25(3):223-237, 1992. URL: https://doi.org/10.1007/BF01374526.
  18. Elmar Böhler, Nadia Creignou, Steffen Reith, and Heribert Vollmer. Playing with boolean blocks, part i: Post’s lattice with applications to complexity theory. SIGACT News, 2003. Google Scholar
  19. Elmar Böhler, Nadia Creignou, Steffen Reith, and Heribert Vollmer. Playing with boolean blocks, part ii: Constraint satisfaction problems. ACM SIGACT-Newsletter, 35, 2004. Google Scholar
  20. Bruno Pasqualotto Cavalar, Mrinal Kumar, and Benjamin Rossman. Monotone circuit lower bounds from robust sunflowers. In LATIN 2020: Theoretical Informatics - 14th Latin American Symposium, São Paulo, Brazil, January 5-8, 2021, Proceedings, volume 12118 of Lecture Notes in Computer Science, pages 311-322. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-61792-9_25.
  21. Arkadev Chattopadhyay, Rajit Datta, and Partha Mukhopadhyay. Lower bounds for monotone arithmetic circuits via communication complexity. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 786-799. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451069.
  22. Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. In Symposium on Theory of Computing (STOC), pages 670-683, 2016. Google Scholar
  23. Lijie Chen, Shuichi Hirahara, Igor C. Oliveira, Ján Pich, Ninad Rajgopal, and Rahul Santhanam. Beyond Natural Proofs: Hardness Magnification and Locality. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151, pages 70:1-70:48, 2020. Google Scholar
  24. Xi Chen, Igor C. Oliveira, and Rocco A. Servedio. Addition is exponentially harder than counting for shallow monotone circuits. In STOC'17 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 1232-1245. ACM, New York, 2017. URL: https://doi.org/10.1145/3055399.3055425.
  25. Nadia Creignou, Sanjeev Khanna, and Madhu Sudan. Complexity classifications of Boolean constraint satisfaction problems, volume 7 of SIAM monographs on discrete mathematics and applications. SIAM, 2001. Google Scholar
  26. Susanna F. de Rezende, Mika Göös, and Robert Robere. Guest column: Proofs, circuits, and communication. SIGACT News, 53(1):59-82, 2022. URL: https://doi.org/10.1145/3532737.3532746.
  27. Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput., 28(1):57-104, 1998. URL: https://doi.org/10.1137/S0097539794266766.
  28. Oded Goldreich, Shafi Goldwasser, Eric Lehman, and Dana Ron. Testing monotonicity. In Symposium on Foundations of Computer Science, (FOCS), pages 426-435, 1998. Google Scholar
  29. Oded Goldreich and Rani Izsak. Monotone circuits: One-way functions versus pseudorandom generators. Theory Comput., 8(1):231-238, 2012. URL: https://doi.org/10.4086/toc.2012.v008a010.
  30. Mika Göös, Rahul Jain, and Thomas Watson. Extension complexity of independent set polytopes. SIAM J. Comput., 47(1):241-269, 2018. URL: https://doi.org/10.1137/16M109884X.
  31. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in monotone complexity and TFNP. In 10th Innovations in Theoretical Computer Science, volume 124 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 38, 19. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019. Google Scholar
  32. Michelangelo Grigni and Michael Sipser. Monotone complexity. In Proceedings of the London Mathematical Society Symposium on Boolean Function Complexity, pages 57-75, USA, 1992. Cambridge University Press. Google Scholar
  33. Siyao Guo, Tal Malkin, Igor C. Oliveira, and Alon Rosen. The power of negations in cryptography. In Theory of Cryptography Conference (TCC), pages 36-65, 2015. Google Scholar
  34. Johan Håstad. 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, pages 6-20, 1986. URL: https://doi.org/10.1145/12130.12132.
  35. Johan Håstad, Ingo Wegener, Norbert Wurm, and Sang-Zin Yi. Optimal depth, very small size circuits for symmetric functions in AC⁰. Inf. Comput., 108(2):200-211, 1994. URL: https://doi.org/10.1006/inco.1994.1008.
  36. Shuichi Hirahara. NP-hardness of learning programs and partial MCSP. In Symposium on Foundations of Computer Science (FOCS), 2022. Google Scholar
  37. Christian Ikenmeyer, Balagopal Komarath, Christoph Lenzen, Vladimir Lysikov, Andrey Mokhov, and Karteek Sreenivasaiah. On the complexity of hazard-free circuits. J. ACM, 66(4):25:1-25:20, 2019. URL: https://doi.org/10.1145/3320123.
  38. Peter Jeavons. On the algebraic structure of combinatorial problems. Theor. Comput. Sci., 200(1-2):185-204, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00230-2.
  39. Peter Jeavons, David Cohen, and Marc Gyssens. Closure properties of constraints. J. ACM, 44(4):527-548, July 1997. URL: https://doi.org/10.1145/263867.263489.
  40. Jan Johannsen. The complexity of satisfiability problems with two occurrences, 2003. Unpublished Manuscript. Google Scholar
  41. Neil D. Jones, Y. Edmund Lien, and William T. Laaser. New problems complete for nondeterministic log space. Math. Syst. Theory, 10:1-17, 1976. URL: https://doi.org/10.1007/BF01683259.
  42. Stasys Jukna. Boolean Function Complexity - Advances and Frontiers. Springer, 2012. Google Scholar
  43. Bala Kalyanasundaram and Georg Schnitger. The probabilistic communication complexity of set intersection. SIAM J. Discret. Math., 5(4):545-557, 1992. URL: https://doi.org/10.1137/0405044.
  44. Mauricio Karchmer and Avi Wigderson. On span programs. In Structure in Complexity Theory Conference (CCC), pages 102-111, 1993. URL: https://doi.org/10.1109/SCT.1993.336536.
  45. Jan Krajíček. Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. J. Symbolic Logic, 62(2):457-486, 1997. Google Scholar
  46. Denis Kuperberg. Positive first-order logic on words. In Symposium on Logic in Computer Science (LICS), pages 1-13, 2021. Google Scholar
  47. Denis Kuperberg. Positive first-order logic on words and graphs. CoRR, abs/2201.11619, 2022. URL: https://arxiv.org/abs/2201.11619.
  48. Benoît Larose and Pascal Tesson. Universal algebra and hardness results for constraint satisfaction problems. Theor. Comput. Sci., 410(18):1629-1647, 2009. URL: https://doi.org/10.1016/j.tcs.2008.12.048.
  49. E. A. Okol'nishnikova. The effect of negations on the complexity of realization of monotone Boolean functions by formulas of bounded depth. Metody Diskret. Analiz., pages 74-80, 1982. Google Scholar
  50. Igor C. Oliveira. Unconditional lower bounds in complexity theory, 2015. (PhD Thesis, Columbia University). Google Scholar
  51. Igor C. Oliveira, Rahul Santhanam, and Srikanth Srinivasan. Parity helps to compute majority. In 34th Computational Complexity Conference, CCC 2019, July 18-20, 2019, New Brunswick, NJ, USA, pages 23:1-23:17, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.23.
  52. Toniann Pitassi and Robert Robere. Strongly exponential lower bounds for monotone computation. In Symposium on Theory of Computing (STOC), pages 1246-1255, 2017. URL: https://doi.org/10.1145/3055399.3055478.
  53. Emil L. Post. The Two-Valued Iterative Systems of Mathematical Logic. (AM-5). Princeton University Press, 1941. URL: http://www.jstor.org/stable/j.ctt1bgzb1r.
  54. Pavel Pudlák. Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symb. Log., 62(3):981-998, 1997. URL: https://doi.org/10.2307/2275583.
  55. W. V. Quine. Two theorems about truth functions. Bol. Soc. Mat. Mexicana, 10:64-70, 1953. Google Scholar
  56. Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. Combinatorica, 19(3):403-435, 1999. URL: https://doi.org/10.1007/s004930050062.
  57. Ran Raz and Avi Wigderson. Monotone circuits for matching require linear depth. J. ACM, 39(3):736-744, 1992. URL: https://doi.org/10.1145/146637.146684.
  58. A. A. Razborov. Lower bounds on the monotone complexity of some Boolean functions. Dokl. Akad. Nauk SSSR, 281(4):798-801, 1985. Google Scholar
  59. Alexander A Razborov. Lower bounds on monotone complexity of the logical permanent. Mathematical Notes of the Academy of Sciences of the USSR, 37(6):485-493, 1985. Google Scholar
  60. Omer Reingold. Undirected st-connectivity in log-space. In Harold N. Gabow and Ronald Fagin, editors, Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 22-24, 2005, pages 376-385. ACM, 2005. URL: https://doi.org/10.1145/1060590.1060647.
  61. Susanna Rezende. Personal communication, 2023. Google Scholar
  62. Robert Robere, Toniann Pitassi, Benjamin Rossman, and Stephen A. Cook. Exponential lower bounds for monotone span programs. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 406-415. IEEE Computer Society, 2016. URL: https://doi.org/10.1109/FOCS.2016.51.
  63. Benjamin Rossman. Homomorphism preservation theorems. J. ACM, 55(3):Art. 15, 53, 2008. URL: https://doi.org/10.1145/1379759.1379763.
  64. Benjamin Rossman. On the constant-depth complexity of k-clique. In Symposium on Theory of Computing (STOC), pages 721-730, 2008. Google Scholar
  65. Benjamin Rossman. An entropy proof of the switching lemma and tight bounds on the decision-tree size of AC⁰, 2017. Google Scholar
  66. Benjamin Rossman. An improved homomorphism preservation theorem from lower bounds in circuit complexity. In Innovations in Theoretical Computer Science Conference (ITCS), pages 27:1-27:17, 2017. Google Scholar
  67. Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, May 1-3, 1978, San Diego, California, USA, pages 216-226. ACM, 1978. URL: https://doi.org/10.1145/800133.804350.
  68. Alexei P. Stolboushkin. Finitely monotone properties. In Symposium on Logic in Computer Science (LICS), pages 324-330, 1995. Google Scholar
  69. É. Tardos. The gap between monotone and nonmonotone circuit complexity is exponential. Combinatorica, 8(1):141-142, 1988. URL: https://doi.org/10.1007/BF02122563.
  70. Alasdair Urquhart. Hard examples for resolution. J. ACM, 34(1):209-219, 1987. URL: https://doi.org/10.1145/7531.8928.
  71. Leslie G. Valiant. Negation can be exponentially powerful. Theor. Comput. Sci., 12:303-314, 1980. Google Scholar
  72. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 331-342. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/FOCS.2017.38.
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 Schloss Dagstuhl - LZI GmbH Imprint Privacy Contact