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Lifting for Constant-Depth Circuits and Applications to MCSP

Authors Marco Carmosino, Kenneth Hoover, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova

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Author Details

Marco Carmosino
  • Department of Computer Science, Boston University, MA, USA
Kenneth Hoover
  • Department of Computer Science, University of California, San Diego, CA, USA
Russell Impagliazzo
  • Department of Computer Science, University of California, San Diego, CA, USA
Valentine Kabanets
  • School of Computing Science, Simon Fraser University, Burnaby, Canada
Antonina Kolokolova
  • Department of Computer Science, Memorial University of Newfoundland, St. John’s, Canada

Funding

  • Carmosino, Marco: Work supported by NSF Computing Innovation Fellowship.
  • Hoover, Kenneth: Work supported by the Simons Foundation and NSF grant CCF-1909634.
  • Impagliazzo, Russell: Work supported by the Simons Foundation and NSF grant CCF-1909634.
  • Kabanets, Valentine: Work supported in part by an NSERC Discovery grant.
  • Kolokolova, Antonina: Work supported in part by an NSERC Discovery grant.

Acknowledgements

We thank the anonymous reviewers for their helpful and insightful comments.

Cite AsGet BibTex

Marco Carmosino, Kenneth Hoover, Russell Impagliazzo, Valentine Kabanets, and Antonina Kolokolova. Lifting for Constant-Depth Circuits and Applications to MCSP. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 44:1-44:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.44

Abstract

Lifting arguments show that the complexity of a function in one model is essentially that of a related function (often the composition of the original function with a small function called a gadget) in a more powerful model. Lifting has been used to prove strong lower bounds in communication complexity, proof complexity, circuit complexity and many other areas. We present a lifting construction for constant depth unbounded fan-in circuits. Given a function f, we construct a function g, so that the depth d+1 circuit complexity of g, with a certain restriction on bottom fan-in, is controlled by the depth d circuit complexity of f, with the same restriction. The function g is defined as f composed with a parity function. With some quantitative losses, average-case and general depth-d circuit complexity can be reduced to circuit complexity with this bottom fan-in restriction. As a consequence, an algorithm to approximate the depth d (for any d > 3) circuit complexity of given (truth tables of) Boolean functions yields an algorithm for approximating the depth 3 circuit complexity of functions, i.e., there are quasi-polynomial time mapping reductions between various gap-versions of AC⁰-MCSP. Our lifting results rely on a blockwise switching lemma that may be of independent interest. We also show some barriers on improving the efficiency of our reductions: such improvements would yield either surprisingly efficient algorithms for MCSP or stronger than known AC⁰ circuit lower bounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Circuit complexity
Keywords
  • circuit complexity
  • constant-depth circuits
  • lifting theorems
  • Minimum Circuit Size Problem
  • reductions
  • Switching Lemma

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References

  1. Eric Allender. The new complexity landscape around circuit minimization. In Alberto Leporati, Carlos Martín-Vide, Dana Shapira, and Claudio Zandron, editors, Language and Automata Theory and Applications, pages 3-16, Cham, 2020. Springer International Publishing. Google Scholar
  2. Eric Allender, Harry Buhrman, Michal Koucký, Dieter van Melkebeek, and Detlef Ronneburger. Power from random strings. SIAM J. Comput., 35(6):1467-1493, 2006. URL: https://doi.org/10.1137/050628994.
  3. 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.
  4. Eric Allender and Shuichi Hirahara. New insights on the (non-)hardness of circuit minimization and related problems. ACM Transactions on Computation Theory (ToCT), 11(4):1-27, 2019. Google Scholar
  5. Eric Allender, Dhiraj Holden, and Valentine Kabanets. The minimum oracle circuit size problem. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, pages 21-33, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.21.
  6. Alexander E. Andreev. On a method for obtaining more than quadratic effective lower bounds for the complexity of π-schemes. Moscow Univ. Math. Bull., 42:63–-66, 1987. Google Scholar
  7. Paul Beame. A switching lemma primer, 1994. Google Scholar
  8. Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow—resolution made simple. Journal of the ACM, 48(2):149-169, 2001. Google Scholar
  9. Vàclav Chvátal. A greedy heuristic for the set-covering problem. Math. Oper. Res., 4(3):233–235, 1979. URL: https://doi.org/10.1287/moor.4.3.233.
  10. Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, and Marc Vinyals. Lifting with simple gadgets and applications to circuit and proof complexity. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 24-30. IEEE, 2020. Google Scholar
  11. Vitaly Feldman. Hardness of approximate two-level logic minimization and PAC learning with membership queries. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC '06, page 363–372, New York, NY, USA, 2006. Association for Computing Machinery. URL: https://doi.org/10.1145/1132516.1132569.
  12. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. SIAM Journal on Computing, 47(6):2435-2450, 2018. Google Scholar
  13. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, pages 6-20. ACM, 1986. URL: https://doi.org/10.1145/12130.12132.
  14. Johan Håstad, Benjamin Rossman, Rocco A. Servedio, and Li-Yang Tan. An average-case depth hierarchy theorem for boolean circuits. J. ACM, 64(5):35:1-35:27, 2017. URL: https://doi.org/10.1145/3095799.
  15. Shuichi Hirahara, Igor C. Oliveira, and Rahul Santhanam. NP-hardness of minimum circuit size problem for OR-AND-MOD circuits. In 33rd Computational Complexity Conference (CCC 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  16. Shuichi Hirahara and Osamu Watanabe. Limits of minimum circuit size problem as oracle. In 31st Conference on Computational Complexity, CCC, pages 18:1-18:20, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.18.
  17. John M. Hitchcock and A. Pavan. On the NP-completeness of the minimum circuit size problem. In 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS, pages 236-245, 2015. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2015.236.
  18. Rahul Ilango. Constant depth formula and partial function versions of MCSP are hard. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 424-433. IEEE, 2020. Google Scholar
  19. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. In search of an easy witness: Exponential time vs. probabilistic polynomial time. J. of Computer and System Sciences, 65(4):672-694, 2002. Google Scholar
  20. Valentine Kabanets and Jin-Yi Cai. Circuit minimization problem. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pages 73-79, 2000. Google Scholar
  21. Richard M. Karp, F. E. McFarlin, J. P. Roth, and J. R. Wilts. A computer program for the synthesis of combinational switching circuits. In 2nd Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 1961), pages 182-194, 1961. URL: https://doi.org/10.1109/FOCS.1961.1.
  22. Subhash Khot and Rishi Saket. Hardness of minimizing and learning DNF expressions. In 2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 231-240, Los Alamitos, CA, USA, October 2008. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2008.37.
  23. William J. Masek. Some NP-complete set covering problems, 1979. Google Scholar
  24. Cody D. Murray and Ryan R. Williams. On the (non) NP-hardness of computing circuit complexity. In 30th Conference on Computational Complexity, CCC, pages 365-380, 2015. URL: https://doi.org/10.4230/LIPIcs.CCC.2015.365.
  25. Toniann Pitassi and Robert Robere. Lifting nullstellensatz to monotone span programs over any field. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1207-1219, 2018. Google Scholar
  26. Ran Raz and Pierre McKenzie. Separation of the monotone NC hierarchy. In Proceedings 38th Annual Symposium on Foundations of Computer Science, pages 234-243. IEEE, 1997. Google Scholar
  27. Alexander A. Razborov. Bounded arithmetic and lower bounds in boolean complexity. In Peter Clote and Jeffrey B. Remmel, editors, Feasible Mathematics II, pages 344-386, Boston, MA, 1995. Birkhäuser Boston. Google Scholar
  28. Alexander A. Razborov and Steven Rudich. Natural proofs. J. of Computer and System Sciences, 55(1):24-35, 1997. Google Scholar
  29. Michael Saks and Rahul Santhanam. Circuit lower bounds from NP-hardness of MCSP under Turing reductions. In 35th Computational Complexity Conference (CCC 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  30. Boris A. Trakhtenbrot. A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms. IEEE Annals of the History of Computing, 6(4):384-400, 1984. URL: https://doi.org/10.1109/MAHC.1984.10036.
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