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Lifting Dichotomies

Authors Yaroslav Alekseev , Yuval Filmus , Alexander Smal

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Author Details

Yaroslav Alekseev
  • Technion - Israel Institute of Technology, Haifa, Israel
Yuval Filmus
  • Technion - Israel Institute of Technology, Haifa, Israel
Alexander Smal
  • Technion - Israel Institute of Technology, Haifa, Israel

Funding

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC. Alexander Smal has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 852870-ERC-SUBMODULAR.

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Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.9

Abstract

Lifting theorems are used for transferring lower bounds between Boolean function complexity measures. Given a lower bound on a complexity measure A for some function f, we compose f with a carefully chosen gadget function g and get essentially the same lower bound on a complexity measure B for the lifted function f ⋄ g. Lifting theorems have a number of applications in many different areas such as circuit complexity, communication complexity, proof complexity, etc. One of the main question in the context of lifting is how to choose a suitable gadget g. Generally, to get better results, i.e., to minimize the losses when transferring lower bounds, we need the gadget to be of a constant size (number of inputs). Unfortunately, in many settings we know lifting results only for gadgets of size that grows with the size of f, and it is unclear whether it can be improved to a constant size gadget. This motivates us to identify the properties of gadgets that make lifting possible. In this paper, we systematically study the question "For which gadgets does the lifting result hold?" in the following four settings: lifting from decision tree depth to decision tree size, lifting from conjunction DAG width to conjunction DAG size, lifting from decision tree depth to parity decision tree depth and size, and lifting from block sensitivity to deterministic and randomized communication complexities. In all the cases, we prove the complete classification of gadgets by exposing the properties of gadgets that make lifting results hold. The structure of the results shows that there is no intermediate cases - for every gadget there is either a polynomial lifting or no lifting at all. As a byproduct of our studies, we prove the log-rank conjecture for the class of functions that can be represented as f ⋄ OR ⋄ XOR for some function f. In this extended abstract, the proofs are omitted. Full proofs are given in the full version [Yaroslav Alekseev et al., 2024].

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
Keywords
  • decision trees
  • log-rank conjecture
  • lifting
  • parity decision trees

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References

  1. Michael Alekhnovich, Jan Johannsen, Toniann Pitassi, and Alasdair Urquhart. An exponential separation between regular and general resolution. Theory Comput., 3:81-102, 2007. URL: https://doi.org/10.4086/toc.2007.v003a005.
  2. Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting dichotomies. Electron. Colloquium Comput. Complex., pages TR24-037, 2024. URL: https://eccc.weizmann.ac.il/report/2024/037.
  3. Paul Beame and Sajin Koroth. On disperser/lifting properties of the index and inner-product functions. In Yael Tauman Kalai, editor, 14th Innovations in Theoretical Computer Science Conference, ITCS 2023, January 10-13, 2023, MIT, Cambridge, Massachusetts, USA, volume 251 of LIPIcs, pages 14:1-14:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ITCS.2023.14.
  4. Eli Ben-Sasson and Avi Wigderson. Short proofs are narrow—resolution made simple. J. ACM, 48(2):149-169, March 2001. URL: https://doi.org/10.1145/375827.375835.
  5. Arkadev Chattopadhyay, Michal Koucký, Bruno Loff, and Sagnik Mukhopadhyay. Simulation theorems via pseudo-random properties. Comput. Complex., 28(4):617-659, December 2019. URL: https://doi.org/10.1007/s00037-019-00190-7.
  6. Arkadev Chattopadhyay, Nikhil S. Mande, Swagato Sanyal, and Suhail Sherif. Lifting to parity decision trees via stifling, 2022. URL: https://doi.org/10.48550/arXiv.2211.17214.
  7. Susanna de Rezende, Or Meir, Jakob Nordstrom, Toniann Pitassi, Robert Robere, and Marc Vinyals. Lifting with simple gadgets and applications to circuit and proof complexity. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 24-30, November 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00011.
  8. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from Resolution. Theory Comput., 16:Paper No. 13, 30, 2020. URL: https://doi.org/10.4086/toc.2020.v016a013.
  9. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. SIAM Journal on Computing, 45(5):1835-1869, 2016. URL: https://doi.org/10.1137/15M103145X.
  10. Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. SIAM J. Comput., 47(5):1778-1806, 2018. URL: https://doi.org/10.1137/16M1082007.
  11. Mika Göös, Toniann Pitassi, and Thomas Watson. Deterministic communication vs. partition number. SIAM Journal on Computing, 47(6):2435-2450, 2018. URL: https://doi.org/10.1137/16M1059369.
  12. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-communication lifting for BPP. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 132-143, 2017. URL: https://doi.org/10.1109/FOCS.2017.21.
  13. Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Structure of protocols for XOR functions. SIAM J. Comput., 47(1):208-217, 2018. URL: https://doi.org/10.1137/17M1136869.
  14. Trinh Huynh and Jakob Nordström. On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity [extended abstract]. In STOC'12 - Proceedings of the 2012 ACM Symposium on Theory of Computing, pages 233-247. ACM, New York, 2012. URL: https://doi.org/10.1145/2213977.2214000.
  15. Alexander Knop, Shachar Lovett, Sam McGuire, and Weiqiang Yuan. Log-rank and lifting for AND-functions. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 197-208, New York, NY, USA, 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3450999.
  16. Shachar Lovett. Communication is bounded by root of rank. J. ACM, 63(1), February 2016. URL: https://doi.org/10.1145/2724704.
  17. Shachar Lovett, Raghu Meka, Ian Mertz, Toniann Pitassi, and Jiapeng Zhang. Lifting with Sunflowers. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), volume 215 of Leibniz International Proceedings in Informatics (LIPIcs), pages 104:1-104:24, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.104.
  18. László Lovász and Michael Saks. Lattices, mobius functions and communications complexity. In [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science, pages 81-90, 1988. URL: https://doi.org/10.1109/SFCS.1988.21924.
  19. Ran Raz and Pierre Mckenzie. Separation of the monotone NC hierarchy. Combinatorica, 19, September 1999. URL: https://doi.org/10.1007/s004930050062.
  20. Alexander A. Sherstov. On quantum-classical equivalence for composed communication problems. Quantum Inf. Comput., 10(5-6):435-455, 2010. Google Scholar
  21. Alasdair Urquhart. The depth of resolution proofs. Studia Logica: An International Journal for Symbolic Logic, 99(1/3):349-364, 2011. URL: http://www.jstor.org/stable/41475208.
  22. Shengyu Zhang. On the tightness of the Buhrman-Cleve-Wigderson simulation. In Proceedings of the 20th International Symposium on Algorithms and Computation, ISAAC '09, pages 434-440, Berlin, Heidelberg, 2009. Springer-Verlag. URL: https://doi.org/10.1007/978-3-642-10631-6_45.
  23. Zhiqiang Zhang and Yaoyun Shi. On the parity complexity measures of Boolean functions. Theoretical Computer Science, 411(26):2612-2618, 2010. URL: https://doi.org/10.1016/j.tcs.2010.03.027.
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