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Lifting to Randomized Parity Decision Trees

Authors Farzan Byramji , Russell Impagliazzo

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  • Theory of computation → Oracles and decision trees
Keywords
  • Parity decision trees
  • composition

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Abstract

We prove a lifting theorem from randomized decision tree depth to randomized parity decision tree (PDT) size. We use the same property of the gadget, stifling, which was introduced by Chattopadhyay, Mande, Sanyal and Sherif [ITCS 23] to prove a lifting theorem for deterministic PDTs. Moreover, even the milder condition that the gadget has minimum parity certificate complexity at least 2 suffices for lifting to randomized PDT size.
To improve the dependence on the gadget g in the lower bounds for composed functions, we consider a related problem g_* whose inputs are certificates of g. It is implicit in the work of Chattopadhyay et al. that for any function f, lower bounds for the *-depth of f_* give lower bounds for the PDT size of f. We make this connection explicit in the deterministic case and show that it also holds for randomized PDTs. We then combine this with composition theorems for *-depth, which follow by adapting known composition theorems for decision trees. As a corollary, we get tight lifting theorems when the gadget is Indexing, Inner Product or Disjointness.

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Farzan Byramji and Russell Impagliazzo. Lifting to Randomized Parity Decision Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 55:1-55:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.55

Author Details

Farzan Byramji
  • University of California, San Diego, CA, USA
Russell Impagliazzo
  • University of California, San Diego, CA, USA

Funding

  • Byramji, Farzan: Supported by NSF Award AF: Medium 2212136.
  • Impagliazzo, Russell: Supported by NSF Award AF: Medium 2212136.

Acknowledgements

FB thanks Sreejata Kishor Bhattacharya, Eric Blais, Zachary Chase, Arkadev Chattopadhyay, Jyun-Jie Liao, Shachar Lovett, Jackson Morris and Anthony Ostuni for discussions and feedback at various stages of this work. The authors would like to thank the anonymous reviewers for helpful comments.

References

  1. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 863-876, 2016. URL: https://doi.org/10.1145/2897518.2897644.
  2. Yaroslav Alekseev, Yuval Filmus, and Alexander Smal. Lifting Dichotomies. In Rahul Santhanam, editor, 39th Computational Complexity Conference (CCC 2024), volume 300 of Leibniz International Proceedings in Informatics (LIPIcs), pages 9:1-9:18, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.9.
  3. Yaroslav Alekseev and Dmitry Itsykson. Lifting to bounded-depth and regular resolutions over parities via games. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC '25, pages 584-595, New York, NY, USA, 2025. Association for Computing Machinery. URL: https://doi.org/10.1145/3717823.3718150.
  4. Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, and Swagato Sanyal. A composition theorem for randomized query complexity. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2017.10.
  5. Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, and Li-Yang Tan. The power of many samples in query complexity. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.9.
  6. 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), volume 251 of Leibniz International Proceedings in Informatics (LIPIcs), pages 14:1-14:17, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.14.
  7. Shalev Ben-David and Eric Blais. A new minimax theorem for randomized algorithms. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 403-411. IEEE, 2020. Google Scholar
  8. Shalev Ben-David, Eric Blais, Mika Göös, and Gilbert Maystre. Randomised composition and small-bias minimax. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 624-635. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00065.
  9. Shalev Ben-David and Robin Kothari. Randomized query complexity of sabotaged and composed functions. Theory of Computing, 14(5):1-27, 2018. URL: https://doi.org/10.4086/toc.2018.v014a005.
  10. Tyler Besselman, Mika Göös, Siyao Guo, Gilbert Maystre, and Weiqiang Yuan. Direct sums for parity decision trees. In 40th Computational Complexity Conference (CCC 2025), Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, 2025. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2025.16.
  11. Sreejata Kishor Bhattacharya, Arkadev Chattopadhyay, and Pavel Dvořák. Exponential Separation Between Powers of Regular and General Resolution over Parities. In Rahul Santhanam, editor, 39th Computational Complexity Conference (CCC 2024), volume 300 of Leibniz International Proceedings in Informatics (LIPIcs), pages 23:1-23:32, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.23.
  12. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00144-X.
  13. Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the composition of randomized query complexity and approximate degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.63.
  14. Arkadev Chattopadhyay, Yuval Filmus, Sajin Koroth, Or Meir, and Toniann Pitassi. Query-to-communication lifting using low-discrepancy gadgets. SIAM Journal on Computing, 50(1):171-210, 2021. URL: https://doi.org/10.1137/19M1310153.
  15. Arkadev Chattopadhyay, Michal Kouckỳ, Bruno Loff, and Sagnik Mukhopadhyay. Simulation theorems via pseudo-random properties. computational complexity, 28:617-659, 2019. URL: https://doi.org/10.1007/S00037-019-00190-7.
  16. Arkadev Chattopadhyay, Nikhil S Mande, Swagato Sanyal, and Suhail Sherif. Lifting to parity decision trees via stifling. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Schloss Dagstuhl - Leibniz Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.33.
  17. Arjan Cornelissen, Nikhil S Mande, and Subhasree Patro. Improved quantum query upper bounds based on classical decision trees. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Schloss Dagstuhl - Leibniz Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2022.15.
  18. Yogesh Dahiya. Exploring Size Complexity and Randomness in the Query Model. PhD thesis, HBNI, 2024. URL: https://www.imsc.res.in/xmlui/handle/123456789/881.
  19. Yogesh Dahiya and Meena Mahajan. On (simple) decision tree rank. Theoretical Computer Science, 978:114177, 2023. URL: https://doi.org/10.1016/J.TCS.2023.114177.
  20. Klim Efremenko, Michal Garlík, and Dmitry Itsykson. Lower bounds for regular resolution over parities. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 640-651, 2024. URL: https://doi.org/10.1145/3618260.3649652.
  21. Andrzej Ehrenfeucht and David Haussler. Learning decision trees from random examples. Information and Computation, 82(3):231-246, 1989. URL: https://doi.org/10.1016/0890-5401(89)90001-1.
  22. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 902-911, 2018. URL: https://doi.org/10.1145/3188745.3188838.
  23. Dmytro Gavinsky, Troy Lee, Miklos Santha, and Swagato Sanyal. Optimal composition theorem for randomized query complexity. Theory of Computing, 19(1):1-35, 2023. URL: https://doi.org/10.4086/TOC.2023.V019A009.
  24. Mika Göös, TS Jayram, Toniann Pitassi, and Thomas Watson. Randomized communication versus partition number. ACM Transactions on Computation Theory (TOCT), 10(1):1-20, 2018. URL: https://doi.org/10.1145/3170711.
  25. Mika Göös, Pritish Kamath, Toniann Pitassi, and Thomas Watson. Query-to-communication lifting for p np. computational complexity, 28:113-144, 2019. URL: https://doi.org/10.1007/S00037-018-0175-5.
  26. Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. SIAM Journal on Computing, 47(5):1778-1806, 2018. URL: https://doi.org/10.1137/16M1082007.
  27. Mika Goos, 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.
  28. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-communication lifting for bpp. SIAM Journal on Computing, 49(4):FOCS17-441, 2020. URL: https://doi.org/10.1137/17M115339X.
  29. Dmitry Itsykson and Dmitry Sokolov. Resolution over linear equations modulo two. Annals of Pure and Applied Logic, 171(1):102722, 2020. URL: https://doi.org/10.1016/J.APAL.2019.102722.
  30. Rahul Jain, Hartmut Klauck, and Miklos Santha. Optimal direct sum results for deterministic and randomized decision tree complexity. Information Processing Letters, 110(20):893-897, 2010. URL: https://doi.org/10.1016/J.IPL.2010.07.020.
  31. 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.
  32. Ashley Montanaro. A composition theorem for decision tree complexity. Chicago Journal of Theoretical Computer Science, 2014(6), 2014. URL: https://doi.org/10.4086/cjtcs.2014.006.
  33. 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. URL: https://doi.org/10.1145/3188745.3188914.
  34. Vladimir Podolskii and Alexander Shekhovtsov. Randomized lifting to semi-structured communication complexity via linear diversity. In 16th Innovations in Theoretical Computer Science Conference (ITCS), LIPIcs. Schloss Dagstuhl, 2025. URL: https://doi.org/10.4230/LIPIcs.ITCS.2025.78.
  35. Pavel Pudlák and Russell Impagliazzo. A lower bound for dll algorithms for k-sat (preliminary version). In Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms, pages 128-136, 2000. URL: http://dl.acm.org/citation.cfm?id=338219.338244.
  36. 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. URL: https://doi.org/10.1109/SFCS.1997.646112.
  37. Swagato Sanyal. Randomized query composition and product distributions. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.STACS.2024.56.
  38. Petr Savický. On determinism versus unambiquous nondeterminism for decision trees. Technical Report TR02-009, Electronic Colloquium on Computational Complexity (ECCC), 2002. URL: https://eccc.weizmann.ac.il//report/2002/009/.
  39. Suhail Sherif. Lifting to parity decision trees via stifling (with applications to proof complexity). URL: https://www.youtube.com/watch?v=PeZVs6WUf-4.
  40. Avishay Tal. Properties and applications of boolean function composition. In Proceedings of the 4th Conference on Innovations in Theoretical Computer Science (ITCS), pages 441-454, 2013. URL: https://doi.org/10.1145/2422436.2422485.
  41. Alasdair Urquhart. The depth of resolution proofs. Studia Logica, 99:349-364, 2011. URL: https://doi.org/10.1007/S11225-011-9356-9.
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