On the Composition of Randomized Query Complexity and Approximate Degree

Authors Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Swagato Sanyal, Nitin Saurabh

Thumbnail PDF


  • Filesize: 0.8 MB
  • 23 pages

Document Identifiers

Author Details

Sourav Chakraborty
  • Indian Statistical Institute, Kolkata, India
Chandrima Kayal
  • Indian Statistical Institute, Kolkata, India
Rajat Mittal
  • Indian Institute of Technology Kanpur, India
Manaswi Paraashar
  • Aarhus University, Denmark
Swagato Sanyal
  • Indian Institute of Technology Kharagpur, India
Nitin Saurabh
  • Indian Institute of Technology Hyderabad, India

Cite AsGet BibTex

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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 63:1-63:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


For any Boolean functions f and g, the question whether R(f∘g) = Θ̃(R(f) ⋅ R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg̃(f∘g) = Θ̃(deg̃(f)⋅deg̃(g)). These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg̃ compose. A recent landmark result (Ben-David and Blais, 2020) showed that R(f∘g) = Ω(noisyR(f)⋅ R(g)). This implies that composition holds whenever noisyR(f) = Θ̃(R(f)). We show two results: 1. When R(f) = Θ(n), then noisyR(f) = Θ(R(f)). In other words, composition holds whenever the randomized query complexity of the outer function is full. 2. If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg̃(f∘g) = Ω(M(f) ⋅ deg̃(g)) (for some non-trivial complexity measure M(⋅)) was known to the best of our knowledge. We prove that deg̃(f∘g) = Ω̃(√{bs(f)} ⋅ deg̃(g)), where bs(f) is the block sensitivity of f. This implies that deg̃ composes when deg̃(f) is asymptotically equal to √{bs(f)}. It is already known that both R and deg̃ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Approximate degree
  • Boolean functions
  • Composition Theorem
  • Partial functions
  • Randomized Query Complexity


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


  1. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In STOC, pages 863-876, 2016. URL: https://doi.org/10.1145/2897518.2897644.
  2. Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao, and Avishay Tal. Degree vs. approximate degree and quantum implications of Huang’s sensitivity theorem. In STOC, pages 1330-1342, 2021. URL: https://doi.org/10.1145/3406325.3451047.
  3. Andris Ambainis. Polynomial degree and lower bounds in quantum complexity: Collision and element distinctness with small range. Theory Comput., 1(1):37-46, 2005. URL: https://doi.org/10.4086/toc.2005.v001a003.
  4. Andris Ambainis, Kaspars Balodis, Aleksandrs Belovs, Troy Lee, Miklos Santha, and Juris Smotrovs. Separations in query complexity based on pointer functions. Journal of the ACM, 64(5):32:1-32:24, 2017. URL: https://doi.org/10.1145/3106234.
  5. Andris Ambainis, Martins Kokainis, and Robin Kothari. Nearly optimal separations between communication (or query) complexity and partitions. In CCC, volume 50, pages 4:1-4:14, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.4.
  6. 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 FSTTCS, volume 93, pages 10:1-10:13, 2017. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2017.10.
  7. Andrew Bassilakis, Andrew Drucker, Mika Göös, Lunjia Hu, Weiyun Ma, and Li-Yang Tan. The power of many samples in query complexity. In ICALP, volume 168, pages 9:1-9:18, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.9.
  8. Shalev Ben-David and Eric Blais. A tight composition theorem for the randomized query complexity of partial functions. arXiv:2002.10809, 2020. Google Scholar
  9. Shalev Ben-David and Eric Blais. A tight composition theorem for the randomized query complexity of partial functions: Extended abstract. In FOCS, pages 240-246, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00031.
  10. 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. Google Scholar
  11. Shalev Ben-David, Adam Bouland, Ankit Garg, and Robin Kothari. Classical lower bounds from quantum upper bounds. In FOCS, pages 339-349, 2018. URL: https://doi.org/10.1109/FOCS.2018.00040.
  12. Shalev Ben-David, Mika Göös, Robin Kothari, and Thomas Watson. When Is Amplification Necessary for Composition in Randomized Query Complexity? In APPROX/RANDOM 2020, volume 176, pages 28:1-28:16, 2020. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.28.
  13. Shalev Ben-David and Robin Kothari. Randomized query complexity of sabotaged and composed functions. Theory Comput., 14(1):1-27, 2018. URL: https://doi.org/10.4086/toc.2018.v014a005.
  14. Eric Blais, Amit Weinstein, and Yuichi Yoshida. Partially symmetric functions are efficiently isomorphism testable. SIAM J. Comput., 44(2):411-432, 2015. URL: https://doi.org/10.1137/140971877.
  15. 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.
  16. Mark Bun, Robin Kothari, and Justin Thaler. Quantum algorithms and approximating polynomials for composed functions with shared inputs. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 662-678. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.42.
  17. Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and Markov-Bernstein inequalities. In ICALP, volume 7965, pages 303-314, 2013. URL: https://doi.org/10.1007/978-3-642-39206-1_26.
  18. Sourav Chakraborty. On the sensitivity of cyclically-invariant boolean functions. Discret. Math. Theor. Comput. Sci., 13(4):51-60, 2011. Google Scholar
  19. Sourav Chakraborty, Eldar Fischer, David García-Soriano, and Arie Matsliah. Junto-symmetric functions, hypergraph isomorphism and crunching. In CCC, pages 148-158, 2012. URL: https://doi.org/10.1109/CCC.2012.28.
  20. Sourav Chakraborty, Chandrima kayal, Rajat Mittal, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the composition of randomized query complexity and approximate degree. Electron. Colloquium Comput. Complex., 1(1):37-46, 2023. URL: https://doi.org/TR23-099.
  21. Sourav Chakraborty, Chandrima Kayal, and Manaswi Paraashar. Separations between combinatorial measures for transitive functions. In ICALP, volume 229, pages 36:1-36:20, 2022. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.36.
  22. Arkadev Chattopadhyay, Nikhil S. Mande, and Suhail Sherif. The log-approximate-rank conjecture is false. Journal of the ACM, 67, 2020. URL: https://eccc.weizmann.ac.il/report/2018/176/.
  23. Andrew Drucker. Block sensitivity of minterm-transitive functions. Theor. Comput. Sci., 412(41):5796-5801, 2011. URL: https://doi.org/10.1016/j.tcs.2011.06.025.
  24. Dmitry Gavinsky, Troy Lee, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity. In ICALP, volume 132, pages 64:1-64:13, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.64.
  25. Justin Gilmer, Michael E. Saks, and Srikanth Srinivasan. Composition limits and separating examples for some Boolean function complexity measures. Combinatorica, 36(3):265-311, 2016. URL: https://doi.org/10.1007/s00493-014-3189-x.
  26. Mika Göös, T. S. Jayram, Toniann Pitassi, and Thomas Watson. Randomized communication versus partition number. ACM Trans. Comput. Theory, 10(1), 2018. URL: https://doi.org/10.1145/3170711.
  27. Hao Huang. Induced subgraphs of hypercubes and a proof of the sensitivity conjecture. Annals of Mathematics, 190(3):949-955, 2019. URL: https://doi.org/10.4007/annals.2019.190.3.6.
  28. Shelby Kimmel. Quantum adversary (upper) bound. Chicago Journal of Theoretical Computer Science, 2013(4), April 2013. URL: https://doi.org/10.4086/cjtcs.2013.004.
  29. Troy Lee, Rajat Mittal, Ben W. Reichardt, Robert Špalek, and Mario Szegedy. Quantum query complexity of state conversion. In FOCS, pages 344-353, 2011. URL: https://doi.org/10.1109/FOCS.2011.75.
  30. Yaqiao Li. Conflict complexity is lower bounded by block sensitivity. Theor. Comput. Sci., 856:169-172, 2021. URL: https://doi.org/10.1016/j.tcs.2020.12.038.
  31. Ashley Montanaro. A composition theorem for decision tree complexity. Chicago Journal of Theoretical Computer Science, 2014(6), July 2014. Google Scholar
  32. Noam Nisan. Crew prams and decision trees. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 327-335, 1989. Google Scholar
  33. Noam Nisan and Mario Szegedy. On the degree of boolean functions as real polynomials. Comput. Complex., 4:301-313, 1994. URL: https://doi.org/10.1007/BF01263419.
  34. Ramamohan Paturi. On the degree of polynomials that approximate symmetric boolean functions. In STOC, pages 468-474, 1992. URL: https://doi.org/10.1145/129712.129758.
  35. Ben W. Reichardt. Reflections for quantum query algorithms, pages 560-569. SODA. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611973082.44.
  36. David Rubinstein. Sensitivity vs. block sensitivity of Boolean functions. Combinatorica, 15(2):297-299, 1995. URL: https://doi.org/10.1007/BF01200762.
  37. Petr Savický. On determinism versus unambiquous nondeterminism for decision trees. Electron. Colloquium Comput. Complex., 9(009), 2002. URL: https://eccc.weizmann.ac.il//report/2002/009/.
  38. A. A. Sherstov. Strong direct product theorems for quantum communication and query complexity. SIAM J. Comput., 41(5):1122-1165, 2012. URL: https://doi.org/10.1137/110842661.
  39. Alexander A. Sherstov. Approximating the AND-OR tree. Theory Comput., 9:653-663, 2013. URL: https://doi.org/10.4086/toc.2013.v009a020.
  40. Alexander A. Sherstov. The intersection of two halfspaces has high threshold degree. SIAM J. Comput., 42(6):2329-2374, 2013. URL: https://doi.org/10.1137/100785260.
  41. Alexander A. Sherstov. Making polynomials robust to noise. Theory Comput., 9:593-615, 2013. URL: https://doi.org/10.4086/toc.2013.v009a018.
  42. Xiaoming Sun. Block sensitivity of weakly symmetric functions. Theor. Comput. Sci., 384(1):87-91, 2007. URL: https://doi.org/10.1016/j.tcs.2007.05.020.
  43. Xiaoming Sun, Andrew Chi-Chih Yao, and Shengyu Zhang. Graph properties and circular functions: How low can quantum query complexity go? In CCC, pages 286-293, 2004. URL: https://doi.org/10.1109/CCC.2004.1313851.
  44. Avishay Tal. Properties and applications of boolean function composition. In ITCS, pages 441-454, 2013. URL: https://doi.org/10.1145/2422436.2422485.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail