On the Pseudo-Deterministic Query Complexity of NP Search Problems

Authors Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, Rahul Santhanam

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Author Details

Shafi Goldwasser
  • University of California, Berkeley, CA, USA
Russell Impagliazzo
  • University of California, San Diego, CA, USA
Toniann Pitassi
  • University of Toronto, Canada
  • Columbia University, New York, NY, USA
  • Institute of Advanced Study, Princeton, NJ, USA
Rahul Santhanam
  • University of Oxford, UK


We thank Ofer Grossman, Ran Raz, Avi Wigderson and Ryan Williams for helpful discussions. The quantum query upper bound for FIND1 was pointed out to the fourth author by Igor Oliveira. We also thank the anonymous CCC reviewers for very helpful comments.

Cite AsGet BibTex

Shafi Goldwasser, Russell Impagliazzo, Toniann Pitassi, and Rahul Santhanam. On the Pseudo-Deterministic Query Complexity of NP Search Problems. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 36:1-36:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study pseudo-deterministic query complexity - randomized query algorithms that are required to output the same answer with high probability on all inputs. We prove Ω(√n) lower bounds on the pseudo-deterministic complexity of a large family of search problems based on unsatisfiable random CNF instances, and also for the promise problem (FIND1) of finding a 1 in a vector populated with at least half one’s. This gives an exponential separation between randomized query complexity and pseudo-deterministic complexity, which is tight in the quantum setting. As applications we partially solve a related combinatorial coloring problem, and we separate random tree-like Resolution from its pseudo-deterministic version. In contrast to our lower bound, we show, surprisingly, that in the zero-error, average case setting, the three notions (deterministic, randomized, pseudo-deterministic) collapse.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Proof complexity
  • Pseudo-determinism
  • Query complexity
  • Proof complexity


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  1. Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 863-876. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897644.
  2. Scott Aaronson, Shalev Ben-David, Robin Kothari, and Avishay Tal. Quantum implications of huang’s sensitivity theorem. CoRR, abs/2004.13231, 2020. URL: http://arxiv.org/abs/2004.13231.
  3. Michael Alekhnovich and Alexander A. Razborov. Lower bounds for polynomial calculus: Non-binomial case. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, Las Vegas, Nevada, USA, pages 190-199, 2001. Google Scholar
  4. Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and Ronald de Wolf. Quantum lower bounds by polynomials. J. ACM, 48(4):778-797, 2001. URL: https://doi.org/10.1145/502090.502097.
  5. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theor. Comput. Sci., 288(1):21-43, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00144-X.
  6. Samuel R. Buss, Dima Grigoriev, Russell Impagliazzo, and Toniann Pitassi. Linear gaps between degrees for the polynomial calculus modulo distinct primes. J. Comput. Syst. Sci., 62(2):267-289, 2001. Google Scholar
  7. Samuel R. Buss, Leszek Aleksander Kolodziejczyk, and Neil Thapen. Fragments of approximate counting. J. Symb. Log., 79(2):496-525, 2014. URL: https://doi.org/10.1017/jsl.2013.37.
  8. Siu On Chan, James R. Lee, Prasad Raghavendra, and David Steurer. Approximate constraint satisfaction requires large LP relaxations. J. ACM, 63(4):34:1-34:22, 2016. URL: https://doi.org/10.1145/2811255.
  9. Richard Cleve. An introduction to quantum complexity theory. Quantum Computation and Quantum Information Theory, page 103–127, January 2001. URL: https://doi.org/10.1142/9789810248185_0004.
  10. Noah Fleming, Pravesh Kothari, and Toniann Pitassi. Semialgebraic proofs and efficient algorithm design. Found. Trends Theor. Comput. Sci., 14(1-2):1-221, 2019. URL: https://doi.org/10.1561/0400000086.
  11. Ankit Garg, Mika Göös, Pritish Kamath, and Dmitry Sokolov. Monotone circuit lower bounds from resolution. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 902-911. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188838.
  12. Eran Gat and Shafi Goldwasser. Probabilistic search algorithms with unique answers and their cryptographic applications. Electron. Colloquium Comput. Complex., 18:136, 2011. URL: http://eccc.hpi-web.de/report/2011/136.
  13. Oded Goldreich, Shafi Goldwasser, and Dana Ron. On the possibilities and limitations of pseudodeterministic algorithms. In 4th Innovations in Theoretical Computer Science Conference, ITCS, pages 127-138, 2013. Google Scholar
  14. Shafi Goldwasser and Ofer Grossman. Bipartite perfect matching in pseudo-deterministic NC. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, pages 87:1-87:13, 2017. Google Scholar
  15. Shafi Goldwasser, Ofer Grossman, and Dhiraj Holden. Pseudo-deterministic proofs. In 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, pages 17:1-17:18, 2018. Google Scholar
  16. Shafi Goldwasser, Ofer Grossman, Sidhanth Mohanty, and David P. Woodruff. Pseudo-deterministic streaming. CoRR, abs/1911.11368, 2019. URL: http://arxiv.org/abs/1911.11368.
  17. Shafi Goldwasser, Ofer Grossman, Sidhanth Mohanty, and David P. Woodruff. Pseudo-deterministic streaming. In Thomas Vidick, editor, 11th Innovations in Theoretical Computer Science Conference, ITCS 2020, January 12-14, 2020, Seattle, Washington, USA, volume 151 of LIPIcs, pages 79:1-79:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ITCS.2020.79.
  18. 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.
  19. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in monotone complexity and TFNP. In Avrim Blum, editor, 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, volume 124 of LIPIcs, pages 38:1-38:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.38.
  20. 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.
  21. Dima Grigoriev. Tseitin’s tautologies and lower bounds for nullstellensatz proofs. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 648-652, 1998. Google Scholar
  22. Ofer Grossman and Yang P. Liu. Reproducibility and pseudo-determinism in log-space. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 606-620, 2019. Google Scholar
  23. Lov K. Grover. A fast quantum mechanical algorithm for database search. In Gary L. Miller, editor, Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, May 22-24, 1996, pages 212-219. ACM, 1996. URL: https://doi.org/10.1145/237814.237866.
  24. Pavel Hrubes. On ε-sensitive monotone computations. Comput. Complex., 29(2):6, 2020. URL: https://doi.org/10.1007/s00037-020-00196-6.
  25. Hao Huang. Induced subgraphs of hypercubes and a proof of the sensitivity conjecture. CoRR, abs/1907.00847, 2019. URL: http://arxiv.org/abs/1907.00847.
  26. Trinh Huynh and Jakob Nordström. On the virtue of succinct proofs: amplifying communication complexity hardness to time-space trade-offs in proof complexity. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 233-248. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214000.
  27. László Lovász, Moni Naor, Ilan Newman, and Avi Wigderson. Search problems in the decision tree model. SIAM J. Discret. Math., 8(1):119-132, 1995. URL: https://doi.org/10.1137/S0895480192233867.
  28. Noam Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999-1007, 1991. Google Scholar
  29. 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.
  30. Igor Carboni Oliveira and Rahul Santhanam. Pseudodeterministic constructions in subexponential time. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 665-677. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055500.
  31. A. A. Razborov. Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic. Izvestiya RAN. Ser. Mat., pages 201-224, 1995. Google Scholar
  32. Dmitry Sokolov. Dag-like communication and its applications. In Pascal Weil, editor, Computer Science - Theory and Applications - 12th International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8-12, 2017, Proceedings, volume 10304 of Lecture Notes in Computer Science, pages 294-307. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-58747-9_26.