Document Open Access Logo

Total Search Problems in ZPP

Authors Noah Fleming , Stefan Grosser , Siddhartha Jain , Jiawei Li , Hanlin Ren , Morgan Shirley , Weiqiang Yuan

PDF

File

Thumbnail PDF
PDF
LIPIcs.ITCS.2026.60.pdf
  • Filesize: 1.07 MB
  • 26 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Oracles and decision trees
  • Theory of computation → Proof complexity
Keywords
  • TFNP
  • lossy code
  • randomized proof systems
  • query complexity

Metrics

Abstract

We initiate a systematic study of TFZPP, the class of total NP search problems solvable by polynomial time randomized algorithms. TFZPP contains a variety of important search problems such as Bertrand-Chebyshev (finding a prime between N and 2N), refuter problems for many circuit lower bounds, and Lossy-Code. The Lossy-Code problem has found prominence due to its fundamental connections to derandomization, catalytic computing, and the metamathematics of complexity theory, among other areas.
While TFZPP collapses to FP under standard derandomization assumptions in the white-box setting, we are able to separate TFZPP from the major TFNP subclasses in the black-box setting. In fact, we are able to separate it from every uniform TFNP class assuming that NP is not in quasi-polynomial time. To do so, we extend the connection between proof complexity and black-box TFNP to randomized proof systems and randomized reductions.
Next, we turn to developing a taxonomy of TFZPP problems. We highlight a problem called Nephew, originating from an infinity axiom in set theory. We show that Nephew is in PWPP∩ TFZPP and conjecture that it is not reducible to Lossy-Code. Intriguingly, except for some artificial examples, most other black-box TFZPP problems that we are aware of reduce to Lossy-Code:  
- We define a problem called Empty-Child capturing finding a leaf in a rooted (binary) tree, and show that this problem is equivalent to Lossy-Code. We also show that a variant of Empty-Child with "heights" is complete for the intersection of SOPL and Lossy-Code. 
- We strengthen Lossy-Code with several combinatorial inequalities such as the AM-GM inequality. Somewhat surprisingly, we show the resulting new problems are still reducible to Lossy-Code. A technical highlight of this result is that they are proved by formalizations in bounded arithmetic, specifically in Jeřábek’s theory APC₁ (JSL 2007). 
- Finally, we show that the Dense-Linear-Ordering problem reduces to Lossy-Code.

Cite As Get BibTex

Noah Fleming, Stefan Grosser, Siddhartha Jain, Jiawei Li, Hanlin Ren, Morgan Shirley, and Weiqiang Yuan. Total Search Problems in ZPP. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 60:1-60:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.60

Author Details

Noah Fleming
  • Lund University, Sweden
  • Columbia University, New York, NY, USA
Stefan Grosser
  • McGill University, Montreal, Canada
Siddhartha Jain
  • UT Austin, TX, USA
Jiawei Li
  • UT Austin, TX, USA
Hanlin Ren
  • Institute for Advanced Study, Princeton, NJ, USA
Morgan Shirley
  • Lund University, Sweden
Weiqiang Yuan
  • EPFL, Lausanne, Switzerland

Funding

  • Fleming, Noah: Supported by an NSERC Discovery grant and the Swedish Research Council under grant number 2025-06762.
  • Grosser, Stefan: Supported by the NSERC CGS D fellowship.
  • Jain, Siddhartha: Supported by Scott Aaronson’s Berkeley CIQC grant and an Amazon AI Fellowship.
  • Li, Jiawei: Supported by Scott Aaronson’s Open Philanthropy grant.
  • Shirley, Morgan: Supported by NSERC and by Knut and Alice Wallenberg grant KAW 2023.0116.
  • Yuan, Weiqiang: Supported by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00026.

Acknowledgements

We thank Robert Robere, Yuhao Li, and Ben Davis for extensive discussions about the Nephew problem and TFZPP. As well, we thank the reviewers for suggestions which improved the presentation of this paper.

References

  1. Aryan Agarwala and Ian Mertz. Bipartite matching is in catalytic logspace. In FOCS, 2025. To appear. URL: https://doi.org/10.48550/arXiv.2504.09991.
  2. Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. PRIMES is in P. Annals of Mathematics, 160(2):781-793, 2004. URL: https://doi.org/10.4007/annals.2004.160.781.
  3. Albert Atserias and Víctor Dalmau. A combinatorial characterization of resolution width. J. Comput. Syst. Sci., 74(3):323-334, 2008. URL: https://doi.org/10.1016/J.JCSS.2007.06.025.
  4. Albert Atserias and Neil Thapen. The ordering principle in a fragment of approximate counting. ACM Trans. Comput. Log., 15(4):29:1-29:11, 2014. URL: https://doi.org/10.1145/2629555.
  5. Paul Beame, Stephen A. Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. The relative complexity of NP search problems. J. Comput. Syst. Sci., 57(1):3-19, 1998. URL: https://doi.org/10.1006/JCSS.1998.1575.
  6. Huck Bennett, Surendra Ghentiyala, and Noah Stephens-Davidowitz. The more the merrier! On total coding and lattice problems and the complexity of finding multicollisions. In 16th Innovations in Theoretical Computer Science Conference, volume 325 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 14, 22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/lipics.itcs.2025.14.
  7. Maria Luisa Bonet and Nicola Galesi. Optimality of size-width tradeoffs for resolution. Comput. Complex., 10(4):261-276, 2001. URL: https://doi.org/10.1007/S000370100000.
  8. Egon Börger, Erich Grädel, and Yuri Gurevich. The classical decision problem. Springer Science & Business Media, 2001. Google Scholar
  9. Harry Buhrman, Richard Cleve, Michal Koucký, Bruno Loff, and Florian Speelman. Computing with a full memory: catalytic space. In STOC, pages 857-866. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591874.
  10. Joshua Buresh-Oppenheim. On the TFNP complexity of factoring. Unpublished, 2006. Google Scholar
  11. Sam Buss, Noah Fleming, and Russell Impagliazzo. TFNP characterizations of proof systems and monotone circuits. 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 30:1-30:40. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.30.
  12. Samuel R Buss and Alan S Johnson. Propositional proofs and reductions between np search problems. Annals of Pure and Applied Logic, 163(9):1163-1182, 2012. URL: https://doi.org/10.1016/J.APAL.2012.01.015.
  13. Samuel R. Buss, Leszek Aleksander Kołodziejczyk, and Neil Thapen. Fragments of approximate counting. J. Symb. Log., 79(2):496-525, 2014. URL: https://doi.org/10.1017/JSL.2013.37.
  14. Lijie Chen, Shuichi Hirahara, and Hanlin Ren. Symmetric exponential time requires near-maximum circuit size. In STOC'24 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1990-1999. ACM, New York, [2024] ©2024. URL: https://doi.org/10.1145/3618260.3649624.
  15. Lijie Chen, Ce Jin, Rahul Santhanam, and Ryan Williams. Constructive separations and their consequences. TheoretiCS, 3, 2024. URL: https://doi.org/10.46298/THEORETICS.24.3.
  16. Lijie Chen, Jiatu Li, and Igor C. Oliveira. Reverse mathematics of complexity lower bounds. In FOCS, pages 505-527. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00040.
  17. Lijie Chen, Xin Lyu, and R. Ryan Williams. Almost-everywhere circuit lower bounds from non-trivial derandomization. In FOCS, pages 1-12. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00009.
  18. Lijie Chen, Roei Tell, and Ryan Williams. Derandomization vs refutation: A unified framework for characterizing derandomization. In FOCS, pages 1008-1047. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00062.
  19. Xi Chen and Xiaotie Deng. Settling the complexity of two-player Nash equilibrium. In FOCS, pages 261-272. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/FOCS.2006.69.
  20. Yeyuan Chen, Yizhi Huang, Jiatu Li, and Hanlin Ren. Range avoidance, remote point, and hard partial truth table via satisfying-pairs algorithms. In STOC, pages 1058-1066. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585147.
  21. Mario Chiari and Jan Krajícek. Witnessing functions in bounded arithmetic and search problems. J. Symb. Log., 63(3):1095-1115, 1998. URL: https://doi.org/10.2307/2586729.
  22. Jonas Conneryd, Susanna F. de Rezende, Jakob Nordström, Shuo Pang, and Kilian Risse. Graph colouring is hard on average for Polynomial Calculus and Nullstellensatz. In FOCS, pages 1-11. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00007.
  23. James Cook, Jiatu Li, Ian Mertz, and Edward Pyne. The structure of catalytic space: Capturing randomness and time via compression. In STOC, pages 554-564. ACM, 2025. URL: https://doi.org/10.1145/3717823.3718112.
  24. Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM J. Comput., 39(1):195-259, 2009. URL: https://doi.org/10.1137/070699652.
  25. Ben Davis and Robert Robere. Colourful TFNP and propositional proofs. In Amnon Ta-Shma, editor, 38th Computational Complexity Conference, CCC 2023, July 17-20, 2023, Warwick, UK, volume 264 of LIPIcs, pages 36:1-36:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.CCC.2023.36.
  26. John Fearnley, Paul Goldberg, Alexandros Hollender, and Rahul Savani. The complexity of gradient descent: CLS = PPAD ∩ PLS. J. ACM, 70(1):7:1-7:74, 2023. URL: https://doi.org/10.1145/3568163.
  27. Noah Fleming, Stefan Grosser, Toniann Pitassi, and Robert Robere. Black-box PPP is not Turing-closed. In Bojan Mohar, Igor Shinkar, and Ryan O'Donnell, editors, Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, Vancouver, BC, Canada, June 24-28, 2024, pages 1405-1414. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649769.
  28. Noah Fleming, Deniz Imrek, and Christophe Marciot. Provably total functions in the polynomial hierarchy. In Srikanth Srinivasan, editor, 40th Computational Complexity Conference, CCC 2025, August 5-8, 2025, Toronto, Canada, volume 339 of LIPIcs, pages 28:1-28:40. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.CCC.2025.28.
  29. 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.
  30. Lukáš Folwarczný, Mika Göös, Pavel Hubáček, Gilbert Maystre, and Weiqiang Yuan. One-way functions vs. TFNP: simpler and improved. In 15th Innovations in Theoretical Computer Science Conference, volume 287 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 50, 14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/lipics.itcs.2024.50.
  31. Karthik Gajulapalli, Alexander Golovnev, Satyajeet Nagargoje, and Sidhant Saraogi. Range avoidance for constant depth circuits: Hardness and algorithms. In APPROX/RANDOM, volume 275 of LIPIcs, pages 65:1-65:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.65.
  32. Surendra Ghentiyala and Zeyong Li. Hierarchies within TFNP: building blocks and collapses. CoRR, 2025. URL: https://doi.org/10.48550/arXiv.2507.21550.
  33. Oded Goldreich, Shafi Goldwasser, and Silvio Micali. How to construct random functions. J. ACM, 33(4):792-807, 1986. URL: https://doi.org/10.1145/6490.6503.
  34. Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao. Further collapses in TFNP. SIAM J. Comput., 53(3):573-587, 2024. URL: https://doi.org/10.1137/22M1498346.
  35. 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.
  36. Svyatoslav Gryaznov. Notes on resolution over linear equations. In CSR, volume 11532 of Lecture Notes in Computer Science, pages 168-179. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-19955-5_15.
  37. Venkatesan Guruswami, Xin Lyu, and Xiuhan Wang. Range avoidance for low-depth circuits and connections to pseudorandomness. ACM Trans. Comput. Theory, 17(2):14:1-14:23, 2025. URL: https://doi.org/10.1145/3718745.
  38. Mika Göös, Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao. Separations in proof complexity and TFNP. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 1150-1161, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00111.
  39. Jiří Hanika. Search Problems and Bounded Arithmetic. PhD thesis, Charles University, Prague, 2004. Google Scholar
  40. Edward A. Hirsch and Ilya Volkovich. Upper and lower bounds for the linear ordering principle. CoRR, 2025. URL: https://doi.org/10.48550/arXiv.2503.19188.
  41. Max Hopkins and Ting-Chun Lin. Explicit lower bounds against Ω(n)-rounds of sum-of-squares. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 662-673. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00069.
  42. Pavel Hubácek, Erfan Khaniki, and Neil Thapen. TFNP intersections through the lens of feasible disjunction. In Venkatesan Guruswami, editor, 15th Innovations in Theoretical Computer Science Conference, ITCS 2024, January 30 to February 2, 2024, Berkeley, CA, USA, volume 287 of LIPIcs, pages 63:1-63:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ITCS.2024.63.
  43. Pavel Hubáček, Chethan Kamath, Karel Král, and Veronika Slívová. On average-case hardness in TFNP from one-way functions. In Theory of cryptography. Part III, volume 12552 of Lecture Notes in Comput. Sci., pages 614-638. Springer, Cham, [2020] ©2020. URL: https://doi.org/10.1007/978-3-030-64381-2_22.
  44. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In STOC, pages 220-229. ACM, 1997. URL: https://doi.org/10.1145/258533.258590.
  45. Emil Jeřábek. Dual weak pigeonhole principle, Boolean complexity, and derandomization. Ann. Pure Appl. Log., 129(1-3):1-37, 2004. URL: https://doi.org/10.1016/j.apal.2003.12.003.
  46. Emil Jeřábek. Weak pigeonhole principle and randomized computation. PhD thesis, Charles University in Prague, 2005. Google Scholar
  47. Emil Jeřábek. Approximate counting in bounded arithmetic. J. Symb. Log., 72(3):959-993, 2007. URL: https://doi.org/10.2178/JSL/1191333850.
  48. Emil Jeřábek. On independence of variants of the weak pigeonhole principle. J. Log. Comput., 17(3):587-604, 2007. URL: https://doi.org/10.1093/LOGCOM/EXM017.
  49. Emil Jeřábek. Integer factoring and modular square roots. J. Comput. Syst. Sci., 82(2):380-394, 2016. URL: https://doi.org/10.1016/J.JCSS.2015.08.001.
  50. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? J. Comput. Syst. Sci., 37(1):79-100, 1988. URL: https://doi.org/10.1016/0022-0000(88)90046-3.
  51. Robert Kleinberg, Oliver Korten, Daniel Mitropolsky, and Christos H. Papadimitriou. Total functions in the polynomial hierarchy. In ITCS, volume 185 of LIPIcs, pages 44:1-44:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.44.
  52. Leszek Aleksander Kolodziejczyk and Neil Thapen. Approximate counting and NP search problems. J. Math. Log., 22(3):2250012:1-2250012:31, 2022. URL: https://doi.org/10.1142/S021906132250012X.
  53. Oliver Korten. The hardest explicit construction. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science - FOCS 2021, pages 433-444. IEEE Computer Soc., Los Alamitos, CA, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00051.
  54. Oliver Korten. Derandomization from time-space tradeoffs. In CCC, volume 234 of LIPIcs, pages 37:1-37:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.37.
  55. Oliver Korten. Range avoidance and the complexity of explicit constructions. Bull. EATCS, 145, 2025. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/825.
  56. Oliver Korten and Toniann Pitassi. Strong vs. weak range avoidance and the linear ordering principle. In FOCS, pages 1388-1407. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00089.
  57. Michal Koucký, Ian Mertz, Edward Pyne, and Sasha Sami. Collapsing catalytic classes. In FOCS, 2025. To appear. URL: https://doi.org/10.48550/arXiv.2504.08444.
  58. Jan Krajíček. Dual weak pigeonhole principle, pseudo-surjective functions, and provability of circuit lower bounds. J. Symb. Log., 69(1):265-286, 2004. URL: https://doi.org/10.2178/jsl/1080938841.
  59. Balakrishnan Krishnamurthy. Short proofs for tricky formulas. Acta Informatica, 22(3):253-275, 1985. URL: https://doi.org/10.1007/BF00265682.
  60. Jiatu Li, Edward Pyne, and Roei Tell. Distinguishing, predicting, and certifying: On the long reach of partial notions of pseudorandomness. In FOCS, pages 1-13. IEEE, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00095.
  61. Jiawei Li, Yuhao Li, and Hanlin Ren. Metamathematics of resolution lower bounds: A TFNP perspective. CoRR, abs/2411.15515, 2024. URL: https://doi.org/10.48550/arXiv.2411.15515.
  62. Yuhao Li, William Pires, and Robert Robere. Intersection classes in TFNP and proof complexity. In Venkatesan Guruswami, editor, 15th Innovations in Theoretical Computer Science Conference, ITCS 2024, January 30 to February 2, 2024, Berkeley, CA, USA, volume 287 of LIPIcs, pages 74:1-74:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ITCS.2024.74.
  63. Zeyong Li. Symmetric exponential time requires near-maximum circuit size: Simplified, truly uniform. In STOC, pages 2000-2007. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649615.
  64. Ralph C. Merkle. A digital signature based on a conventional encryption function. In CRYPTO, volume 293 of Lecture Notes in Computer Science, pages 369-378. Springer, 1987. URL: https://doi.org/10.1007/3-540-48184-2_32.
  65. Ian Mertz. Reusing space: Techniques and open problems. Bull. EATCS, 141, 2023. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/780.
  66. Moritz Müller. Typical forcings, NP search problems and an extension of a theorem of riis. Ann. Pure Appl. Log., 172(4):102930, 2021. URL: https://doi.org/10.1016/J.APAL.2020.102930.
  67. Noam Nisan and Avi Wigderson. Hardness vs randomness. J. Comput. Syst. Sci., 49(2):149-167, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80043-1.
  68. Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci., 48(3):498-532, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80063-7.
  69. Jeff B. Paris, A. J. Wilkie, and Alan R. Woods. Provability of the pigeonhole principle and the existence of infinitely many primes. J. Symb. Log., 53(4):1235-1244, 1988. URL: https://doi.org/10.1017/S0022481200028061.
  70. Amol Pasarkar, Christos H. Papadimitriou, and Mihalis Yannakakis. Extremal combinatorics, iterated pigeonhole arguments and generalizations of PPP. 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 88:1-88:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.88.
  71. Aaron Potechin. Sum of squares bounds for the ordering principle. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 38:1-38:37. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.38.
  72. Pavel Pudlák. On the complexity of finding falsifying assignments for Herbrand disjunctions. Arch. Math. Log., 54(7-8):769-783, 2015. URL: https://doi.org/10.1007/S00153-015-0439-6.
  73. Pavel Pudlák and Neil Thapen. Random resolution refutations. Comput. Complex., 28(2):185-239, 2019. URL: https://doi.org/10.1007/S00037-019-00182-7.
  74. Edward Pyne. Derandomizing logspace with a small shared hard drive. In CCC, volume 300 of LIPIcs, pages 4:1-4:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.4.
  75. Edward Pyne, Ran Raz, and Wei Zhan. Certified hardness vs. randomness for log-space. In FOCS, pages 989-1007. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00061.
  76. Alexander A Razborov. Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Mathematical Notes of the Academy of Sciences of the USSR, 41(4):333-338, 1987. Google Scholar
  77. Hanlin Ren, Rahul Santhanam, and Zhikun Wang. On the range avoidance problem for circuits. In FOCS, pages 640-650. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00067.
  78. Søren Riis. A complexity gap for tree resolution. Comput. Complex., 10(3):179-209, 2001. URL: https://doi.org/10.1007/S00037-001-8194-Y.
  79. Roman Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In STOC, pages 77-82. ACM, 1987. URL: https://doi.org/10.1145/28395.28404.
  80. Neil Thapen. The weak pigeonhole principle in models of bounded arithmetic. PhD thesis, University of Oxford, 2002. Google Scholar
  81. Neil Thapen. How to fit large complexity classes into TFNP. CoRR, abs/2412.09984, 2024. URL: https://doi.org/10.48550/arXiv.2412.09984.
  82. Christopher Umans. Pseudo-random generators for all hardnesses. J. Comput. Syst. Sci., 67(2):419-440, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00046-1.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact