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Upper and Lower Bounds for the Linear Ordering Principle

Authors Edward A. Hirsch , Ilya Volkovich

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LIPIcs.STACS.2026.52.pdf
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ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Circuit complexity
Keywords
  • Complexity Classes
  • Structural Complexity Theory
  • Linear Ordering Principle
  • Symmetric Alternation
  • Merlin-Arthur Protocols
  • Karp-Lipton Collapse

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Abstract

Korten and Pitassi (FOCS, 2024) defined a new complexity class L₂^P as the polynomial-time Turing closure of the Linear Ordering Principle (a total function extending finding the minimum of an order [M. Chiari and J. Krajíček, 1998] to the case where the order is not linear). They put it between MA (Merlin-Arthur protocols) and S₂^P (the second symmetric level of the polynomial hierarchy).
In this paper we sandwich L₂^P between P^prMA and P^prSBP. (The oracles here are promise problems, and SBP is the only known class between MA and AM.) The containment in P^prSBP is proved via an iterative process that uses a prSBP oracle to estimate the average order rank of a subset and find the minimum of a linear order.
Another containment result of this paper is P^prO₂^P ⊆ O₂^P (where O₂^P is the input-oblivious version of S₂^P). These containment results altogether have several byproducts:  
- We give an affirmative answer to an open question posed by Chakaravarthy and Roy (Computational Complexity, 2011) whether P^prMA ⊆ S₂^P, thereby settling the relative standing of the existing (non-oblivious) Karp–Lipton–style collapse results of [V. T. Chakaravarthy and S. Roy, 2011] and [J.-Y. Cai, 2007],
- We give an affirmative answer to an open question of Korten and Pitassi whether a Karp-Lipton-style collapse can be proven for L₂^P,
- We show that the Karp-Lipton-style collapse to P^prOMA is actually better than both known collapses to P^prMA due to Chakaravarthy and Roy (Computational Complexity, 2011) and to O₂^P also due to Chakaravarthy and Roy (STACS, 2006). Thus we resolve the controversy between previously incomparable Karp-Lipton collapses stemming from these two lines of research.

Cite As Get BibTex

Edward A. Hirsch and Ilya Volkovich. Upper and Lower Bounds for the Linear Ordering Principle. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 52:1-52:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.52

Author Details

Edward A. Hirsch
  • Department of Computer Science, Ariel University, Israel
Ilya Volkovich
  • Boston College, Chestnut Hill, MA, USA

Acknowledgements

The authors are grateful to Yaroslav Alekseev for discussing and to Dmitry Itsykson for discussing and proofreading a preliminary version of this paper. This research was conducted with the support of the State of Israel, the Ministry of Immigrant Absorption, and the Center for the Absorption of Scientists.

References

  1. S. Aaronson, R. Kothari, W. Kretschmer, and J. Thaler. Quantum lower bounds for approximate counting via Laurent polynomials. In 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 7:1-7:47. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.7.
  2. S. Aaronson and P. Rall. Quantum approximate counting, simplified. In Martin Farach-Colton and Inge Li Gørtz, editors, 3rd Symposium on Simplicity in Algorithms, SOSA 2020, pages 24-32. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611976014.5.
  3. V. Arvind, J. Köbler, U. Schöning, and R. Schuler. If NP has polynomial-size circuits, then MA=AM. Theor. Comput. Sci., 137(2):279-282, 1995. URL: https://doi.org/10.1016/0304-3975(95)91133-B.
  4. B. Aydinlio~glu, D. Gutfreund, J. M. Hitchcock, and A. Kawachi. Derandomizing arthur-merlin games and approximate counting implies exponential-size lower bounds. Comput. Complex., 20(2):329-366, 2011. URL: https://doi.org/10.1007/s00037-011-0010-8.
  5. L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991. URL: https://doi.org/10.1007/BF01200056.
  6. R. Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational Complexity, 4:339-349, 1994. URL: https://doi.org/10.1007/BF01263422.
  7. E. Böhler, C. Glaßer, and D. Meister. Error-bounded probabilistic computations between MA and AM. J. Comput. Syst. Sci., 72(6):1043-1076, 2006. URL: https://doi.org/10.1016/J.JCSS.2006.05.001.
  8. N. H. Bshouty, R. Cleve, R. Gavaldà, S. Kannan, and C. Tamon. Oracles and queries that are sufficient for exact learning. J. Comput. Syst. Sci., 52(3):421-433, 1996. URL: https://doi.org/10.1006/JCSS.1996.0032.
  9. H. Buhrman and L. Fortnow. One-sided versus two-sided error in probabilistic computation. In STACS, pages 100-109, 1999. Google Scholar
  10. H. Buhrman, L. Fortnow, and T. Thierauf. Nonrelativizing separations. In Proceedings of the 13th Annual IEEE Conference on Computational Complexity (CCC), pages 8-12, 1998. Google Scholar
  11. H. Buhrman and S. Homer. Superpolynomial circuits, almost sparse oracles and the exponential hierarchy. In Foundations of Software Technology and Theoretical Computer Science, 12th Conference, New Delhi, India, December 18-20, 1992, Proceedings, pages 116-127, 1992. URL: https://doi.org/10.1007/3-540-56287-7_99.
  12. J.-Y. Cai. S₂P ⊆ ZPP^NP. Journal of Computer and System Sciences, 73(1):25-35, 2007. Google Scholar
  13. R. Canetti. More on BPP and the polynomial-time hierarchy. Inf. Process. Lett., 57(5):237-241, 1996. URL: https://doi.org/10.1016/0020-0190(96)00016-6.
  14. V. T. Chakaravarthy and S. Roy. Oblivious symmetric alternation. In STACS, pages 230-241, 2006. Google Scholar
  15. V. T. Chakaravarthy and S. Roy. Arthur and Merlin as oracles. Comput. Complex., 20(3):505-558, 2011. URL: https://doi.org/10.1007/s00037-011-0015-3.
  16. L. Chen, S. Hirahara, and H. Ren. Symmetric exponential time requires near-maximum circuit size. In Proceedings of the 56th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2024, to appear. Association for Computing Machinery, 2024. Google Scholar
  17. L. Chen, D. M. McKay, C. D. Murray, and R. R. Williams. Relations and equivalences between circuit lower bounds and Karp-Lipton theorems. In CCC-2019, LIPICS, pages 30:1-21, 2019. Google Scholar
  18. M. Chiari and J. Krajíček. Witnessing functions in bounded arithmetic and search problems. The Journal of Symbolic Logic, 63(3):1095-1115, 1998. URL: https://doi.org/10.2307/2586729.
  19. John Fearnley, Spencer Gordon, Ruta Mehta, and Rahul Savani. Unique end of potential line. Journal of Computer and System Sciences, 114:1-35, 2020. URL: https://doi.org/10.1016/J.JCSS.2020.05.007.
  20. L. Fortnow and N. Reingold. PP is closed under truth-table reductions. Inf. Comput., 124(1):1-6, 1996. URL: https://doi.org/10.1006/inco.1996.0001.
  21. K. Gajulapalli, S. Ghentiyala, Z. Li, and S. Saraogi. Downward self-reducibility in the total function polynomial hierarchy. Electron. Colloquium Comput. Complex., TR25-121, 2025. URL: https://eccc.weizmann.ac.il/report/2025/121.
  22. K. Gajulapalli, Z. Li, and I. Volkovich. Oblivious complexity classes revisited: Lower bounds and hierarchies. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2024, volume 323 of LIPIcs, pages 23:1-23:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2024.23.
  23. S. Goldwasser and M. Sipser. Private coins versus public coins in interactive proof systems. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC), pages 59-68, 1986. Google Scholar
  24. J. Grollmann and A. L. Selman. Complexity measures for public-key cryptosystems. SIAM J. Comput., 17(2):309-335, 1988. URL: https://doi.org/10.1137/0217018.
  25. Y. Han, L. A. Hemaspaandra, and T. Thierauf. Threshold computation and cryptographic security. SIAM J. Comput., 26(1):59-78, 1997. URL: https://doi.org/10.1137/S0097539792240467.
  26. R. Impagliazzo, V. Kabanets, and I. Volkovich. Synergy between circuit obfuscation and circuit minimization. In APPROX/RANDOM 2023, volume 275 of LIPIcs, pages 31:1-31:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.31.
  27. R. Impagliazzo, V. Kabanets, and A. Wigderson. In search of an easy witness: exponential time vs. probabilistic polynomial time. J. of Computer and System Sciences, 65(4):672-694, 2002. URL: https://doi.org/10.1016/S0022-0000(02)00024-7.
  28. E. Jeřábek. Dual weak pigeonhole principle, boolean complexity, and derandomization. Annals of Pure and Applied Logic, 129:1-37, 2004. URL: https://doi.org/10.1016/J.APAL.2003.12.003.
  29. E. Jeřábek. Approximate counting in bounded arithmetic. Journal of Symbolic Logic, 72(3):959-993, 2007. URL: https://doi.org/10.2178/JSL/1191333850.
  30. M. Jerrum, L. G. Valiant, and V. V. Vazirani. Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci., 43:169-188, 1986. URL: https://doi.org/10.1016/0304-3975(86)90174-X.
  31. R. Kannan. Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control, 55(1-3):40-56, 1982. URL: https://doi.org/10.1016/S0019-9958(82)90382-5.
  32. R. M. Karp and R. J. Lipton. Some connections between nonuniform and uniform complexity classes. In Proceedings of the 12th Annual ACM Symposium on Theory of Computing, April 28-30, 1980, Los Angeles, California, USA, pages 302-309, 1980. URL: https://doi.org/10.1145/800141.804678.
  33. R. Kleinberg, O. Korten, D. Mitropolsky, and C. Papadimitriou. Total Functions in the Polynomial Hierarchy. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 of Leibniz International Proceedings in Informatics (LIPIcs), pages 44:1-44:18, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.44.
  34. J. Köbler and O. Watanabe. New collapse consequences of NP having small circuits. SIAM J. Comput., 28(1):311-324, 1998. URL: https://doi.org/10.1137/S0097539795296206.
  35. O. Korten. The hardest explicit construction. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 433-444. IEEE, 2022. Google Scholar
  36. O. Korten and T. Pitassi. Strong vs. Weak Range Avoidance and the Linear Ordering Principle. Electron. Colloquium Comput. Complex., TR24-076, 2024. URL: https://eccc.weizmann.ac.il/report/2024/076.
  37. J. Krajíček. Proof Complexity Generators. London Mathematical Society Lecture Note Series. Cambridge University Press, 2025. Google Scholar
  38. Z. Li. Symmetric exponential time requires near-maximum circuit size: Simplified, truly uniform. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 2000-2007. ACM, 2024. URL: https://doi.org/10.1145/3618260.3649615.
  39. S. C. Marshall, S. Aaronson, and V. Dunjko. Improved separation between quantum and classical computers for sampling and functional tasks. In 40th Computational Complexity Conference, CCC 2025, August 5-8, 2025, Toronto, Canada, volume 339 of LIPIcs, pages 5:1-5:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.CCC.2025.5.
  40. P. B. Miltersen, N. V. Vinodchandran, and O. Watanabe. Super-polynomial versus half-exponential circuit size in the exponential hierarchy. In COCOON, pages 210-220, 1999. Google Scholar
  41. R. O'Donnell and A. C. C. Say. The weakness of CTC qubits and the power of approximate counting. ACM Trans. Comput. Theory, 10(2):5:1-5:22, 2018. URL: https://doi.org/10.1145/3196832.
  42. J. Paris, A. Wilkie, and A. 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.
  43. A. Russell and R. Sundaram. Symmetric alternation captures BPP. Comput. Complex., 7(2):152-162, 1998. URL: https://doi.org/10.1007/s000370050007.
  44. R. Santhanam. Circuit lower bounds for Merlin-Arthur classes. SIAM J. Comput., 39(3):1038-1061, 2009. URL: https://doi.org/10.1137/070702680.
  45. U. Schöning. A low and a high hierarchy within NP. Journal of Computer and System Sciences, 27:14-28, 1983. URL: https://doi.org/10.1016/0022-0000(83)90027-2.
  46. R. Shaltiel and C. Umans. Pseudorandomness for approximate counting and sampling. Comput. Complex., 15(4):298-341, 2006. URL: https://doi.org/10.1007/s00037-007-0218-9.
  47. M. Sipser. A complexity theoretic approach to randomness. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 330-335. ACM, 1983. URL: https://doi.org/10.1145/800061.808762.
  48. L. J. Stockmeyer. On approximation algorithms for #P. SIAM J. Comput., 14(4):849-861, 1985. URL: https://doi.org/10.1137/0214060.
  49. S. Toda. PP is as hard as the polynomial time hierarchy. SIAM J. Comput., 20(5):865-877, 1991. URL: https://doi.org/10.1137/0220053.
  50. N. V. Vinodchandran. A note on the circuit complexity of PP. Theor. Comput. Sci., 347(1-2):415-418, 2005. URL: https://doi.org/10.1016/j.tcs.2005.07.032.
  51. I. Volkovich. On learning, lower bounds and (un)keeping promises. In Proceedings of the 41st ICALP, pages 1027-1038, 2014. Google Scholar
  52. I. Volkovich. The untold story of SBP. In Henning Fernau, editor, The 15th International Computer Science Symposium in Russia, CSR, volume 12159 of Lecture Notes in Computer Science, pages 393-405. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-50026-9_29.
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