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Degrees of Second and Higher-Order Polynomials

Authors Donghyun Lim , Martin Ziegler

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LIPIcs.FSTTCS.2025.42.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing
  • Theory of computation → Higher order logic
  • Theory of computation → Type theory
Keywords
  • Logic in Computer Science
  • Higher Order Program Analysis
  • Asymptotic Type Theory

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Abstract

Second-order polynomials generalize classical (=first-order) ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory, extending for instance discrete classes (like P/FP or PSPACE/FPSPACE) to operators in Analysis [http://doi.org/10.1137/S0097539794263452], [http://doi.org/10.1145/2189778.2189780]. The degree subclassifies ordinary polynomial growth into linear, quadratic, cubic, etc. To similarly classify second-order polynomials, we (well-)define their degree by structural induction as an "arctic" first-order polynomial: a term/expression over integer variable D and operations + and ⋅ and binary max(). This generalized degree turns out to transform nicely under (now two kinds of) polynomial composition. As examples, we collect and determine the degrees of previous and new asymptotic analyses of algorithms and operators receiving function/oracle arguments. Then we motivate and introduce third-order polynomials and their degrees as arctic second-order polynomials, along with their transformations under three kinds of composition. Proceeding to fourth order and beyond yields a hierarchy, with characterization in Simply Typed Lambda Calculus.

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Donghyun Lim and Martin Ziegler. Degrees of Second and Higher-Order Polynomials. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.42

Author Details

Donghyun Lim
  • KAIST, Daejeon, Republic of Korea
Martin Ziegler
  • KAIST, Daejeon, Republic of Korea

Funding

This work was supported by the National Research Foundation of Korea and by the Korean Ministry of Science and ICT with grants NRF-2017R1E1A1A03071032 and NRF-2016K1A3A7A03950702 and by KAIST in-house grant.

Acknowledgements

We thank anonymous reviewers for feedback and guidance.

References

  1. Aras Bacho and Martin Ziegler. Second-order parameterizations for the complexity theory of integrable functions. arXiv CoRR, 2506.11210, 2025. URL: https://doi.org/10.48550/arXiv.2506.11210.
  2. Patrick Baillot, Ugo Dal Lago, Cynthia Kop, and Deivid Vale. On basic feasible functionals and the interpretation method. In Naoki Kobayashi and James Worrell, editors, Proc.II of 27th Int. Conf. Foundations of Software Science and Computation Structures (FoSSaCS2024), held as part of the European Joint Conferences on Theory and Practice of Software (ETAPS2024) in Luxembourg, volume 14575 of Lecture Notes in Computer Science, pages 70-91. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-57231-9_4.
  3. Eyvind Martol Briseid. Fixed point of generalized contractive mappings. Journal of Nonlinear and Convex Analysis, 9:2:181-204, 2008. Google Scholar
  4. Eyvind Martol Briseid. Logical aspects of rates of convergence in metric spaces. The Journal of Symbolic Logic, 74(4):1401-1428, 2009. URL: https://doi.org/10.2178/jsl/1254748697.
  5. Michael Codish, Yoav Fekete, Carsten Fuhs, Jürgen Giesl, and Johannes Waldmann. Exotic Semiring Constraints. In SMT 2012. 10th International Workshop on Satisfiability Modulo Theories, volume 20 of EasyChair Proceedings in Computing, pages 88-97. EasyChair, 2013. URL: https://doi.org/10.29007/QQVT.
  6. Martin Dietzfelbinger. Primality Testing in Polynomial Time, volume 3000 of Lecture Notes in Computer Science. Springer, 2004. URL: https://doi.org/10.1007/B12334.
  7. Hugo Férée and Mathieu Hoyrup. Higher-order complexity in analysis. HAL Archive, July 2013. Proc 10th Int.Conf. Computability and Complexity in Analysis (CCA). URL: https://hal.inria.fr/hal-00915973.
  8. Anka Gajentaan and Mark H Overmars. On a class of O(n²) problems in computational geometry. Computational Geometry, 5(3):165-185, 1995. URL: https://doi.org/10.1016/0925-7721(95)00022-2.
  9. Mikhail Gromov. Groups of polynomial growth and expanding maps. Publications Mathématiques de L’Institut des Hautes Scientifiques, 53:53-78, 1981. URL: https://doi.org/10.1007/BF02698687.
  10. Emmanuel Hainry, Bruce M. Kapron, Jean-Yves Marion, and Romain Péchoux. Declassification policy for program complexity analysis. In Pawel Sobocinski, Ugo Dal Lago, and Javier Esparza, editors, Proc. 39th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2024, Tallinn, Estonia, pages 41:1-41:14. ACM, 2024. URL: https://doi.org/10.1145/3661814.3662100.
  11. Emmanuel Hainry and Romain Péchoux. Higher order interpretation for higher order complexity. In Thomas Eiter and David Sands, editors, LPAR-21, 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Maun, Botswana, May 7-12, 2017, volume 46 of EPiC Series in Computing, pages 269-285. EasyChair, 2017. URL: https://doi.org/10.29007/1TKW.
  12. Peter Hertling. Topological complexity with continuous operations. Journal of Complexity, 12(4):315-338, 1996. URL: https://doi.org/10.1006/jcom.1996.0021.
  13. Bruce M. Kapron and Stephen A. Cook. A new characterization of type-2 feasibility. SIAM J. Comput., 25(1):117-132, 1996. URL: https://doi.org/10.1137/S0097539794263452.
  14. Bruce M. Kapron and Florian Steinberg. Type-two polynomial-time and restricted lookahead. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '18, pages 579-588, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3209108.3209124.
  15. Akitoshi Kawamura and Stephen A. Cook. Complexity theory for operators in analysis. ACM Trans. Comput. Theory, 4(2):5:1-5:24, 2012. URL: https://doi.org/10.1145/2189778.2189780.
  16. Akitoshi Kawamura, Florian Steinberg, and Holger Thies. Second-order linear-time computability with applications to computable analysis. In Proc. 15th Conf. on Theory and Applications of Models of Computation (TAMC), pages 337-358, Berlin, Heidelberg, 2019. Springer-Verlag. URL: https://doi.org/10.1007/978-3-030-14812-6_21.
  17. Ker-I Ko. Complexity Theory of Real Functions, volume 3 of Progress in theoretical computer science. Birkhäuser / Springer, 1991. URL: https://doi.org/10.1007/978-1-4684-6802-1.
  18. Ivan Koswara, Gleb Pogudin, Svetlana Selivanova, and Martin Ziegler. Bit-complexity of classical solutions of linear evolutionary systems of partial differential equations. Journal of Complexity, 76:101727, 2023. URL: https://doi.org/10.1016/j.jco.2022.101727.
  19. David A. Levin and Yuval Peres. Markov chains and mixing times. American Mathematical Society, Providence, Rhode Island, 2009. Google Scholar
  20. Donghyun Lim. Degrees of second and higher-order polynomials. 4th Workshop on Mathematical Logic and its Applications, March 2021. URL: http://jaist.ac.jp/event/mla2021.
  21. Donghyun Lim and Martin Ziegler. Degrees of second and higher-order polynomials. arXiv CoRR, 2305.03439, 2023. URL: https://doi.org/10.48550/arXiv.2305.03439.
  22. Donghyun Lim and Martin Ziegler. Quantitative coding and complexity theory of Continuous data: Part I. J. ACM, 72(1):4:1-4:39, 2025. URL: https://doi.org/10.1145/3705609.
  23. George G. Lorentz. Metric entropy and approximation, volume 72. AMS, 1996. Google Scholar
  24. Kurt Mehlhorn. Polynomial and abstract subrecursive classes. Journal of Computer and System Sciences, 12(2):147-178, 1976. URL: https://doi.org/10.1016/S0022-0000(76)80035-9.
  25. Eike Neumann and Florian Steinberg. Parametrised second-order complexity theory with applications to the study of interval computation. Theor. Comput. Sci., 806:281-304, 2020. URL: https://doi.org/10.1016/j.tcs.2019.05.009.
  26. Jürgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry. Idempotent Mathematics and Mathematical Physics, 377:289-317, 2005. Google Scholar
  27. Jon G. Riecke and Ramesh Subrahmanyam. Conditions for the completeness of functional and algebraic equational reasoning. Mathemat.Structures Computer Science, 9(6):651-685, 1999. URL: https://doi.org/10.1017/S0960129599002807.
  28. Ernst Specker, Norbert Hungerbühler, and Micha Wasem. Polyfunctions over commutative rings. J. Algebra and Applications, 23(01):2450014, 2024. URL: https://doi.org/10.1142/S0219498824500142.
  29. Volker Strassen. Gaussian elimination is not optimal. Numer.Mathematik, 13:354-356, 1969. Google Scholar
  30. Mike Townsend. Complexity for type-2 relations. Notre Dame Formal Logic, 31(2):241-262, 1990. URL: https://doi.org/10.1305/NDJFL/1093635419.
  31. Martin Ziegler. Hyper-degrees of 2nd-order polynomial-time reductions. Technical report, Dagstuhl Seminar #15392, 2016. abstract §3.20. URL: https://doi.org/10.4230/DagRep.5.9.77.
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