Document Open Access Logo

On Equivalent Characterizations of the Polynomial Hierarchy in Abstract Models of Computation

Authors Jeremy C. Kirn , Lucas Meijer , Tillmann Miltzow , Hans L. Bodlaender

PDF

File

Thumbnail PDF
PDF
LIPIcs.MFCS.2026.63.pdf
  • Filesize: 0.89 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Abstract machines
Keywords
  • Machines over a first-order structure
  • BSS machines
  • Cook Levin
  • Fagin
  • NP
  • existential theory of the reals
  • polynomial hierarchy
  • metafinite model theory
  • descriptive complexity
  • oracles

Metrics

Abstract

We investigate machine models similar to Turing machines that are augmented with the operations of a first-order structure ℛ, and we show that under weak conditions on ℛ, the complexity class Σ_kℛ may be characterized in four equivalent ways: (1) by polynomial-time algorithms implemented on ℛ-machines together with witness strings, (2) by the Σ_k ℛ-complete problem Σ_k SAT(ℛ), (3) by k-th existential fragment second-order metafinite logic over ℛ via descriptive complexity, and (4) via oracles. By characterizing Σ_k ℛ in these four ways, we extend previous work and embed it in one coherent framework. In addition, we derive similar results for ∃_k ℛ, the constant-free Boolean part of Σ_k ℛ, by showing that ∃_k ℛ may be characterized in four analogous ways. 
Some conditions on ℛ must be assumed in order to achieve the above quaternity because there are infinite-vocabulary structures for which NP(ℛ) = Σ₁ ℛ does not have a complete problem. Surprisingly, even in these cases, we show that NP(ℛ) does have a characterization in terms of existential second-order metafinite logic, suggesting that descriptive complexity theory is well suited to working with infinite-vocabulary structures, such as real vector spaces.

Cite As Get BibTex

Jeremy C. Kirn, Lucas Meijer, Tillmann Miltzow, and Hans L. Bodlaender. On Equivalent Characterizations of the Polynomial Hierarchy in Abstract Models of Computation. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 63:1-63:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.63

Author Details

Jeremy C. Kirn
  • Utrecht University, The Netherlands
Lucas Meijer
  • Utrecht University, The Netherlands
Tillmann Miltzow
  • Utrecht University, The Netherlands
Hans L. Bodlaender
  • Utrecht University, The Netherlands

Funding

  • Kirn, Jeremy C.: generously supported by the Netherlands Organisation for Scientific Research (NWO) under project no. VI.Vidi.213.150.
  • Meijer, Lucas: generously supported by the Netherlands Organisation for Scientific Research (NWO) under project no. VI.Vidi.213.150.
  • Miltzow, Tillmann: generously supported by the Netherlands Organisation for Scientific Research (NWO) under project no. VI.Vidi.213.150.

Acknowledgements

The first author gratefully acknowledges conversations with Nima Motamed.

References

  1. Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009. URL: https://doi.org/10.1017/CBO9780511804090.
  2. Lenore Blum. Computing over the reals: Where Turing meets Newton. Notices of the AMS, 51(9):1024-1034, 2004. URL: https://www.ams.org/journals/notices/200409/fea-blum.pdf?adat=October%202004&trk=200409fea-blum&cat=none&type=.pdf.
  3. Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale. Complexity and real computation. Springer-Verlag, New York, 1998. With a foreword by Richard M. Karp. URL: https://doi.org/10.1007/978-1-4612-0701-6.
  4. Lenore Blum, Mike Shub, and Steve Smale. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. (N.S.), 21(1):1-46, 1989. URL: https://doi.org/10.1090/S0273-0979-1989-15750-9.
  5. Peter Bürgisser and Felipe Cucker. Counting complexity classes for numeric computations. II. Algebraic and semialgebraic sets. Journal of Complexity, 22(2):147-191, 2006. URL: https://doi.org/10.1016/j.jco.2005.11.001.
  6. Peter Bürgisser and Felipe Cucker. Exotic quantifiers, complexity classes, and complete problems. Foundations of Computational Mathematics, 9(2):135-170, 2009. URL: https://doi.org/10.1007/s10208-007-9006-9.
  7. Olivier Chapuis and Pascal Koiran. Saturation and stability in the theory of computation over the reals. Ann. Pure Appl. Logic, 99(1-3):1-49, 1999. URL: https://doi.org/10.1016/S0168-0072(98)00060-8.
  8. Stephen A. Cook. The complexity of theorem-proving procedures. In Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, pages 151-158. ACM, 1971. URL: https://doi.org/10.1145/800157.805047.
  9. Felipe Cucker and Pascal Koiran. Computing over the reals with addition and order: higher complexity classes. Journal of Complexity, 11(3):358-376, 1995. URL: https://doi.org/10.1006/jcom.1995.1018.
  10. Julian D'Costa, Engel Lefaucheux, Eike Neumann, Joël Ouaknine, and James Worrell. On the complexity of the escape problem for linear dynamical systems over compact semialgebraic sets. In Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021, volume 202 of LIPIcs, pages 33:1-33:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.MFCS.2021.33.
  11. Paulin Jacobé de Naurois. Completeness results and syntactic characterizations of complexity classes over arbitrary structures. PhD thesis, éditeur inconnu, 2004. Google Scholar
  12. Michael Gene Dobbins, Linda Kleist, Tillmann Miltzow, and PawełRzażewski. ∀∃ℝ-completeness and area-universality. In Proceedings of the 44th International Workshop on Graph-theoretic Concepts in Computer Science, WG 2018, volume 11159 of Lecture Notes in Computer Science, pages 164-175. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-00256-5_14.
  13. Jeff Erickson, Ivor van der Hoog, and Tillmann Miltzow. Smoothing the gap between NP and ER. In Proceedigns of the 61st Annual Symposium on Foundations of Computer Science, FOCS 2020, pages 1022-1033, 2020. URL: https://doi.org/10.1137/20M1385287.
  14. Ronald Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In R.Karp, editor, Proceedings Complexity of Computation, volume 7, pages 43-73, 1974. Google Scholar
  15. Solomon Feferman. About and around computing over the reals, chapter 3, pages 55-76. MIT Press, Cambridge, MA, 2013. Google Scholar
  16. Christine Gaßner. On NP-completeness for linear machines. Journal of Complexity, 13(2):259-271, 1997. URL: https://doi.org/10.1006/JCOM.1997.0444.
  17. Christine Gaßner. An introduction to a model of abstract computation. In Contemporary Logic and Computing, volume II of Landscapes in Logic, pages 574-603. College Publications, 2020. Google Scholar
  18. Robert Giegerich, Carsten Meyer, and Peter Steffen. A discipline of dynamic programming over sequence data. Science of Computer Programming, 51(3):215-263, 2004. URL: https://doi.org/10.1016/j.scico.2003.12.005.
  19. John B. Goode. Accessible telephone directories. The Journal of Symbolic Logic, 59(1):92-105, 1994. URL: https://doi.org/10.2307/2275252.
  20. Erich Grädel and Klaus Meer. Descriptive complexity theory over the real numbers. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing, STOC 1995, pages 315-324, 1995. URL: https://doi.org/10.1145/225058.225151.
  21. Erich Graedel and Yuri Gurevich. Metafinite model theory. Information and Computation, 140(1):26-81, 1998. URL: https://doi.org/10.1006/INCO.1997.2675.
  22. Thekla Hamm, Lucas Meijer, Tillmann Miltzow, and Subhasree Patro. Oracle separations for RPH, 2025. URL: https://doi.org/10.48550/arXiv.2502.09279.
  23. Miika Hannula and Jonni Virtema. Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals. Annals of Pure and Applied Logic, 173(10):103108, 2022. URL: https://doi.org/10.1016/j.apal.2022.103108.
  24. Armin Hemmerling. Computability over structures of infinite signature. Mathematical Logic Quarterly, 44:394-416, 1998. URL: https://doi.org/10.1002/MALQ.19980440311.
  25. Armin Hemmerling. Computability of string functions over algebraic structures. Mathematical Logic Quarterly, 44:1-44, 2006. URL: https://doi.org/10.1002/malq.19980440102.
  26. Paul Jungeblut, Linda Kleist, and Tillmann Miltzow. The complexity of the Hausdorff distance. In Proceedings of the 38th International Symposium on Computational Geometry, SoCG 2022, volume 224 of LIPIcs, pages Art. No. 48, 17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/lipics.socg.2022.48.
  27. Tim Janik Junginger. Robustness of the discrete real polynomial hierarchy. Bachelor’s thesis, Karlsruher Institut für Technologie, Karlsruhe, 2023. (last accessed 10/19/2023). URL: https://i11www.iti.kit.edu/_media/teaching/theses/ba-junginger-23.pdf.
  28. Jeremy C Kirn, Lucas Meijer, Tillmann Miltzow, and Hans L Bodlaender. On equivalent characterizations of the polynomial hierarchy in abstract models of computation. arXiv preprint, 2025. URL: https://arxiv.org/abs/2510.05894.
  29. Stephen Cole Kleene. Mathematical logic. Dover Publications, 2002. Google Scholar
  30. Pascal Koiran. Computing over the reals with addition and order. Theoretical Computer Science, 133(1):35-47, 1994. URL: https://doi.org/10.1016/0304-3975(93)00063-B.
  31. Leonid A. Levin. Universal search problems. Problemy Peredachi Informatsii, 9(3):265-266, 1973. Google Scholar
  32. Klaus Meer and Christian Michaux. A survey on real structural complexity theory. Bull. Belg. Math. Soc. Simon Stevin, 4(1):113-148, 1997. URL: http://projecteuclid.org/euclid.bbms/1105730626.
  33. Klaus Meer and Adrian Wurm. Some structural complexity results for ∃ℝ, 2025. URL: https://doi.org/10.48550/arXiv.2502.00680.
  34. Nimrod Megiddo. A general NP-completeness theorem. In From Topology to Computation: Proceedings of the Smalefest, pages 432-442. Springer, 1990. URL: https://doi.org/10.1007/978-1-4612-2740-3_39.
  35. Eike Neumann and Arno Pauly. A topological view on algebraic computation models. Journal of Complexity, 44:1-22, 2018. URL: https://doi.org/10.1016/j.jco.2017.08.003.
  36. Philippos Papayannopoulos. Unrealistic models for realistic computations: how idealisations help represent mathematical structures and found scientific computing. Synthese, 199(1):249-283, 2021. URL: https://doi.org/10.1007/s11229-020-02654-8.
  37. Bruno Poizat. Les Petits Cailloux : Une Approche Modèle-Théorique de L'Algorithmie. Lyon : Aléas, 1995. Google Scholar
  38. Marcus Schaefer, Jean Cardinal, and Tillmann Miltzow. The existential theory of the reals as a complexity class: A compendium, 2024. URL: https://doi.org/10.48550/arxiv.2407.18006.
  39. Marcus Schaefer and Daniel Štefankovič. Fixed points, Nash equilibria, and the existential theory of the reals. Theory of Computing Systems, 60(2):172-193, 2017. URL: https://doi.org/10.1007/s00224-015-9662-0.
  40. Marcus Schaefer and Daniel Štefankovič. Beyond the existential theory of the reals. Theory of Computing Systems, 2023. URL: https://doi.org/10.1007/s00224-023-10151-x.
  41. Marcus Schaefer and Daniel Štefankovič. The complexity of tensor rank, 2024. URL: https://doi.org/10.48550/arXiv.1612.04338.
  42. John C. Shepherdson and Howard E. Sturgis. Computability of recursive functions. Journal of the ACM, 10(2):217-255, 1963. URL: https://doi.org/10.1145/321160.321170.
  43. Larry J Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3(1):1-22, 1976. URL: https://doi.org/10.1016/0304-3975(76)90061-X.
  44. Alfred Tarski. A Decision Method for Elementary Algebra and Geometry. The Rand Corporation, Santa Monica, CA, 1948. Google Scholar
  45. Dirk van Dalen. Logic and Structure. Springer, 1994. URL: https://doi.org/10.1007/978-1-4471-4558-5.
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
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact