Document Open Access Logo

Unifying Boolean and Algebraic Descriptive Complexity

Authors Baptiste Chanus , Damiano Mazza , Morgan Rogers

PDF

File

Thumbnail PDF
PDF
LIPIcs.FSCD.2025.13.pdf
  • Filesize: 0.8 MB
  • 22 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Complexity classes
  • Theory of computation → Finite Model Theory
Keywords
  • Descriptive complexity theory
  • Categorical logic
  • Blum-Shub-Smale complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We introduce ultrarings, which simultaneously generalize commutative rings and Boolean lextensive categories. As such, they allow to blend together standard algebraic notions (from commutative algebra) and logical notions (from categorical logic), providing a unifying descriptive framework in which complexity classes over arbitrary rings (as in the Blum, Schub, Smale model) and usual, Boolean complexity classes may be captured in a uniform way.

Cite As Get BibTex

Baptiste Chanus, Damiano Mazza, and Morgan Rogers. Unifying Boolean and Algebraic Descriptive Complexity. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSCD.2025.13

Author Details

Baptiste Chanus
  • Université Sorbonne Paris Nord, LIPN, CNRS, Villetaneuse, France
Damiano Mazza
  • CNRS, LIPN, Université Sorbonne Paris Nord, Villetaneuse, France
Morgan Rogers
  • Université Sorbonne Paris Nord, LIPN, CNRS, Villetaneuse, France

References

  1. Samson Abramsky, Anuj Dawar, and Pengming Wang. The pebbling comonad in finite model theory. In Proceedings of LICS, pages 1-12. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/LICS.2017.8005129.
  2. Samson Abramsky and Luca Reggio. Arboreal categories: An axiomatic theory of resources. Log. Methods Comput. Sci., 19(3), 2023. URL: https://doi.org/10.46298/LMCS-19(3:14)2023.
  3. Samson Abramsky and Nihil Shah. Relating structure and power: Comonadic semantics for computational resources. J. Log. Comput., 31(6):1390-1428, 2021. URL: https://doi.org/10.1093/LOGCOM/EXAB048.
  4. John C. Baez, Joe Moeller, and Todd Trimble. 2-rig extensions and the splitting principle. arXiv:2410.05598 [math.CT], 2024. Google Scholar
  5. Lenore Blum, Mike Shub, and Steve Smale. On a theory of computation and complexity over the real numbers. Bulletin of the American Mathematical Society, 21(1):1-46, 1989. Google Scholar
  6. Martin Brandenburg. Tensor categorical foundations of algebraic geometry. Ph.d. thesis, University of Münster, 2014. Google Scholar
  7. Martin Brandenburg and Alexandru Chirvasitu. Tensor functors between categories of quasi-coherent sheaves. Journal of Algebra, 399:675-692, 2014. Google Scholar
  8. Aurelio Carboni, Stephen Lack, and R.F.C. Walters. Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra, 84(2):145-198, 1993. Google Scholar
  9. Baptiste Chanus and Damiano Mazza. A categorical approach to describing complexity classes. In preparation. Available on the authors' web page, 2024. Google Scholar
  10. Alexandru Chirvasitu and Theo Johnson-Freyd. The fundamental pro-groupoid of an affine 2-scheme. Appl. Categ. Structures, 21(5):469-522, 2013. URL: https://doi.org/10.1007/S10485-011-9275-Y.
  11. Anuj Dawar, Tomás Jakl, and Luca Reggio. Lovász-type theorems and game comonads. In Proceedings of LICS, pages 1-13. IEEE, 2021. URL: https://doi.org/10.1109/LICS52264.2021.9470609.
  12. Nikolai Durov. New Approach to Arakelov Geometry. arXiv 0704.2030 [math.AG], 2007. Google Scholar
  13. Nikolai Durov. Classifying vectoids and operad kinds. Proc. Steklov Inst. Math., 273:48-63, 2011. Google Scholar
  14. Ronald Fagin. Generalized first-order spectra and polynomial-time recognizable sets. In Complexity of Computation, SIAM-AMS Proccedings, volume 7, pages 43-73, 1974. Google Scholar
  15. Erich Grädel and Klaus Meer. Descriptive complexity theory over the real numbers. In Proceedings of STOC, pages 315-324. ACM Press, 1995. URL: https://doi.org/10.1145/225058.225151.
  16. Neil Immerman. Relational queries computable in polynomial time. Information and Control, 68(1):86-104, 1986. URL: https://doi.org/10.1016/S0019-9958(86)80029-8.
  17. Neil Immerman. Descriptive complexity. Graduate texts in computer science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0539-5.
  18. Bart P. F. Jacobs. Categorical Logic and Type Theory, volume 141 of Studies in logic and the foundations of mathematics. North-Holland, 2001. Google Scholar
  19. Peter T. Johnstone. Sketches of en Elephant. A Topos Theory Compendium. Volume 2. Oxford University Press, 2002. Google Scholar
  20. Melven R. Krom. The decision problem for a class of first-order formulas in which all disjunctions are binary. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 13(1-2):15-20, 1967. Google Scholar
  21. Saunders Mac Lane and Ieke Moerdijk. Sheaves in Geometry and Logic. Springer, 1994. Google Scholar
  22. Yoàv Montacute and Glynn Winskel. Concurrent games over relational structures: The origin of game comonads. In Proceedings of LICS, pages 58:1-58:14. ACM, 2024. URL: https://doi.org/10.1145/3661814.3662075.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

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