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Recursion and Proof Theoretical Characterizations of Small Circuit Classes with Modulo Counting via Discrete Differential Equations

Authors Melissa Antonelli , Arnaud Durand , Rui Li

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LIPIcs.ICALP.2026.162.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Circuit complexity
Keywords
  • Implicit complexity
  • circuit complexity
  • small circuit classes with counting
  • discrete ODEs
  • recursion theory
  • bounded arithmetic

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Abstract

The paper proposes an implicit (i.e., machine-independent) complexity approach to studying computation by polynomial-size, constant-depth circuits with gates counting modulo a constant through the lens of discrete ordinary differential equations (ODEs). So far, recursion-theoretic characterizations have been provided for functions computed by circuits of constant depth, including gates counting modulo 2 and 6 only (i.e., for the classes FAC⁰[2] and FAC⁰[6], resp.). In this paper, it is shown that considering ODE schemas, rather than bounded recursion, allows for a more fine-grained analysis, leading to (uniform) characterizations for all classes FAC⁰[n] (n ∈ ℕ), i.e. functions computed by circuits including counting modulo n gates. Inspired by the syntactic form of the ODE schemas, we go further in this direction and present first-order bounded theories for capturing provably total functions in each of these classes.

Cite As Get BibTex

Melissa Antonelli, Arnaud Durand, and Rui Li. Recursion and Proof Theoretical Characterizations of Small Circuit Classes with Modulo Counting via Discrete Differential Equations. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 162:1-162:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.162

Author Details

Melissa Antonelli
  • CFvW, University of Tübingen, Germany
Arnaud Durand
  • Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75013 Paris, France
Rui Li
  • Institut Polytechnique de Paris, Ecole Polytechnique, LIX and Université Paris Cité, IMJ-PRG, CNRS, France

Funding

The first author thanks CFvW for funding her research since 2026. The first and second authors acknowledge support from the CARACAL project, and all three authors acknowledge support from the ANR grant "Difference" (ANR-20-CE48-0002).

References

  1. B. Allen. Arithmetizing uniform NC. Ann. Pure Appl. Log., 53(1):1-50, 1991. URL: https://doi.org/10.1016/0168-0072(91)90057-S.
  2. E. Allender and K.W. Wagner. Counting hierarchies: polynomial time and constant. Bull. EATCS, 40:182-194, 1990. Google Scholar
  3. M. Antonelli, A. Durand, and J. Kontinen. A New Characterization of FAC⁰ via Discrete Ordinary Differential Equations. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024), volume 306 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.MFCS.2024.10.
  4. M. Antonelli, A. Durand, and J. Kontinen. Characterizing Small Circuit Classes from FAC⁰ to FAC¹ via Discrete Ordinary Differential Equations. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025), volume 345 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1-10:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. URL: https://doi.org/10.4230/LIPIcs.MFCS.2025.10.
  5. T. Barlag, V. Holzapfel, L. Strieker, J. Virtema, and H. Vollmer. Graph neural networks and arithmetic circuits. In Amir Globersons, Lester Mackey, Danielle Belgrave, Angela Fan, Ulrich Paquet, Jakub M. Tomczak, and Cheng Zhang, editors, Advances in Neural Information Processing Systems 38: Annual Conference on Neural Information Processing Systems 2024, NeurIPS 2024, Vancouver, BC, Canada, December 10 - 15, 2024, 2024. Google Scholar
  6. M. Blanc and O. Bournez. A characterization of polynomial time computable functions from the integers to the reals using discrete ordinary differential equations. In Jérôme Durand-Lose and György Vaszil, editors, Machines, Computations, and Universality - 9th International Conference, MCU 2022, Debrecen, Hungary, August 31 - September 2, 2022, Proceedings, volume 13419 of Lecture Notes in Computer Science, pages 58-74. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-13502-6_4.
  7. M. Blanc and O. Bournez. A characterisation of functions computable in polynomial time and space over the reals with discrete ordinary differential equations: Simulation of turing machines with analytic discrete odes. In Jérôme Leroux, Sylvain Lombardy, and David Peleg, editors, 48th International Symposium on Mathematical Foundations of Computer Science, MFCS 2023, Bordeaux, France, August 28 - September 1, 2023, volume 272 of LIPIcs, pages 21:1-21:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.MFCS.2023.21.
  8. M. Blanc and O. Bournez. The complexity of computing in continuous time: Space complexity is precision. In Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson, editors, 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024, Tallinn, Estonia, July 8-12, 2024, volume 297 of LIPIcs, pages 129:1-129:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.ICALP.2024.129.
  9. O. Bournez and A. Durand. Recursion schemes, discrete differential equations and characterization of polynomial time computations. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, Aachen, Germany, August 26-30, 2019, volume 138 of LIPIcs, pages 23:1-23:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.MFCS.2019.23.
  10. O. Bournez and A. Durand. A characterization of functions over the integers computable in polynomial time using discrete ordinary differential equations. Comput. Complex., 32(2):7, 2023. URL: https://doi.org/10.1007/S00037-023-00240-1.
  11. S. Buss. Bounded Arithmetic. PhD thesis, Department of Mathematics, 1986. Google Scholar
  12. P.G. Clote. A sequential characterization of the parallel complexity class NC. Technical report, Boston College, 1988. Google Scholar
  13. P.G. Clote. Sequential, machine-independent characterizations of the parallel complexity classes AlogTIME, AC^k, NC^k and NC, pages 49-69. Progress in Computer Science and Applied Logic. Birkhäuser, Boston, MA, 1990. Google Scholar
  14. P.G. Clote. On polynomial size Frege proofs of certain combinatorial principles. In P. Clote and Krjícek J., editors, Arithmetic, Proof Theory, and Computational Complexity, pages 166-184. Clarendon Press, 1993. Google Scholar
  15. P.G. Clote and G. Takeuti. First order bounded arithmetic and small complexity classes. In P.G. Clote and J.B. Remmel, editors, Feasible Mathematics II, pages 154-218. Springer, 1995. Google Scholar
  16. A. Cobham. The intrinsic computational difficulty of functions. In Logic, Methodology and Phylosophy of Science: Proc. 1964 International Congress, pages 24-30. North-Holland Publishing Company, 1965. Google Scholar
  17. K.J. Compton and C. Laflamme. An algebra and a logic for NC¹. In Proceedings of the Third Annual Symposium on Logic in Computer Science (LICS '88), Edinburgh, Scotland, UK, July 5-8, 1988, pages 12-21. IEEE Computer Society, 1988. URL: https://doi.org/10.1109/LICS.1988.5096.
  18. S. Cook and P. Nguyen. Theories for TC⁰ and other small circuit classes. Log. Meth. Comput. Sci., 2(1-40), 2006. Google Scholar
  19. A. Durand, A. Haak, and H. Vollmer. Model-theoretic characterization of boolean and arithmetic circuit classes of small depth. In Anuj Dawar and Erich Grädel, editors, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 354-363. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209179.
  20. M. L. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. In 22nd Annual Symposium on Foundations of Computer Science, Nashville, Tennessee, USA, 28-30 October 1981, pages 260-270. IEEE Computer Society, 1981. URL: https://doi.org/10.1109/SFCS.1981.35.
  21. A.O. Gelfand. Calcul des différences finies. Dunod, 1963. Google Scholar
  22. L. Hella, J. Kontinen, and K. Luosto. Regular representations of uniform TC^0. ACM Trans. Comput. Log., 26(4):21:1-21:46, 2025. URL: https://doi.org/10.1145/3750044.
  23. William Hesse. Division is in uniform TC⁰. In Fernando Orejas, Paul G. Spirakis, and Jan van Leeuwen, editors, Automata, Languages and Programming, pages 104-114, Berlin, Heidelberg, 2001. Springer Berlin Heidelberg. Google Scholar
  24. N. Immerman. Languages that capture complexity classes. SIAM J. Comput., 16:760-778, 1987. URL: https://doi.org/10.1137/0216051.
  25. N. Immerman. Expressibility and parallel complexity. SIAM J. Comput., 18(3):625-638, 1989. URL: https://doi.org/10.1137/0218043.
  26. C. Jordan and K. Jordán. Calculus of finite differences, volume 33. American Mathematical Soc., 1965. Google Scholar
  27. W. Maass, G. Schnitger, and E.D. Sontag. On the computational power of sigmoid versus boolean threshold circuits. In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, October 1-4, 1991, pages 767-776. IEEE Computer Society, 1991. URL: https://doi.org/10.1109/SFCS.1991.185447.
  28. I. Parberry. Circuit Complexity and Neural Networks. MIT Press, 1994. Google Scholar
  29. R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Alfred V. Aho, editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing, 1987, New York, New York, USA, pages 77-82. ACM, 1987. URL: https://doi.org/10.1145/28395.28404.
  30. H. Vollmer. Introduction to Circuit Complexity: A Uniform Approach. Springer, 1999. Google Scholar
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