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Bounded Analog Complexity

Authors Ho-Lin Chen , Xiang Huang

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LIPIcs.DNA.32.14.pdf
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Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Analog computation
  • GPAC
  • bounded complexity
  • chemical reaction networks
  • polynomial ODE

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Abstract

Current analog complexity theory, built on the General-Purpose Analog Computer (GPAC) model and polynomial ODEs, allows unbounded state variables - an assumption that is physically unrealistic for chemical reaction networks and other laboratory-scale analog computers. We develop a bounded analog complexity theory in which all state variables remain in compact intervals and physical time is the only diverging resource.
Our main technical contribution is bounded surrogate compilation, a compilation framework that transforms unbounded polynomial ODE systems into bounded ones while preserving computational limits and time-to-precision guarantees. We prove that on compact domains, physical time and trajectory length differ by at most constant factors; combined with the Bournez-Graça-Pouly characterization, this yields: bounded-GPAC polynomial time equals 𝐏 over the reals.
We exhibit concrete constructions demonstrating fine-grained bounded time complexity - a tunable polynomial-degree family, a Lambert-W-based system achieving Θ(rlog r) time-to-precision (where r is the desired precision parameter, in nats: |x(t)-α| < e^{-r}), and an iterated-logarithm tower realizing arbitrarily high complexity classes - all for the task of computing the constant 1. We show that bounded GPACs are closed under exponentiation (α^β) with time complexity equal to the harder input, and that the full GPAC-to-CRN compilation pipeline preserves time complexity class via a low-pass filter analysis of readout modules.

Cite As Get BibTex

Ho-Lin Chen and Xiang Huang. Bounded Analog Complexity. In 32nd International Conference on DNA Computing and Molecular Programming (DNA 32). Leibniz International Proceedings in Informatics (LIPIcs), Volume 387, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DNA.32.14

Author Details

Ho-Lin Chen
  • Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan
Xiang Huang
  • Department of Computer Science, University of Illinois Springfield, IL, USA

Funding

  • Chen, Ho-Lin: Supported in part by NSTC (Taiwan) grant 113-2221-E-002-204-MY3.
  • Huang, Xiang: Supported in part by Department of Energy EXPRESS grant DE-SC0024278.

Acknowledgements

We thank the anonymous DNA32 reviewers for their detailed and constructive feedback.

Supplementary Materials

References

  1. David F. Anderson and Badal Joshi. Chemical mass-action systems as analog computers: implementing arithmetic computations at specified speed. arXiv preprint, 2024. arXiv:2404.04396. Google Scholar
  2. Dana Angluin, James Aspnes, David Eisenstat, and Eric Ruppert. The computational power of population protocols. Distributed Computing, 20(4):279-304, 2007. URL: https://doi.org/10.1007/S00446-007-0040-2.
  3. Olivier Bournez, Pierre Fraigniaud, and Xavier Koegler. Computing with large populations using interactions. In Mathematical Foundations of Computer Science (MFCS), volume 7464 of Lecture Notes in Computer Science, pages 234-246. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32589-2_23.
  4. Olivier Bournez, Daniel S. Graça, and Amaury Pouly. Polynomial time corresponds to solutions of polynomial ordinary differential equations of polynomial length. Journal of the ACM, 64(6):38:1-38:76, 2017. URL: https://doi.org/10.1145/3127496.
  5. Olivier Bournez and Amaury Pouly. A universal ordinary differential equation. In Proc. 45th International Colloquium on Automata, Languages, and Programming (ICALP), volume 107 of LIPIcs, pages 116:1-116:14, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.116.
  6. Ho-Lin Chen, David Doty, and David Soloveichik. Deterministic function computation with chemical reaction networks. Natural Computing, 13(4):517-534, 2014. URL: https://doi.org/10.1007/S11047-013-9393-6.
  7. Ho-Lin Chen, David Doty, and David Soloveichik. Rate-independent computation in continuous chemical reaction networks. Journal of the ACM, 70(3):1-61, 2023. arXiv:2107.13681. Google Scholar
  8. François Fages, Guillaume Le Guludec, Olivier Bournez, and Amaury Pouly. Strong Turing completeness of continuous chemical reaction networks and compilation of mixed analog-digital programs. In Computational Methods in Systems Biology (CMSB 2017), volume 10545 of Lecture Notes in Computer Science, pages 108-127. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-67471-1_7.
  9. Daniel S. Graça, Manuel L. Campagnolo, and Jorge Buescu. Computability with polynomial differential equations. Advances in Applied Mathematics, 40(3):330-349, 2008. URL: https://doi.org/10.1016/J.AAM.2007.02.003.
  10. Daniel S. Graça and José Félix Costa. Analog computers and recursive functions over the reals. Journal of Complexity, 19(5):644-664, 2003. URL: https://doi.org/10.1016/S0885-064X(03)00034-7.
  11. Nicholas Haisler, Xiang Huang, Andrei N. Migunov, Khalid Mohammed, and Garrett Provence. A selective dual-railing technique for general-purpose analog computers. In Unconventional Computation and Natural Computation (UCNC 2025), volume 16364 of Lecture Notes in Computer Science. Springer, 2025. URL: https://doi.org/10.1007/978-3-032-15641-9_26.
  12. Xiang Huang and Rachel N. Huls. Computing real numbers with large-population protocols having a continuum of equilibria. In 28th International Conference on DNA Computing and Molecular Programming (DNA 28), volume 238 of LIPIcs, pages 6:1-6:17. Schloss Dagstuhl, 2022. URL: https://doi.org/10.4230/LIPIcs.DNA.28.7.
  13. Xiang Huang, Titus H. Klinge, and James I. Lathrop. Real-time equivalence of chemical reaction networks and analog computers. In 25th International Conference on DNA Computing and Molecular Programming (DNA 25), volume 11648 of Lecture Notes in Computer Science, pages 37-53. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-26807-7_3.
  14. Ker-I Ko. Complexity Theory of Real Functions. Birkhäuser, 1991. Google Scholar
  15. Leonard Lipshitz and Lee A. Rubel. A differentially algebraic replacement theorem, and analog computability. Proceedings of the American Mathematical Society, 99(2):367-372, 1987. Google Scholar
  16. Marian B. Pour-El. Abstract computability and its relation to the general purpose analog computer. Transactions of the American Mathematical Society, 199:1-28, 1974. Google Scholar
  17. Claude E. Shannon. Mathematical theory of the differential analyzer. Journal of Mathematics and Physics, 20(1-4):337-354, 1941. Google Scholar
  18. David Soloveichik, Matthew Cook, Erik Winfree, and Jehoshua Bruck. Computation with finite stochastic chemical reaction networks. Natural Computing, 7(4):615-633, 2008. URL: https://doi.org/10.1007/S11047-008-9067-Y.
  19. Alan M. Turing. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42(1):230-265, 1936. URL: https://doi.org/10.1112/PLMS/S2-42.1.230.
  20. Klaus Weihrauch. Computable analysis: an introduction. Springer Science & Business Media, 2000. Google Scholar
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