Computing with reaction networks at input-independent speed: exponential and logarithmic functions

[2604.08859] Computing with reaction networks at input-independent speed: exponential and logarithmic functions Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2604.08859 (math) [Submitted on 10 Apr 2026 ( v1 ), last revised 26 Aug 2026 (this version, v3)] Title: Computing with reaction networks at input-independent speed: exponential and logarithmic functions Authors: David F. Anderson , Badal Joshi , Tung D. Nguyen View PDF HTML (experimental) Abstract: The concept of \textit{input-independent computational speed} for chemistry-based analog computers was introduced in Anderson-Joshi (2025), where it was shown that arithmetic operations can be computed with a convergence rate bounded below independently of the input values. Here, by inputs we mean the numerical values encoded by the initial concentrations of designated input species, with the underlying reaction network and rate constants held fixed. Combining these operations via power series to approximate transcendental functions is possible in principle, but the number of chemical species required grows with the number of terms retained, and achieving sufficient accuracy may demand many terms---a burden that is especially severe for slowly converging series such as the power series for the logarithm. In this paper, we begin the program of directly computing transcendental functions by chemical reaction networks by focusing on the exponential and logarithmic functions, two widely used transcendental functions, and constructing reaction network modules that compute them without relying on truncated power series. We show that the resulting modules are mass-action systems, and prove that they achieve arbitrary accuracy given sufficient time while operating at input-independent speed. Moreover, these modules can be combined with each other and with the arithmetic modules of Anderson-Joshi (2025): any such composite computation also runs at input-independent speed, with a convergence rate that does not depend on the length of the feedforward chain. Logarithm and exponential functions serve as foundational cases, and the constructions developed here are intended to serve as templates for the direct computation of more general transcendental functions by chemical reaction networks. Subjects: Dynamical Systems (math.DS) Cite as: arXiv:2604.08859 [math.DS] (or arXiv:2604.08859v3 [math.DS] for this version) https://doi.org/10.48550/arXiv.2604.08859 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Badal Joshi [ view email ] [v1] Fri, 10 Apr 2026 01:46:15 UTC (19 KB) [v2] Tue, 14 Apr 2026 04:46:17 UTC (20 KB) [v3] Wed, 26 Aug 2026 18:27:36 UTC (25 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-04 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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