Towards New Characterizations of Small Circuit Classes via Discrete Ordinary Differential Equations

Melissa Antonelli; Arnaud Durand; Juha Kontinen

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arXiv · 2508.19392

Towards New Characterizations of Small Circuit Classes via Discrete Ordinary Differential Equations

Melissa Antonelli·Arnaud Durand·Juha Kontinen

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Abstract

Implicit computational complexity is a lively area of theoretical computer science, which aims to provide machine-independent characterizations of relevant complexity classes. % for uniformity with subsequent uses >> 1960s (but feel free to modify it) % One of the seminal works in this field appeared in the 1960s, when Cobham introduced a function algebra closed under bounded recursion on notation to capture polynomial time computable functions ($FP$). Later on, several complexity classes have been characterized using \emph{limited} recursion schemas. In this context, an original approach has been recently introduced, showing that ordinary differential equations (ODEs) offer a natural tool for algorithmic design and providing a characterization of $FP$ by a new ODE-schema. In the present paper we generalize this approach by presenting original ODE-characterizations for the small circuit classes $AC^0$ and $FTC^0$.

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Melissa Antonelli, Arnaud Durand, Juha Kontinen. 2025-08-26. Towards New Characterizations of Small Circuit Classes via Discrete Ordinary Differential Equations. https://arxiv.org/abs/2508.19392

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