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

TL;DR

This paper introduces novel characterizations of small circuit classes $AC^0$ and $FTC^0$ using ordinary differential equations, expanding the application of ODE-based methods in implicit computational complexity.

Contribution

It generalizes the use of ODE schemas to characterize small circuit classes, providing new machine-independent descriptions.

Findings

ODE schemas characterize $AC^0$ and $FTC^0$ classes.

Provides a new perspective on complexity class characterizations.

Extends the framework of implicit computational complexity.

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 $FTC^0$.