Characterizing Small Circuit Classes from FAC^0 to FAC^1 via Discrete Ordinary Differential Equations

TL;DR

This paper introduces a unified framework using ordinary differential equations to analyze small circuit classes, establishing new characterizations and bounds for classes like FAC^0, FAC^1, and others, revealing deep connections between ODE constraints and circuit complexity.

Contribution

It provides the first ODE-based characterizations for classes such as FACC[2] and FNC1, linking differential equations with circuit class bounds and complexity hierarchies.

Findings

Established ODE-based characterizations for FACC[2] and FNC1.

Linked linearity constraints in ODEs to circuit class bounds.

Provided tools for analyzing circuit complexity via differential equations.

Abstract

In this paper, we provide a uniform framework for investigating small circuit classes and bounds through the lens of ordinary differential equations (ODEs). Following an approach recently introduced to capture the class of polynomial-time computable functions via ODE-based recursion schemas and later applied to the context of functions computed by unbounded fan-in circuits of constant depth (FAC^0), we study multiple relevant small circuit classes. In particular, we show that natural restrictions on linearity and derivation along functions with specific growth rate correspond to kinds of functions that can be proved to be in various classes, ranging from FAC^0 to FAC^1. This reveals an intriguing link between constraints over linear-length ODEs and circuit computation, providing new tools to tackle the complex challenge of establishing bounds for classes in the circuit hierarchies and possibly enhancing our understanding of the role of counters in this setting. Additionally, we establish several completeness results, in particular obtaining the first ODE-based characterizations for the classes of functions computable in constant depth with unbounded fan-in and Mod 2 gates (FACC[2]) and in logarithmic depth with bounded fan-in Boolean gates (FNC1).