Descriptive Complexity of $\#\textrm{AC}^0$ Functions

TL;DR

This paper develops a descriptive complexity framework for arithmetic functions, establishing a hierarchy that includes #AC^0 and #P, and clarifies their relationships within computational complexity.

Contribution

It introduces a new descriptive complexity hierarchy for arithmetic functions, unconditionally placing #AC^0 within this hierarchy and comparing it to existing model-theoretic approaches.

Findings

#AC^0 is properly placed within a strict hierarchy of #P.

The hierarchy unconditionally includes #P and #AC^0.

The framework offers a better characterization of arithmetic circuit classes.

Abstract

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC^0 appear as classes of this hierarchy. In this way, we unconditionally place #AC^0 properly in a strict hierarchy of arithmetic classes within #P. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. We argue that our approach is better suited to study arithmetic circuit classes such as #AC^0 which can be descriptively characterized as a class in our framework.