Model-Theoretic Characterizations of Boolean and Arithmetic Circuit Classes of Small Depth

TL;DR

This paper provides a logical framework to characterize small-depth Boolean and arithmetic circuit classes, extending descriptive complexity theory with inductive operators and semantic game strategies.

Contribution

It introduces a new logical operator to characterize small-depth circuit classes and links arithmetic classes to semantic game strategies, extending existing descriptive complexity results.

Findings

Logical characterization of NC^1, SAC^1, and AC^1 classes.

Arithmetic classes characterized by counting winning strategies.

Extension of Immerman's formulas with inductive operators.

Abstract

In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\textrm{NC}^1$, $\textrm{SAC}^1$ and $\textrm{AC}^1$ as well as their arithmetic counterparts $\#\textrm{NC}^1$, $\#\textrm{SAC}^1$ and $\#\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulas of first-order logic, i.e., $\textrm{AC}^0 = \textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering slight variations of the new operator, we obtain uniform characterizations of the three just mentioned Boolean classes. The arithmetic classes can then be characterized by functions counting winning strategies in semantic games for formulas characterizing languages in the corresponding Boolean class.