Slicewise definability in first-order logic with bounded quantifier rank

TL;DR

This paper investigates the definability of graph properties in bounded quantifier rank first-order logic with built-in arithmetic, linking it to circuit complexity and parameterized complexity classes, and demonstrating that certain problems can be expressed with fixed quantifier rank.

Contribution

It shows that some graph problems can be defined in FO with bounded quantifier rank independent of parameters, connecting logical definability to circuit complexity and parameterized classes.

Findings

FO_q is strictly contained in FO_{q+1} with built-in addition and multiplication.

Certain hypergraph problems are definable in FO_q with fixed q, independent of size parameters.

The results connect logical definability, circuit complexity, and parameterized complexity classes.

Abstract

For every $q\in \mathbb N$ let $\textrm{FO}_q$ denote the class of sentences of first-order logic FO of quantifier rank at most $q$. If a graph property can be defined in $\textrm{FO}_q$, then it can be decided in time $O(n^q)$. Thus, minimizing $q$ has favorable algorithmic consequences. Many graph properties amount to the existence of a certain set of vertices of size $k$. Usually this can only be expressed by a sentence of quantifier rank at least $k$. We use the color-coding method to demonstrate that some (hyper)graph problems can be defined in $\textrm{FO}_q$ where $q$ is independent of $k$. This property of a graph problem is equivalent to the question of whether the corresponding parameterized problem is in the class $\textrm{para-AC}^0$. It is crucial for our results that the FO-sentences have access to built-in addition and multiplication. It is known that then FO corresponds to the circuit complexity class uniform $\textrm{AC}^0$. We explore the connection between the quantifier rank of FO-sentences and the depth of $\textrm{AC}^0$-circuits, and prove that $\textrm{FO}_q \subsetneq \textrm{FO}_{q+1}$ for structures with built-in addition and multiplication.