A parameterized halting problem, $\Delta_0$ truth and the MRDP theorem

TL;DR

This paper investigates the parameterized complexity of deciding whether a large number satisfies a given $ ext{Delta}_0$-formula, revealing complexity class separations and linking to the MRDP theorem.

Contribution

It introduces a parameterized complexity analysis of the $ ext{Delta}_0$-formula satisfaction problem and connects it to classical complexity class separations via the halting problem.

Findings

The problem is not in the parameterized $ ext{AC}^0$ class.

Certain upper bounds imply classical complexity class separations.

Assumptions about $I ext{Delta}_0$ prove the MRDP theorem in a weak sense.

Abstract

We study the parameterized complexity of the problem to decide whether a given natural number $n$ satisfies a given $\Delta_0$-formula $\varphi(x)$; the parameter is the size of $\varphi$. This parameterization focusses attention on instances where $n$ is large compared to the size of $\varphi$. We show unconditionally that this problem does not belong to the parameterized analogue of $\mathsf{AC}^0$. From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameterized halting problem. Some of these upper bounds follow assuming that $I\Delta_0$ proves the MRDP theorem in a certain weak sense.