The Polynomial Hierarchy and $\omega$-categorical CSPs

TL;DR

This paper demonstrates that for every level of the Polynomial Hierarchy, there exist corresponding $$-categorical CSPs, extending previous results and introducing new construction tools.

Contribution

It generalizes the existence of $$-categorical CSPs to all levels of the Polynomial Hierarchy using MSO logic and develops a new method for constructing MSO sentences with specific preservation properties.

Findings

$$-categorical CSPs are complete for all PH levels

New MSO sentence construction tool developed

Extended previous results to all PH levels

Abstract

In 2008, Bodirsky and Grohe showed that for every $\Pi_n^{\mathrm{P}}$-level of the Polynomial Hierarchy (PH) there are $\omega$-categorical Constraint Satisfaction Problems (CSPs) complete for this level. We show that, in fact, there are $\omega$-categorical CSPs complete for any level of the PH. To this end, we use a recent result of Bodirsky, Kn\"{a}uer, and Rudolph for constructing $\omega$-categorical CSPs from sentences of Monadic Second-Order logic (MSO) with certain preservation properties. As a secondary contribution, we develop a new tool for producing MSO sentences satisfying said preservation properties.