The Polynomial Hierarchy and $\ddot{I}‰$-categorical CSPs

Abstract

In 2008, Bodirsky and Grohe showed that for every ${\hat{I}}_n^{\mathrm{P}}$-level of the Polynomial Hierarchy (PH) there are $\ddot{I}‰$-categorical Constraint Satisfaction Problems (CSPs) complete for this level. We show that, in fact, there are $\ddot{I}‰$-categorical CSPs complete for any level of the PH. To this end, we use a recent result of Bodirsky, Knäuer, and Rudolph for constructing $\ddot{I}‰$-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.

20 pages