Tue, 21 Apr 2026, 1:25 pm MDT
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, Knauer, 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. In the present talk, I give an overview of the above-mentioned results and motivate several new questions.
This is joint work with Santiago Guzmán-Pro.