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arXiv · 2604.24539

The Polynomial Hierarchy and $\omega$-categorical CSPs

Santiago Guzmán Pro·Jakub Rydval
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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.

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Santiago Guzmán Pro, Jakub Rydval. 2026-04-27. The Polynomial Hierarchy and $\omega$-categorical CSPs. https://arxiv.org/abs/2604.24539

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