The Polynomial Hierarchy and $ω$-categorical CSPs
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162K/year
🤖 AI Summary
This work investigates whether there exist ω-categorical constraint satisfaction problems (CSPs) that are complete for arbitrary levels of the polynomial hierarchy (PH). By integrating monadic second-order logic (MSO), ω-categorical model theory, and computational complexity theory, the authors develop a novel approach to construct MSO sentences with specific preservation properties. Using these sentences, they define a family of ω-categorical CSP instances that are complete for every level of PH. This result extends the earlier findings of Bodirsky and Grohe (2008) from individual complexity classes to the entire polynomial hierarchy, thereby establishing—for the first time—the existence of such complete problems across all levels of PH.
Application Category
📝 Abstract
In 2008, Bodirsky and Grohe showed that for every $Π_n^{\mathrm{P}}$-level of the Polynomial Hierarchy (PH) there are $ω$-categorical Constraint Satisfaction Problems (CSPs) complete for this level. We show that, in fact, there are $ω$-categorical CSPs complete for any level of the PH. To this end, we use a recent result of Bodirsky, Knäuer, and Rudolph for constructing $ω$-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.
Problem
Research questions and friction points this paper is trying to address.
Polynomial Hierarchy
ω-categorical
Constraint Satisfaction Problems
Monadic Second-Order logic
Innovation
Methods, ideas, or system contributions that make the work stand out.
Polynomial Hierarchy
ω-categorical CSPs
Monadic Second-Order logic
preservation properties
computational complexity
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