The Polynomial Hierarchy and $ω$-categorical CSPs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911626585702400 |
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| author | Pro, Santiago Guzmán Rydval, Jakub |
| author_facet | Pro, Santiago Guzmán Rydval, Jakub |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_24539 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Polynomial Hierarchy and $ω$-categorical CSPs Pro, Santiago Guzmán Rydval, Jakub Logic in Computer Science 03B70 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. |
| title | The Polynomial Hierarchy and $ω$-categorical CSPs |
| topic | Logic in Computer Science 03B70 |
| url | https://arxiv.org/abs/2604.24539 |