We study Lindström quantifiers that satisfy certain closure properties which are motivated by the study of polymorphisms in the context of constraint satisfaction problems (CSP). When the algebra of polymorphisms of a finite structure 𝔅 satisfies certain equations, this gives rise to a natural closure condition on the class of structures that map homomorphically to 𝔅. The collection of quantifiers that satisfy closure conditions arising from a fixed set of equations are rather more general than those arising as CSP. For any such conditions 𝒫, we define a pebble game that delimits the distinguishing power of the infinitary logic with all quantifiers that are 𝒫-closed. We use the pebble game to show that the problem of deciding whether a system of linear equations is solvable in ℤ / 2ℤ is not expressible in the infinitary logic with all quantifiers closed under a near-unanimity condition.
@InProceedings{dawar_et_al:LIPIcs.CSL.2024.23, author = {Dawar, Anuj and Hella, Lauri}, title = {{Quantifiers Closed Under Partial Polymorphisms}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {23:1--23:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.23}, URN = {urn:nbn:de:0030-drops-196662}, doi = {10.4230/LIPIcs.CSL.2024.23}, annote = {Keywords: generalized quantifiers, constraint satisfaction problems, pebble games, finite variable logics, descriptive complexity theory} }
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