A generalized quantifier Q_𝒦 is called a CSP-quantifier if its defining class 𝒦 consists of all structures that can be homomorphically mapped to a fixed finite template structure. For all positive integers n ≥ 2 and k, we define a pebble game that characterizes equivalence of structures with respect to the logic L^k_{∞ω}(CSP^+_n), where CSP^+_n is the union of the class Q₁ of all unary quantifiers and the class CSP_n of all CSP-quantifiers with template structures that have at most n elements. Using these games we prove that for every n ≥ 2 there exists a CSP-quantifier with template of size n+1 which is not definable in L^ω_{∞ω}(CSP^+_n). The proof of this result is based on a new variation of the well-known Cai-Fürer-Immerman construction.
@InProceedings{hella:LIPIcs.CSL.2023.25, author = {Hella, Lauri}, title = {{The Expressive Power of CSP-Quantifiers}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {25:1--25:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.25}, URN = {urn:nbn:de:0030-drops-174867}, doi = {10.4230/LIPIcs.CSL.2023.25}, annote = {Keywords: generalized quantifiers, constraint satisfaction problems, pebble games, finite variable logics, descriptive complexity theory} }
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