@inproceedings{11e1f012925543da8468758cfcb05d50,

title = "Extensions and Limits of the Specker-Blatter Theorem",

abstract = "The original Specker-Blatter Theorem (1983) was formulated for classes of structures C of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set [n] is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker-Blatter Theorem does not hold for one quaternary relation (2003). If the vocabulary allows a constant symbol c, there are n possible interpretations on [n] for c. We say that a constant c is hard-wired if c is always interpreted by the same element j ∈ [n]. In this paper we show: (i) The Specker-Blatter Theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case. (ii) The Specker-Blatter Theorem does not hold already for C with one ternary relation definable in First Order Logic FOL. This was left open since 1983. Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers Br,A, restricted Stirling numbers of the second kind Sr,A or restricted Lah-numbers Lr,A. Here r is an non-negative integer and A is an ultimately periodic set of non-negative integers.",

keywords = "MC-finiteness, Monadic Second Order Logic, Specker-Blatter Theorem",

author = "Eldar Fischer and Makowsky, {Johann A.}",

note = "Publisher Copyright: {\textcopyright} Eldar Fischer and Johann A. Makowsky.; 32nd EACSL Annual Conference on Computer Science Logic, CSL 2024 ; Conference date: 19-02-2024 Through 23-02-2024",

year = "2024",

month = feb,

doi = "https://doi.org/10.4230/LIPIcs.CSL.2024.26",

language = "الإنجليزيّة",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

editor = "Aniello Murano and Alexandra Silva",

booktitle = "32nd EACSL Annual Conference on Computer Science Logic, CSL 2024",

}