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Extensions and Limits of the Specker-Blatter Theorem

Authors Eldar Fischer, Johann A. Makowsky

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Author Details

Eldar Fischer
  • Faculty of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel
Johann A. Makowsky
  • Faculty of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel

Funding

  • Fischer, Eldar: Israel Science Foundation ISF 879/22

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Eldar Fischer and Johann A. Makowsky. Extensions and Limits of the Specker-Blatter Theorem. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 26:1-26:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CSL.2024.26

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 B_{r,A}, restricted Stirling numbers of the second kind S_{r,A} or restricted Lah-numbers L_{r,A}. Here r is an non-negative integer and A is an ultimately periodic set of non-negative integers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Mathematics of computing → Enumeration
Keywords
  • Specker-Blatter Theorem
  • Monadic Second Order Logic
  • MC-finiteness

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References

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