Document

# Extensions and Limits of the Specker-Blatter Theorem

## File

LIPIcs.CSL.2024.26.pdf
• Filesize: 0.68 MB
• 20 pages

## Cite As

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
• MC-finiteness

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. C. Blatter and E. Specker. Le nombre de structures finies d'une théorie à charactère fini. Sciences Mathématiques, Fonds Nationale de la recherche Scientifique, Bruxelles, pages 41-44, 1981.
2. C. Blatter and E. Specker. Modular periodicity of combinatorial sequences. Abstracts of the AMS, 4:313, 1983.
3. C. Blatter and E. Specker. Recurrence relations for the number of labeled structures on a finite set. In E. Börger, G. Hasenjaeger, and D. Rödding, editors, In Logic and Machines: Decision Problems and Complexity, volume 171 of Lecture Notes in Computer Science, pages 43-61. Springer, 1984.
4. Andrei Z Broder. The r-stirling numbers. Discrete Mathematics, 49(3):241-259, 1984.
5. B. Courcelle and J. Engelfriet. Graph Structure and Monadic Second-order Logic, a Language Theoretic Approach. Cambridge University Press, 2012.
6. Anuj Dawar, Martin Grohe, Stephan Kreutzer, and Nicole Schweikardt. Model theory makes formulas large. In International Colloquium on Automata, Languages, and Programming, pages 913-924. Springer, 2007.
7. R. Diestel. Graph Theory. Graduate Texts in Mathematics. Springer, 3 edition, 2005.
8. H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer Verlag, 1995.
9. Graham Everest, Alfred J van der Poorten, Igor Shparlinski, Thomas Ward, et al. Recurrence sequences, volume 104. American Mathematical Society Providence, RI, 2003.
10. S. Feferman and R. Vaught. The first order properties of algebraic systems. Fundamenta Mathematicae, 47:57-103, 1959.
11. Yuval Filmus, Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita. MC-finiteness of restricted set partitions, 2023. URL: https://arxiv.org/abs/2302.08265.
12. E. Fischer. The Specker-Blatter theorem does not hold for quaternary relations. Journal of Combinatorial Theory, Series A, 103:121-136, 2003.
13. E. Fischer. The Specker-Blatter theorem does not hold for quaternary relations. Journal of Combinatorial Theory, Series A, 103:121-136, 2003.
14. E. Fischer, T. Kotek, and J.A. Makowsky. Application of logic to combinatorial sequences and their recurrence relations. In M. Grohe and J.A. Makowsky, editors, Model Theoretic Methods in Finite Combinatorics, volume 558 of Contemporary Mathematics, pages 1-42. American Mathematical Society, 2011.
15. E. Fischer and J. A. Makowsky. The Specker-Blatter theorem revisited. In COCOON, volume 2697 of Lecture Notes in Computer Science, pages 90-101. Springer, 2003.
16. Eldar Fischer, Tomer Kotek, and Johann A Makowsky. Application of logic to combinatorial sequences and their recurrence relations. Model Theoretic Methods in Finite Combinatorics, 558:1-42, 2011.
17. Eldar Fischer and Johann A. Makowsky. Extensions and limits of the specker-blatter theorem, 2022. URL: https://arxiv.org/abs/2206.12135.
18. Ronald L Graham, Donald E Knuth, and Oren Patashnik. Concrete mathematics: a foundation for computer science. Addison-Wesley, 1989.
19. Manuel Kauers and Paule. The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates. Springer, 2011.
20. Thomas Koshy. Catalan numbers with applications. Oxford University Press, 2008.
21. L. Libkin. Elements of Finite Model Theory. Springer, 2004.
22. J.A. Makowsky. Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic, 126.1-3:159-213, 2004.
23. Götz Pfeiffer. Counting transitive relations. Journal of Integer Sequences, 7(2):3, 2004.
24. James A Reeds and Neil JA Sloane. Shift register synthesis (modulo m). SIAM Journal on Computing, 14(3):505-513, 1985.
25. E. Specker. Modular counting and substitution of structures. Combinatorics, Probability and Computing, 14:203-210, 2005.
26. Ernst Specker. Application of logic and combinatorics to enumeration problems. In Ernst Specker Selecta, pages 324-350. Springer, 1990.