Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-05-06T05:10:52.423Z Has data issue: false hasContentIssue false
  • English
  • Français

EXTENSIONS AND LIMITS OF THE SPECKER–BLATTER THEOREM

Part of: Graph theory Combinatorics Model theory

Published online by Cambridge University Press:  21 March 2024

ELDAR FISCHER  and
JOHANN A. MAKOWSKY Open the ORCID record for JOHANN A. MAKOWSKY [Opens in a new window]
ELDAR FISCHER
Affiliation:
FACULTY OF COMPUTER SCIENCE ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL E-mail: eldar@cs.technion.ac.il
JOHANN A. MAKOWSKY*
Affiliation:
FACULTY OF COMPUTER SCIENCE ISRAEL INSTITUTE OF TECHNOLOGY HAIFA, ISRAEL E-mail: eldar@cs.technion.ac.il
*
E-mail: jmakowsky@bluewin.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

The original Specker–Blatter theorem (1983) was formulated for classes of structures $\mathcal {C}$ 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 $j \in [n]$. In this paper we show:

  1. (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.

  2. (ii) The Specker–Blatter theorem does not hold already for $\mathcal {C}$ 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 a non-negative integer and A is an ultimately periodic set of non-negative integers.

Keywords

finite model theorycombinatorial countingmonadic second order logic with modular countingSpecker–Blatter theorem

MSC classification

Primary: 05A18: Partitions of sets 05C30: Enumeration in graph theory 03C13: Finite structures 03C98: Applications of model theory
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 29
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Blatter, C. and Specker, E., Le nombre de structures finies d’une théorie à charactère fini , Sciences Mathématiques, Fonds Nationale de la recherche Scientifique, Brussels, 1981, pp. 41–44.Google Scholar
Blatter, C. and Specker, E., Modular periodicity of combinatorial sequences . Abstracts of the AMS, vol. 4 (1983), p. 313.Google Scholar
Blatter, C. and Specker, E., Recurrence relations for the number of labeled structures on a finite set , Logic and Machines: Decision Problems and Complexity (Börger, E., Hasenjaeger, G., and Rödding, D., editors), Lecture Notes in Computer Science, vol. 171, Springer, Berlin–Heidelberg, 1984, pp. 43–61.CrossRefGoogle Scholar
Broder, A. Z., The r-Stirling numbers . Discrete Mathematics, vol. 49 (1984), no. 3, pp. 241–259.CrossRefGoogle Scholar
Courcelle, B., The monadic second-order logic of graphs. I. Recognizable sets of finite graphs . Information and Computation, vol. 85 (1990), no. 1, pp. 12–75.CrossRefGoogle Scholar
Courcelle, B. and Engelfriet, J., Graph Structure and Monadic Second-Order Logic: A Language Theoretic Approach, Cambridge University Press, Cambridge, 2012.CrossRefGoogle Scholar
Dawar, A., Grohe, M., Kreutzer, S., and Schweikardt, N., Model theory makes formulas large , International Colloquium on Automata, Languages, and Programming, Springer, Berlin–Heidelberg, 2007, pp. 913–924.CrossRefGoogle Scholar
Diestel, R., Graph Theory, third ed., Graduate Texts in Mathematics, vol. 173, Springer, Berlin–Heidelberg, 2005.Google Scholar
Ebbinghaus, H.-D. and Flum, J., Finite Model Theory, Springer, Berlin–Heidelberg–New York, 1995.Google Scholar
Everest, G., van der Poorten, A. J, Shparlinski, I., and Ward, T., Recurrence Sequences, Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
Feferman, S. and Vaught, R., The first order properties of algebraic systems . Fundamenta Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
Filmus, Y., Fischer, E., Makowsky, J. A., and Rakita, V., MC-finiteness of restricted set partition functions . Journal of Integer Sequences, vol. 26 (2023), no. 2, p. 3.Google Scholar
Fischer, E., The Specker–Blatter theorem does not hold for quaternary relations . Journal of Combinatorial Theory, Series A, vol. 103 (2003), pp. 121–136.CrossRefGoogle Scholar
Fischer, E., Kotek, T., and Makowsky, J. A., Application of logic to combinatorial sequences and their recurrence relations , Model Theoretic Methods in Finite Combinatorics (Grohe, M. and Makowsky, J. A., editors), Contemporary Mathematics, vol. 558, American Mathematical Society, Providence, RI, 2011, pp. 1–42.CrossRefGoogle Scholar
Fischer, E., Kotek, T., and Makowsky, J. A., Application of logic to combinatorial sequences and their recurrence relations . Model Theoretic Methods in Finite Combinatorics, vol. 558 (2011), pp. 1–42.CrossRefGoogle Scholar
Fischer, E. and Makowsky, J. A., The Specker–Blatter theorem revisited , Cocoon, Lecture Notes in Computer Science, vol. 2697, Springer, Berlin–Heidelberg, 2003, pp. 90–101.Google Scholar
Fischer, E. and Makowsky, J. A., Extensions and limits of the Specker–Blatter theorem, preprint, 2022, arXiv:2206.12135.Google Scholar
Fischer, E. and Makowsky, J. A., Extensions and limits of the Specker–Blatter theorem , 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024), Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Wadern, 2024, Article 26.Google Scholar
Kauers, M. and Paule, P.. The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates, Springer, Vienna, 2011.CrossRefGoogle Scholar
Kummer, E., Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen . Journal für die reine und angewandte Mathematik, vol. 1852 (1852), no. 44, pp. 93–146.Google Scholar
Koshy, T., Catalan Numbers with Applications, Oxford University Press, Oxford, 2008.CrossRefGoogle Scholar
Libkin, L., Elements of Finite Model Theory, Springer, Berlin–Heidelberg, 2004.CrossRefGoogle Scholar
Makowsky, J. A., Algorithmic uses of the Feferman–Vaught theorem . Annals of Pure and Applied Logic, vol. 126 (2004), nos. 1–3, pp. 159–213.CrossRefGoogle Scholar
Pfeiffer, G., Counting transitive relations . Journal of Integer Sequences, vol. 7 (2004), no. 2, p. 3.Google Scholar
Reeds, J. A. and Sloane, N. J. A., Shift register synthesis (modulo m) . SIAM Journal on Computing, vol. 14 (1985), no. 3, pp. 505–513.CrossRefGoogle Scholar
Specker, E., Application of logic and combinatorics to enumeration problems , Ernst Specker Selecta, Birkhäuser, Basel, 1990, pp. 324–350.CrossRefGoogle Scholar
Specker, E., Modular counting and substitution of structures . Combinatorics, Probability and Computing, vol. 14 (2005), pp. 203–210.CrossRefGoogle Scholar