Extensions and Limits of the Specker-Blatter Theorem

Eldar Fischer, Johann A. Makowsky

Research output: Contribution to journalArticlepeer-review

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 a non-negative integer and A is an ultimately periodic set of non-negative integers.

Original languageEnglish
JournalJournal of Symbolic Logic
DOIs
StateAccepted/In press - 2024

Keywords

  • Combinatorial counting
  • Finite model theory
  • Monadic second order logic with modular counting
  • Specker-Blatter theorem

All Science Journal Classification (ASJC) codes

  • Philosophy
  • Logic

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