Extensions and Limits of the Specker-Blatter Theorem

TL;DR

This paper investigates the Specker-Blatter Theorem's applicability to structures with constants and higher-arity relations, extending its scope and identifying its limitations in logical definability and counting functions.

Contribution

It proves the theorem holds for CMSOL with hard-wired constants and demonstrates its failure for certain ternary relations in First Order Logic, expanding understanding of counting functions.

Findings

The theorem holds for CMSOL with hard-wired constants.

The theorem does not hold for structures with a ternary relation in FOL.

Certain restricted partition functions are proven to be MC-finite.

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 {\em hard-wired} if $c$ is always interpreted by the same element $j \in [n]$. In this paper we show: 1. 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. 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 an non-negative integer and $A$ is an ultimately periodic set of non-negative integers.