Effective MC-finiteness

TL;DR

This work analyzes when the MC-finiteness of integer sequences extends to an effectively computable residue function $F:(m,n)\mapsto a_n\bmod m$, introducing fixed-parameter tractability as a natural criterion. It shows that polynomial recurrence sequences (PRS) yield MC-finite behavior with $a_n\bmod m$ computable in time $T(m)\cdot n^{O(1)}$, via the finite-state dynamics of the modulo-$m$ map $F^{(m)}$. The core framework, Specker's method, connects density functions of combinatorial structure classes to DU-rank and G-degree, proving MC-finiteness of density under finite DU-rank and bounded G-degree, and derives explicit linear recurrences modulo $m$ that govern these counts. The paper also charts limitations in unbounded G-degree, presents examples of ternary structures, and outlines strategies to extend MC-finiteness results to ternary properties through CMSOL/MSOL definability, Feferman–Vaught decompositions, and potential ternary analogues of the main theorem. Overall, it advances understanding of which combinatorial and logical conditions guarantee effectively MC-finite behavior and computability of residue functions across modular regimes.

Abstract

An integer sequence $(a_n)_{n \in \mathbb{N}}$ is \emph{MC-finite} if for all $m$, the sequence $a_n \bmod m$ is eventually periodic. There are MC-finite sequences $(a_n)_{n \in \mathbb{N}}$ such that the function $F: (m,n) \mapsto a_n \bmod m$ is not computable. In \cite{filmus2023mc} we presented concrete examples of MC-finite sequences taken from the Online Encyclopedia of Integer Sequences (OEIS) without discussing the computability of $F$. In this paper we discuss cases when this $F$ is effectively computable.

Theorems & Definitions (17)

• Theorem 1
• proof
• Theorem 2
• proof
• Proposition 3
• Theorem 5: fischer2003speckerFiMa-ETH-2003
• Lemma 8
• proof
• Lemma 10
• proof