Decidability and k-Regular Sequences

TL;DR

This paper investigates various decision problems related to k-regular sequences, demonstrating that many such problems, including growth bounds and recognizability, are undecidable.

Contribution

It establishes the undecidability of several natural decision problems involving k-regular sequences, advancing understanding of their computational limits.

Findings

Decidability of growth bounds is undecidable.

Recognizability of preimages by finite automata is undecidable.

Problems involving factors like squares and palindromes are undecidable.

Abstract

In this paper we consider a number of natural decision problems involving k-regular sequences. Specifically, they arise from - lower and upper bounds on growth rate; in particular boundedness, - images, - regularity (recognizability by a deterministic finite automaton) of preimages, and - factors, such as squares and palindromes of such sequences. We show that the decision problems are undecidable.