# Undecidability and Finite Automata

## TL;DR

This paper demonstrates that many decision problems related to finite automata are undecidable using a new rewriting problem, while also identifying some problems that remain decidable, and applies these findings to k-automatic sets of rationals.

## Contribution

Introduces a novel rewriting problem to establish undecidability results for several finite automata decision problems and proves some related problems are decidable.

## Key findings

- Several natural decision problems about finite automata are undecidable.
- Some related problems about finite automata are decidable.
- Undecidability of a problem about k-automatic sets of rational numbers.

## Abstract

Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). In contrast, we also prove three related problems are decidable. We apply one result to prove the undecidability of a related problem about k-automatic sets of rational numbers.