How to prove that a sequence is not automatic

TL;DR

This paper reviews various methods to demonstrate that certain sequences are not automatic, and discusses how these methods can also establish the transcendence of related formal power series over finite fields.

Contribution

It provides a comprehensive survey of techniques for proving non-automaticity and their applications to transcendence in finite fields.

Findings

Multiple methods for proving non-automaticity are summarized.

These methods can also show transcendence of formal power series over finite fields.

The paper highlights the connection between sequence properties and algebraic independence.

Abstract

Automatic sequences have many properties that other sequences (in particular, non-uniformly morphic sequences) do not necessarily share. In this paper we survey a number of different methods that can be used to prove that a given sequence is not automatic. When the sequences take their values in the finite field ${\mathbb F}_q$, this also permits proving that the associated formal power series are transcendental over ${\mathbb F}_q(X)$.