# Planar digraphs for automatic complexity

## TL;DR

This paper demonstrates that the digraphs representing nondeterministic finite automata for automatic complexity can always be made planar, and studies the counting function of words with fixed complexity, showing it stabilizes and is computable.

## Contribution

It proves planarity of automaton digraphs for automatic complexity and analyzes the asymptotic behavior of the number of words with fixed complexity.

## Key findings

- Automaton digraphs for automatic complexity can always be planar.
- The function counting words with fixed complexity stabilizes and is computable.
- Planarity can fail in total transition functions studied previously.

## Abstract

We show that the digraph of a nondeterministic finite automaton witnessing the automatic complexity of a word can always be taken to be planar. In the case of total transition functions studied by Shallit and Wang, planarity can fail. Let $s_q(n)$ be the number of binary words $x$ of length $n$ having nondeterministic automatic complexity $A_N(x)=q$. We show that $s_q$ is eventually constant for each $q$ and that the eventual constant value of $s_q$ is computable.