# Binary expression of ancestors in the Collatz graph

## TL;DR

This paper explores the binary structure of the Collatz graph, revealing how message passing behaviors relate to the complexity of the Collatz process, and introduces an efficient regular expression construction for ancestor sets.

## Contribution

It presents a new exponential construction of regular expressions for ancestor sets in the Collatz graph, improving upon previous methods and generalizing prior work to all natural numbers.

## Key findings

- Constructs regular expressions for ancestor sets in the Collatz graph.
- Provides an exponential algorithm for defining ancestor sets.
- Generalizes previous results from the case x=1 to all natural numbers.

## Abstract

The Collatz graph is a directed graph with natural number nodes and where there is an edge from node $x$ to node $T(x)=T_0(x)=x/2$ if $x$ is even, or to node $T(x)=T_1(x)=\frac{3x+1}{2}$ if $x$ is odd. Studying the Collatz graph in binary reveals complex message passing behaviors based on carry propagation which seem to capture the essential dynamics and complexity of the Collatz process. We study the set $\mathcal{E} \text{Pred}_k(x)$ that contains the binary expression of any ancestor $y$ that reaches $x$ with a limited budget of $k$ applications of $T_1$. The set $\mathcal{E} \text{Pred}_k(x)$ is known to be regular, Shallit and Wilson [EATCS 1992]. In this paper, we find that the geometry of the Collatz graph naturally leads to the construction of a regular expression, $\texttt{reg}_k(x)$, which defines $\mathcal{E} \text{Pred}_k(x)$. Our construction, is exponential in $k$ which improves upon the doubly exponentially construction of Shallit and Wilson. Furthermore, our result generalises Colussi's work on the $x = 1$ case [TCS 2011] to any natural number $x$, and gives mathematical and algorithmic tools for further exploration of the Collatz graph in binary.