Binary expression of ancestors in the Collatz graph

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.