The Collatz graph is a directed graph with natural number nodes and where there is an edge from node x to node if x is even, or to node 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 that contains the binary expression of any ancestor y that reaches x with a limited budget of k applications of The set is known to be regular, Shallit and Wilson [EATCS 1992]. In this paper, we find that the structure of the Collatz graph naturally leads to the construction of a regular expression, which defines 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 case [TCS 2011] to any natural number x, and gives mathematical and algorithmic (Code available here: https://github.com/tcosmo/coreli.) tools for further exploration of the Collatz graph in binary.
CITATION STYLE
Stérin, T. (2020). Binary Expression of Ancestors in the Collatz Graph. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 12448 LNCS, pp. 115–130). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-030-61739-4_8
Mendeley helps you to discover research relevant for your work.