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Binary Expression of Ancestors in the Collatz Graph

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Part of the Lecture Notes in Computer Science book series (LNCS, volume 12448)

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 structure 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 (Code available here: https://github.com/tcosmo/coreli.) tools for further exploration of the Collatz graph in binary.

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Notes

Acknowledgement

Many thanks to Jose Capco, Damien Woods, Pierre-Étienne Meunier and Turlough Neary for their kind help, interest and feedback on this project. We also thank Jeffrey C. Lagarias for his surveys on the Collatz problem ( [16] and [17]). We thank the OEIS, always of great help. Finally, sincere thanks to the anonymous reviewers. Their comments were very helpful, for instance making us realise the exponential gain of our construction compared to previous literature.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Hamilton Institute, Department of Computer ScienceMaynooth UniversityCounty KildareIreland

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