The Collatz process embeds a base conversion algorithm
Abstract
The Collatz process is defined on natural numbers by iterating the map $T(x) = T_0(x) = x/2$ when $x\in\mathbb{N}$ is even and $T(x)=T_1(x) =(3x+1)/2$ when $x$ is odd. In an effort to understand its dynamics, and since Generalised Collatz Maps are known to simulate Turing Machines [Conway, 1972], it seems natural to ask what kinds of algorithmic behaviours it embeds. We define a quasicellular automaton that exactly simulates the Collatz process on the square grid: on input $x\in\mathbb{N}$, written horizontally in base 2, successive rows give the Collatz sequence of $x$ in base 2. We show that vertical columns simultaneously iterate the map in base 3. This leads to our main result: the Collatz process embeds an algorithm that converts any natural number from base 3 to base 2. We also find that the evolution of our automaton computes the parity of the number of 1s in any ternary input. It follows that predicting about half of the bits of the iterates $T^i(x)$, for $i = O(\log x)$, is in the complexity class NC$^1$ but outside AC$^0$. Finally, we show that in the extension of the Collatz process to numbers with infinite binary expansions ($2$adic integers) [Lagarias, 1985], our automaton encodes the cyclic Collatz conjecture as a natural reachability problem. These results show that the Collatz process is capable of some simple, but nontrivial, computation in bases 2 and 3, suggesting an algorithmic approach to thinking about existence, prediction and structure of cycles in the Collatz process.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.06979
 Bibcode:
 2020arXiv200706979S
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computational Complexity;
 Computer Science  Formal Languages and Automata Theory;
 F.1.1;
 F.2.1;
 F.4.3;
 G.2
 EPrint:
 28 pages. 8 figures. 2 appendices. Short version accepted to the 14th International Conference on Reachability Problems (RP 2020)