Synchronizable functions on integers
Formal Languages and Automata Theory
2022-05-30 v1
Abstract
For all natural numbers a,b and d > 0, we consider the function f_{a,b,d} which associates n/d to any integer n when it is a multiple of d, and an + b otherwise; in particular f_{3,1,2} is the Collatz function. Coding in base a > 1 with b < a, we realize these functions by input-deterministic letter-to-letter transducers with additional output final words. This particular form allows to explicit, for any integer n, the composition n times of such a transducer to compute f^n_{a,b,d}. We even realize the closure under composition f^*_{a,b,d by an infinite input-deterministic letter-to-letter transducer with a regular set of initial states and a length recurrent terminal function.
Keywords
Cite
@article{arxiv.2205.14018,
title = {Synchronizable functions on integers},
author = {Didier Caucal and Chloé Rispal},
journal= {arXiv preprint arXiv:2205.14018},
year = {2022}
}
Comments
23 pages, 15 figures