Synchronizable functions on integers
Abstract
For all natural numbers a,b and d > 0, we consider the function fa,b,d which associates n/d to any integer n when it is a multiple of d, and an + b otherwise; in particular f3,1,2 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 fna,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.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.