Synchronizable functions on integers

TL;DR

This paper introduces a class of functions on integers, including the Collatz function, and demonstrates how to realize their compositions and closures using input-deterministic transducers with regular initial states.

Contribution

It presents a novel transducer-based framework to explicitly compute and analyze the composition and closure of a broad class of integer functions.

Findings

Explicit transducer models for function compositions

Realization of closure under composition with infinite transducers

Application to functions including the Collatz function

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.