# Preservation of normality by transducers

## TL;DR

This paper investigates how input-deterministic finite state transducers affect the Borel normality of infinite words, providing an algorithm to decide if they preserve normality efficiently.

## Contribution

It introduces an algorithm that determines in cubic time whether such transducers preserve the normality of infinite sequences.

## Key findings

- The output of a normal input sequence has frequencies described by a weighted automaton.
- An algorithm exists to decide normality preservation in cubic time.
- Normality preservation depends on the structure of the transducer and input sequence.

## Abstract

We consider input-deterministic finite state transducers with infinite inputs and infinite outputs, and we consider the property of Borel normality on infinite words. When these transducers are given by a strongly connected set of states, and when the input is a Borel normal sequence, the output is an infinite word such that every word has a frequency given by a weighted automaton over the rationals. We prove that there is an algorithm that decides in cubic time whether an input-deterministic transducer preserves normality.