Notebook 11: Infinite Words & In(α)
Infinite Words and \(In(\alpha)\)
This notebook explores ultimately periodic \(\omega\)-words. These are infinite words of the form \(\alpha = u \cdot v^\omega\), where \(u\) is a finite prefix called the stem, and \(v\) is a finite word called the period that repeats infinitely often.
We also compute \(In(\alpha)\), which is the set of symbols from the alphabet that appear infinitely often in the word \(\alpha\). For an ultimately periodic word, \(In(\alpha)\) is exactly the set of symbols present in the period \(v\).
The running example is an editor that is opened once and then remains active. For instance, \(\texttt{open}\cdot(\texttt{edit}\ \texttt{save})^\omega\) describes an execution in which editing and saving both recur. The one-time open belongs to the stem and therefore does not belong to \(In(\alpha)\).
Widget 1: \(\omega\)-Word Explorer
Enter a stem \(u\) and a period \(v\) (comma or space-separated symbols, e.g., open, edit, save). The word strip below will visualize the infinite word \(\alpha = u \cdot v^\omega\).
open · (edit save)ω