Notebook 13: Vectorial Closure — 🔀W
Vectorial Closure \(\vec{W}\)
The vectorial closure \(\vec{W}\) of a language \(W \subseteq \Sigma^*\) of finite words is the set of all \(\omega\)-words \(\alpha\) such that infinitely many finite prefixes of \(\alpha\) belong to \(W\).
Formally: \[ \alpha \in \vec{W} \iff \text{there exist infinitely many } n \ge 0 \text{ such that } \alpha(0, n) \in W \]
Notice the difference with infinite concatenation \(W^\omega\): to be in \(W^\omega\), a word must be factorable into infinitely many blocks from \(W\). To be in \(\vec{W}\), the word itself must have infinitely many prefixes falling into \(W\).
For the running editor example, let \(W=\{\texttt{open}\,x\,\texttt{save} : x\in\{\texttt{edit},\texttt{idle},\texttt{save}\}^*\}\). Then \(\vec{W}\) contains exactly the well-formed infinite editor traces whose prefixes end immediately after a save infinitely often—equivalently, the traces in which saving recurs forever.
Widget: Prefix Membership Explorer
Select a preset for \(W\) and an ultimately periodic word \(\alpha = u \cdot v^\omega\). The widget shows which prefixes of \(\alpha\) belong to \(W\).