\begin{definition}
A \emph{prefix-free universal Turing machine} is a universal Turing machine $U:\mathcal{B}^\ast \times \mathcal{B}^\ast \rightarrow \mathcal{B}^\ast$ such that, for every $v \in \mathcal{B}^\ast$, the domain $U{v}$ is prefix-free, where $U{v}:\mathcal{B}^\ast \rightarrow \mathcal{B}^\ast$ and $U_{v}(x) = U(x, v)$ for all $x in \mathcal{B}^\ast$.
\end{definition}
Write a paragraph clarifying why we have to fix the imput string $v$ in $U_{v}(x) = U(x, v)$, since this is something that might confuse some readers.
rleiva
commented
7 months ago
In the following definition,
\begin{definition} A \emph{prefix-free universal Turing machine} is a universal Turing machine $U:\mathcal{B}^\ast \times \mathcal{B}^\ast \rightarrow \mathcal{B}^\ast$ such that, for every $v \in \mathcal{B}^\ast$, the domain $U{v}$ is prefix-free, where $U{v}:\mathcal{B}^\ast \rightarrow \mathcal{B}^\ast$ and $U_{v}(x) = U(x, v)$ for all $x in \mathcal{B}^\ast$. \end{definition}
Write a paragraph clarifying why we have to fix the imput string $v$ in $U_{v}(x) = U(x, v)$, since this is something that might confuse some readers.