Any transduction defined by a nested marble transducer can be expressed as an \msomi.
\end{theorem}
\begin{remark}
~
\begin{itemize}
\item Nested marble transducers subsume both pebble and marble transducers.
\item They are closed under postcomposition by polyregular functions. TODO check
\end{itemize}
\end{remark}
\begin{example}
Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup\set{\sharp}\right)^*\rightarrow(A\times\set{0,1}\cup\set{\natural})^*$.
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@@ -252,14 +264,21 @@ An invisible-pebble automaton $\auta$ is given by:
\begin{itemize}
\item a finite set of letters $A$, and two extra letters $\set{\vdash,\dashv}$
\item a finite set of states $Q$
\item a transition function $\delta: A_{\vdash,\dashv}\times Q \rightarrow\set{\mleft,\mright,\push,\pop}$
\item a push function: $\push: A_{\vdash,\dashv}\times Q\rightarrow Q$
\item a pop update function: $\pop: A_{\tiny{\vdash,\dashv}}\times Q\rightarrow(Q\rightarrow Q)$
\item a transition function $\delta: A_{\vdash,\dashv}\times Q \rightarrow\set{\mleft,\mright,\push,\pop}\times Q$
\end{itemize}
Let us explain how the automaton is run over a word $w\in A^*$. The automaton actually runs over the word ${\vdash} w {\dashv}$.
A \emph{configuration}$c$ over a word ${\vdash} w {\dashv}$ of length $n$ is a word over $Q\times\set{1,\ldots,n}$. The configuration is actually a stack of reading heads over the input word. The head of the stack is the \emph{active head}.
Let us explain how the automaton is run over a word $u\in A^*$. The automaton actually runs over the word $v={\vdash} u {\dashv}$.
A \emph{configuration}$c$ over the word $v$ of length $n$ is a word over $Q\times\set{1,\ldots,n}$. The configuration is actually a stack of reading heads over the input word. The head of the stack is the \emph{active head}.
The \emph{initial configuration} of $\auta$ over $v$ is the word $(q_0,1)$. Given a non-empty configuration $c(p,i)$ the \emph{successor configuration}$c'$ is defined according to $\delta(v[i],p)=(t,q)$ in the following way: