diff --git a/main.tex b/main.tex
index 5b551c444b3827434281a463510c64a94cb03450..a78616599ac8f77c77e1effa78c9aacb91a415dd 100644
--- a/main.tex
+++ b/main.tex
@@ -231,6 +231,18 @@ pebble transducer model. candidate:
 \begin{theorem}
    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 $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \set{0,1}\cup \set{\natural})^*$.
@@ -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:
+\begin{itemize}
+\item if $t=\mleft$, and $i>1$ then $c'=c(q,i-1)$
+\item if $t=\mright$ and $i<n$ then $c'=c(q,i+1)$
+\item if $t=\push$ then $c'=c(p,i)(q,i)$
+\item if $t=\pop$ and $c=d(r,j)$ then $c'=d(q,j)$
+\item in all other cases, $c'$ is not defined
+\end{itemize}
 
 
 \section{Some remaining questions}