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main.tex

@@ -254,16 +254,13 @@ The \emph{initial configuration} of $\auta$ over $v$ is the word $(q_0,1)$. Give
 
 
 A \emph{$1$-nested} invisible-pebble automaton $\auta$ is simply an invisible pebble automaton.
-A \emph{$d+1$-nested} invisible-pebble automaton $\auta$ is given by a pair $(\auta,\autb)$
+A \emph{$d+1$-nested} invisible-pebble automaton is given by a pair $(\auta,\autb)$
 
 \begin{theorem}
    Any transduction defined by a nested marble transducer can be expressed as an \msomi. 
 \end{theorem}
 
-\begin{question}
 
-Does $\ipt\subseteq \mt\circ\msot$ hold?
-\end{question}
 
 
 \section{Nested marble transducer}
@@ -272,17 +269,29 @@ We define a model of transducers based on marble transducers.
  A finite nesting height. a marble of height k cannot go through another marble of height k. However it can go through marbles of height $<k$.
     
     
+A marble automaton is a restriction of an invisble-pebble automaton, with two kinds of allowed transitions: $\push$ followed by $\mright$ or $\pop$.
+
+
+
 A marble 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 $\delta: A_{\vdash,\dashv}{\times} Q \rightarrow {\set{\mleft,\mright}}{\times}Q + {\set{\push}}{\times}Q^2 +\set{\pop}$
+\item a transition function $\delta: A_{\vdash,\dashv}{\times} Q \rightarrow {\set{\push}}{\times}Q^2 +\set{\pop}{\times}(Q{\rightarrow}Q)$
 
 
 \end{itemize}
-\begin{theorem}
-   Any transduction defined by a nested marble transducer can be expressed as an \msomi. 
-\end{theorem}
+
+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 $\delta(v[i],p)=\alpha$ in the following way:
+\begin{itemize}
+\item if $\alpha=(\push,q_1,q_2)$ and $i<n$ then $c'=c(q_1,i)(q_2,i+1)$
+\item if $\alpha= (\pop,f)$ and $c=d(r,j)$ then $c'=d(f(r),j)$
+\item in all other cases, $c'$ is not defined
+\end{itemize}
+
+
 \begin{remark}
 ~
 \begin{itemize}
@@ -290,7 +299,9 @@ A marble automaton $\auta$ is given by:
 
 \item Nested marble transducers subsume both pebble and marble transducers.
 
-\item They are closed under postcomposition by polyregular functions. TODO check
+\item They are closed under postcomposition by polyregular functions.
+
+\item They are closed under pre-composition by rational functions
 \end{itemize}
 
 
@@ -310,6 +321,10 @@ Basically, the most significant figure inside a block between two $\sharp$ symbo
 \begin{conjecture}
 The function $f$ above is not definable by a nested marble transducer.
 \end{conjecture}
+\begin{question}
+
+Does $\ipt\subseteq \mt\circ\msot$ hold?
+\end{question}