diff --git a/main.tex b/main.tex
index 390f482a3c90c722340a6e373266d759ad9f34d8..7970e99ad8eb003dd862d890279ea0d826657531 100644
--- a/main.tex
+++ b/main.tex
@@ -32,10 +32,16 @@
 \newcommand{\mtt}{\textsf{MTT}\xspace}
 
 \newcommand{\rev}{\textsf{rev}\xspace}
+\newcommand{\rtm}{\textsf{RTM}\xspace}
+\newcommand{\drtm}{\textsf{DRTM}\xspace}
+\newcommand{\reg}{\textsf{Reg}}
 
 
 
-\newcommand{\dlin}{\textsc{DLinSpace}\xspace}
+\newcommand{\dlinspace}{\textsc{DLinSpace}\xspace}
+\newcommand{\dlogspace}{\textsc{DLogSpace}\xspace}
+\newcommand{\dptime}{\textsc{DPTime}\xspace}
+\newcommand{\dexptime}{\textsc{DExpTime}\xspace}
 
 \newcommand{\ssign}{\mathfrak{S}}
 \newcommand{\tsign}{\mathfrak{T}}
@@ -96,7 +102,7 @@
 \paragraph{Signature}
 A \emph{signature} $\ssign$ is a set $S$ of \emph{symbols}\footnote{We only consider relational signatures.}, together with an \emph{arity function}, which we denote by $\ar:S\rightarrow \nat$. Abusing notations, we will often write $R\in \ssign$ instead of $R\in S$.
 
-\paragraph{Structures} A \emph{structure} $A$ over a signature $\ssign$ is given as a \emph{domain} $\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)-\subseteq \dom(A)^r$, with $r=\ar(R)$.
+\paragraph{Structures} A \emph{structure} (sometimes \emph{model}) $A$ over a signature $\ssign$ is given as a \emph{domain} $\dom_A$ together with an \emph{interpretation function} which maps any symbol $R$ of $\ssign$ to a set denoted $\inter_A(R)$ such that $\inter_A(R)\subseteq \dom(A)^r$, with $r=\ar(R)$.
 
 \paragraph{Operations}
 Any structure over a signature can be seen as a structure over a larger signature where additional symbols are interpreted as empty sets.
@@ -107,9 +113,9 @@ The \emph{product} of two structures $A,B$ over a signature $\ssign$ is a struct
 
 The \emph{ordered coproduct} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigoplus_{i\leq n} A_i$ over the signature $\ssign+\set{\sqsubseteq}$, where $\sqsubseteq$ has arity $2$. The domain of $\bigoplus_{i\leq n} A_i$ is $\bigoplus_{i\leq n} \dom_{A_i}=\bigcup_{i\leq n} \dom_{A_i}{\times}\set{i}$. The interpretation of a symbol $R$ is $\bigoplus_{i\leq n} \inter_{A_i}(R)$. The interpretation of $\sumorder$ is $\bigcup_{i\leq j\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$.
 
-The \emph{ordered product} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigotimes_{i\leq n} A_i$ over the signature $\ssign^1 + \ssign^{>1}\times\set{\sqsubseteq}$. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n} \dom_{A_i}$.
-The interpretation of a symbol $R$ is $\bigoplus_{i\leq n} \inter_{A_i}(R)$. The interpretation of $\sumorder$ is $\bigcup_{i\leq j\leq n}(\dom_{A_i}{\times}\set{i})\times(\dom_{A_j}{\times}\set{j})$.
-
+The \emph{ordered product} of a sequence of structures $A_1,\cdots, A_n$ over the same signature $\ssign$ is a structure $\bigotimes_{i\leq n} A_i$ over the signature $\ssign^{1} + \ssign^{>1}\times\set{\sqsubseteq}$, where $\ssign^{1}$, $\ssign^{>1}$ denote the symbols of arity one and at least two, respectively. The domain of $\bigotimes_{i\leq n} A_i$ is $\bigotimes_{i\leq n} \dom_{A_i}$.
+The interpretation of a symbol $R$ of arity $1$ is $\bigotimes_{i\leq n} \inter_{A_i}(R)$. The interpretation of $(R,\sumorder)$ is: $$\bigcup_{j\leq n}\quad \bigotimes_{i<j}\Delta_{i,k} \times\inter_{A_j}\times\bigotimes_{j<i\leq n}\dom_{A_i}$$
+where $\Delta_{i,k}=\set{(a,\ldots,a)\in \dom_{A_i}^k\mid\ a\in \dom_{A_i}}$ is the $k$-fold diagonal of $\dom_{A_i}$.
 The \emph{powerset structure} of a structure $A$ over signature $\ssign$ is a structure $\pow A$ over signature $\ssign\uplus \set{\subseteq}$, where $\subseteq$ is a binary symbol. The domain of $\pow A$ is $\pow{\dom_A}$. The interpretation of a symbol $R$ is the set $\set{(\set{x_1},\dots,\set{x_k})\mid\ (x_1,\ldots,x_n)\in \inter_A(R)}$.
 The interpretation of $\subseteq$ is the actual set inclusion $\set{(x,y)\mid\ x\subseteq y\subseteq \dom_A}$.
 \paragraph{Word models}
@@ -234,26 +240,6 @@ An \msomi of dimension $d$ has growth $O((2^n)^d)=O(2^{dn})=2^{O(n)}$.
 
 
 
-\section{The case of the successor}
-
-\begin{theorem}
-\msomi with successor is equivalent to \dlin reductions.
-\end{theorem}
-
-\begin{proof}
-Given an \msomi, we can define a rational letter-to-letter relation which realizes the successor and from this obtain a linear bounded automaton.
-From a linear bounded automaton, the next configuration relation can be defined in \mso.
-\end{proof}
-
-\begin{question}
-
-
-Does the model $a_1\cdots a_n\mapsto f\circ f_{a_1}\circ \cdots \circ f_{a_n}(\epsilon)$ compute \dlin ?
-
-\end{question}
-
-
-
 \section{Nested marble transducer}
 
 We define a model of transducers based on marble transducers.
@@ -299,7 +285,7 @@ The \emph{initial configuration} of $\auta$ over $v$ is the word $(q_0,1)$. Give
 \end{remark}
 
 \begin{example}
-Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \cup{0,1}\cup \set{\natural})^*$.
+Let $A=\set{a,b}$. Let us define an \expreg function $f:\left(A\cup \set{\sharp}\right)^*\rightarrow (A\times \cup\set{0,1,\natural})^*$.
 What the function $f$ does is list all subsets of positions (without $\sharp$) and output the word labelled with the corresponding subset. The labelled words are output in a specific order, and separated by $\natural$. Let us decribe the order in which these subsets of positions are listed.
 
 Over a word $u$ without $\sharp$, $f$ outputs the list of all annotations, in the lexicographic order over the subset of positions of $u$, \eg
@@ -352,8 +338,35 @@ Does $\ipt\subseteq \mt\circ\msot$ hold?
 \end{question}
 
 
+\section{Rational Turing Machine}
 
 
+A \emph{Rational Turing Machine} is a Turing Machine with a reachability relation over configurations that is rational.
+
+
+\begin{theorem}
+A function is \expreg if and only if it is definable by a \drtm.
+\end{theorem}
+
+\begin{theorem}
+For rational turing machines, the following hold:
+\begin{enumerate}
+\item \drtm = \dlinspace
+\item \dlogspace= \dptime
+\end{enumerate}
+\end{theorem}
+
+
+\section{The case of the successor}
+
+\begin{theorem}
+\msomi with successor is equivalent to \dlinspace reductions.
+\end{theorem}
+
+\begin{proof}
+Given an \msomi, we can define a rational letter-to-letter relation which realizes the successor and from this obtain a linear bounded automaton.
+From a linear bounded automaton, the next configuration relation can be defined in \mso.
+\end{proof}
 
 
 \section{Some remaining questions}
@@ -386,6 +399,8 @@ Given a (fixed) \msomi realizing $f$, can one compute the image of a word $u$ in
     \item recursive programming language corresponding to or being
       captured by \msomi ?
 	\item Krohn-Rhodes like decomposition \eg $\msoi\circ \mathsf{exp} \circ \msot$, with $\mathsf{exp}$ being some simple class of exponential growth function.
+	
+	\item Does the model $a_1\cdots a_n\mapsto f\circ f_{a_1}\circ \cdots \circ f_{a_n}(\epsilon)$, with sequential functions compute \dlinspace ?
 \end{itemize}