diff --git a/main.tex b/main.tex
index 5068a572300159d083c66027063e52865dbf346b..f1e600098b11f755ca3d53281ef2bea24de1255e 100644
--- a/main.tex
+++ b/main.tex
@@ -1,8 +1,27 @@
 
 \documentclass[a4paper,UKenglish,cleveref, autoref, thm-restate]{article}
 
-\usepackage{amssymb,amsthm,amsmath}
-
+\usepackage{amssymb,amsthm,amsmath,stmaryrd}
+\usepackage{xspace}
+
+\newcommand{\expreg}{exporegular}
+\newcommand{\Expreg}{Exporegular\xspace}
+\newcommand{\mso}{\textsf{MSO}\xspace}
+\newcommand{\fo}{\textsf{FO}\xspace}
+\newcommand{\msot}{\textsf{MSO-T}\xspace}
+\newcommand{\fot}{\textsf{FO-T}\xspace}
+\newcommand{\msoi}{\textsf{MSO-I}\xspace}
+\newcommand{\foi}{\textsf{FO-I}\xspace}
+\newcommand{\msomi}{\textsf{MSO-MI}\xspace}
+\newcommand{\fomi}{\textsf{FO-MI}\xspace}
+\newcommand{\nmsomi}{\textsf{NMSO-MI}\xspace}
+\newcommand{\sst}{\textsf{SST}\xspace}
+\newcommand{\ssign}{\mathfrak{S}}
+\newcommand{\tsign}{\mathfrak{T}}
+
+\newcommand{\set}[1]{\left\{#1\right\}}
+\newcommand{\tuple}[1]{\left(#1\right)}
+\newcommand{\sem}[1]{\left\llbracket #1\right\rrbracket}
 
 \newtheorem{theorem}{Theorem}
 \newtheorem{corollary}{Corollary}
@@ -10,7 +29,7 @@
 
 \bibliographystyle{alpha}% the mandatory bibstyle
 
-\title{Exporegular functions} %TODO Please add
+\title{\Expreg functions} %TODO Please add
 \author{}
 
 
@@ -24,10 +43,10 @@
 \begin{abstract}
     We introduce a new class of functions of exponential growth (at
     most). It captures the class of polyregular functions and of
-    functions definable by SST with copy. We provide diverse
-    characterizations of exporegular functions: monadic-second order
-    interpretations (MSOMI). As an application, we prove that
-    model-checking non-deterministic SST with copy as well as
+    functions definable by \sst with copy. We provide diverse
+    characterizations of \expreg functions: monadic-second order
+    interpretations (\msomi). As an application, we prove that
+    model-checking non-deterministic \sst with copy as well as
     deterministic k-pebble transducers against a first-order logic
     defining properties of their origin graphs is decidable. 
 \end{abstract}
@@ -40,22 +59,31 @@
 \label{sec:prelim}
 \subsection{Words, languages and transductions}
 \subsection{Relational structures and logical interpretations}
-\subsection{Exporegular functions}
+Here define \msoi and \foi.
+\subsection{Monadic interpretation}
 
-An exporegular function is a function $f : \Sigma^*\rightarrow
-\Sigma^*$ defined by a monadic second-order interpretation\footnote{We
+\footnote{We
 could add copy to MSOMI just as in Courcelle to avoid undesirable
-cornercase behaviours with small words.} $T =
-(\phi_{dom}(\overline{X}), \phi_\leq(\overline{X},\overline{Y}),
-\phi_\sigma({\overline{X}}))$ where each of those formulas are MSO
-formulas over signature $\{x\leq y,\sigma(x)\}$. We denote by
-MSOMI(s2s) those MSO-interpretations whose output structure  is a
-string (it is decidable), or just MSOMI if this is clear from the
-context. If instead FO formulas are used, we denote FO interpretations
-with monadic parameters by MSOI.
+cornercase behaviours with small words.}
+
+Given two relational signatures $\ssign,\tsign$, an \mso monadic interpretation $T$ of dimension $d$ from structures over $\ssign$ to structures over $\tsign$ is given by:
+\begin{itemize}
+\item An \mso-formula $\phi_{U}(\bar X)$ defining the universe of the output structure
+\item For each relation $R\in \tsign$ of arity $k$, a formula $\phi_R(\bar X _1,\ldots, \bar X_k )$
+
+\end{itemize}
+where $\bar X$ denotes a $d$-tuple of monadic variables.
+
+Given a structure $A$ over $\ssign$, we define its image by $T$ by a structure $B=\sem T(B)$ over $\tsign$: the universe of $B$ is the set $U=\set{\bar X\mid\ A \models \phi_U(\bar X)}$, given $R\in \tsign$ of arity $k$, we define $R$ in $B$ as the set $\set{\tuple{\bar X_1,\ldots,\bar X_k}\in U^k\mid\ \phi_R(\bar X_1,\ldots,\bar X_k)}$.
+%Given tuples $\bar X_1,\ldots,\bar X_l$, we extend $\sem T$ by $\sem T(A,\bar X_1,\ldots,\bar X_l)=B,\set{\bar X_1},\ldots,\set{\bar X_l}$.
+
+\fomi is defined by restricting formulas to be in \fo.
+
 
+We call string-to-string (monadic) interpretations the particular case when the two signatures have unary symbols and one binary symbol which is a linear order (this can be enforced syntactically).
 
-We denote by \textsf{ExpF} the class of exporegular functions. 
+An \expreg function is a string-to-string monadic interpretation.
+We denote by \textsf{ExpF} the class of \expreg functions. 
 As an example, consider the following function. Let $\Sigma=\{a,b\}$
 and $\Gamma = \Sigma\times \{0,1\}\cup \#$. Let $u\in\Sigma^*$.
 Given a
@@ -72,35 +100,48 @@ It is easy to see that $subsets\in\textsf{ExpF}$.
 
 
 \begin{theorem}[Backward translation theorem]
-    Given an MSOMI $T$ (from $\sigma$-relational structures to
-    $\beta$-relational structures) and some FO formula $\phi$ over
-    $\beta$, $T^{-1}(\phi)$, the set of $\sigma$-relational structures
-    $A$ such that $T(A)\models \phi$, is $MSO[\sigma]$-definable. 
+    Let $T$ be an \msomi $T$ of dimension $d$ from structures over $\ssign$ to
+     structures over $\tsign$ ,and let $\phi(x_1,\dots,x_k)$ be an \fo-formula over
+    $\tsign$.
+    We can define an \mso-formula $\psi(\bar X_1,\ldots,\bar X_k)$  such that for any structure $A$ over $\ssign$, $A\models\psi(\bar S_1,\ldots,\bar S_k) \psi$ if and only if $\sem T(A)\models \phi(\bar S_1,\ldots,\bar S_k)$.\footnote{Note that $\bar S_i$ is a position of the structure $\sem T(A)$}
 \end{theorem}
 
 \begin{proof}
     Idea: any first-order quantification $\exists x$ in $\phi$ is
     replaced by the monadic quantification $\exists \overline{X}$,
-    the atoms $x\leq y$ are replaced by
-    $\phi_\leq(\overline{X},\overline{Y})$, etc. 
+    the atoms $R(x_1,\dots,x_k)$ are replaced by
+    $\phi_R(\bar X_1,\ldots,\bar X_k)$, etc. 
 \end{proof}
 
 \begin{corollary}
-    The inverse image of any regular language by an exporegular
-    function is regular. In particular, exporegular functions have
+    The inverse image of any \fo-definable language by an \expreg
+    function is regular. In particular, \expreg functions have
     regular domains. 
 \end{corollary}
 
-\subsection{NMSOMI relations}
+\subsection{\nmsomi relations}
 
-Just as MSOT can be extended with parameters to add some form of
-non-determinism, MSOMI can also be extended with parameters to define
-exporegular relations, and the backward translation theorem still
-holds. We denote by $\textsf{ExpR}$ the class of exporegular
+Just as \msot can be extended with parameters to add some form of
+non-determinism, \msomi can also be extended with parameters to define
+\expreg relations, and the backward translation theorem still
+holds. We denote by $\textsf{ExpR}$ the class of \expreg
 relations. 
 
 
-\section{Expressiveness of exporegular functions}
+\section{Expressiveness of \expreg\ functions}
+
+\subsection{Closure properties}
+
+\begin{theorem}
+    FOI $\circ$ MSOMI = MSOMI
+\end{theorem}
+
+\begin{proof}
+    Use standard ideas for composition logical transductions: formula
+    substitutions. The first-order parameters of the FOI becomes
+    monadic parameters. 
+\end{proof}
+
 
 \begin{theorem}
     NSST$_{copy}$ $\subseteq$ \textsf{ExpR}. 
@@ -145,7 +186,7 @@ are of the form $x\not\in X_i \wedge x\in Y_i$.
 \end{proof}
 
 
-\subsection{ExpoRegular functions of polynomial growth}
+\subsection{\Expreg functions of polynomial growth}
 
 \begin{theorem}
     $\textsf{ExpF}\cap O(n^d) = \textsf{PolyF}$
@@ -159,17 +200,7 @@ are of the form $x\not\in X_i \wedge x\in Y_i$.
 
 Moreover, it is decidable whether an
 
-\subsection{Closure properties}
 
-\begin{theorem}
-    FOI $\circ$ MSOMI = MSOMI
-\end{theorem}
-
-\begin{proof}
-    Use standard ideas for composition logical transductions: formula
-    substitutions. The first-order parameters of the FOI becomes
-    monadic parameters. 
-\end{proof}
 
 
 \section{FO model-checking of transductions with origins: SSTs and pebble transducers}