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main.tex

@@ -107,7 +107,8 @@ 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 $\ssign\times\set{\sqsubseteq}$. 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 $\ssign^1 + \ssign^{>1}\times\set{\sqsubseteq}$. 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{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 $\set{(x,y)\mid\ x\subseteq y\subseteq \dom_A}$.
@@ -160,7 +161,7 @@ An \expreg function is a word-to-word monadic interpretation.
 
 \begin{proof}
     Idea: any first-order quantification $\exists x$ in $\phi$ is
-    replaced by the monadic quantification $\exists \overline{X}$,
+    replaced by the monadic quantification $\exists \bar{X}$,
     the atoms $R(x_1,\dots,x_k)$ are replaced by
     $\phi_R(\bar X_1,\ldots,\bar X_k)$, etc. 
 \end{proof}
@@ -176,15 +177,7 @@ An \expreg function is a word-to-word monadic interpretation.
 An \msomi can also be seen as an \foi over the powerset model.
 \end{remark}
 
-\begin{question}
-Does $\msot\circ \msomi \subseteq \msomi$ hold? Using Krohn-Rhodes theorem of Miko\l aj, it seems that the only difficult case
-      is to post-compose an \msomi with a reversible Mealy machine. In other words the problem can be formulated as: does $\rev\circ \msomi \subseteq \msomi$ hold? 
-\end{question}
 
-
-\begin{question}
-Given a (fixed) \msomi realizing $f$, can one compute the image of a word $u$ in time $O \big( |u|+|f(u)|\big)$ ?
-\end{question}
 \subsection{Closure properties}
 
 \subsubsection{Postcomposition by \foi}
@@ -365,9 +358,19 @@ Does $\ipt\subseteq \mt\circ\msot$ hold?
 
 \section{Some remaining questions}
 
-\subsection{Expressiveness}
+\subsection{Closure}
 
+\begin{question}
+Does $\msot\circ \msomi \subseteq \msomi$ hold? Using Krohn-Rhodes theorem of Miko\l aj, it seems that the only difficult case
+      is to post-compose an \msomi with a reversible Mealy machine. In other words the problem can be formulated as: does $\rev\circ \msomi \subseteq \msomi$ hold? 
+\end{question}
+It works for commutative groups.
 
+\subsection{Computing image}
+
+\begin{question}
+Given a (fixed) \msomi realizing $f$, can one compute the image of a word $u$ in time $O \big( |u|+|f(u)|\big)$ ?
+\end{question}
 
    
 
@@ -380,7 +383,7 @@ Does $\ipt\subseteq \mt\circ\msot$ hold?
 \begin{itemize}
     \item what about \mtt ? They have doubly-exponential growth, but do
       they capture \msomi  ?
-    \item recursive programming language corresponding or being
+    \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.
 \end{itemize}