From 0f22b2447fd3710bd70a56bdbd39a84f7a985b48 Mon Sep 17 00:00:00 2001
From: "nathan.lhote" <nathan.lhote@lis-lab.fr>
Date: Wed, 15 Feb 2023 11:55:23 +0100
Subject: [PATCH] m
---
main.tex | 27 +++++++++++++++------------
1 file changed, 15 insertions(+), 12 deletions(-)
diff --git a/main.tex b/main.tex
index 839e9f4..390f482 100644
--- a/main.tex
+++ b/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}
--
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