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Commit fcc10224 authored by Nathan Lhote's avatar Nathan Lhote
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......@@ -8,6 +8,7 @@
\newcommand{\Expreg}{Exp-regular\xspace}
\newcommand{\mso}{\textsf{MSO}\xspace}
\newcommand{\fo}{\textsf{FO}\xspace}
\newcommand{\msot}{\textsf{MSO-T}\xspace}
......@@ -17,7 +18,8 @@
\newcommand{\msomi}{\textsf{MSO-MI}\xspace}
\newcommand{\fomi}{\textsf{FO-MI}\xspace}
\newcommand{\nmsomi}{\textsf{NMSO-MI}\xspace}
\newcommand{\pow}[1]{\mathsf{2}^{#1}}
\newcommand{\ar}{\mathsf{ar}}
\newcommand{\sst}{\textsf{SST}\xspace}
\newcommand{\mt}{\textsf{MT}\xspace}
......@@ -37,6 +39,8 @@
\newcommand{\tuple}[1]{\left(#1\right)}
\newcommand{\sem}[1]{\left\llbracket #1\right\rrbracket}
\newcommand{\nat}{\mathbb{N}}
\newcommand{\eg}{\textit{e.g.~}}
\newcommand{\ie}{\textit{i.e.~}}
......@@ -84,11 +88,20 @@
\subsection{Relational structures and logical interpretations}
\subsubsection{Relational structures}
\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} $w$ over a signature $\ssign$ is given as a \emph{domain} $D$ together with a function which maps any symbol $R$ of $\ssign$ to a set denoted $R^w$ such that $R^w\subseteq D^r$, with $r=\ar(R)$.
\paragraph{Operations} The \emph{disjoint union} of two structures over
\paragraph{The orderned model of words}
\paragraph{The powerset model of words}~\\
Any \mso-formula can be seen as an \fo-formula over the powerset model
Any \mso-formula can be seen as an \fo-formula over the powerset model.
Given a structure $A$ we denote the powerset of $A$ the structure $\pow A$.
\subsubsection{Logical interpretation}
Here define \msoi and \foi.
\subsection{Monadic interpretation}
......
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