diff --git a/main.tex b/main.tex index 02d256b5f07090fc0618bcf7ed4bb8529e2fb6ff..32fbf416cf080c3cd37772dad77789283407034f 100644 --- a/main.tex +++ b/main.tex @@ -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}