\phi_\sigma({\overline{X}}))$ where each of those formulas are MSO
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:
formulas over signature $\{x\leq y,\sigma(x)\}$. We denote by
\begin{itemize}
MSOMI(s2s) those MSO-interpretations whose output structure is a
\item An \mso-formula $\phi_{U}(\bar X)$ defining the universe of the output structure
string (it is decidable), or just MSOMI if this is clear from the
\item For each relation $R\in\tsign$ of arity $k$, a formula $\phi_R(\bar X _1,\ldots, \bar X_k )$
context. If instead FO formulas are used, we denote FO interpretations
with monadic parameters by MSOI.
\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 $\Sigma=\{a,b\}$
As an example, consider the following function. Let $\Sigma=\{a,b\}$
and $\Gamma=\Sigma\times\{0,1\}\cup\#$. Let $u\in\Sigma^*$.
and $\Gamma=\Sigma\times\{0,1\}\cup\#$. Let $u\in\Sigma^*$.
Given a
Given a
...
@@ -72,35 +100,48 @@ It is easy to see that $subsets\in\textsf{ExpF}$.
...
@@ -72,35 +100,48 @@ It is easy to see that $subsets\in\textsf{ExpF}$.
\begin{theorem}[Backward translation theorem]
\begin{theorem}[Backward translation theorem]
Given an MSOMI $T$ (from $\sigma$-relational structures to
Let $T$ be an \msomi$T$ of dimension $d$ from structures over $\ssign$ to
$\beta$-relational structures) and some FO formula $\phi$ over
structures over $\tsign$ ,and let $\phi(x_1,\dots,x_k)$ be an \fo-formula over
$\beta$, $T^{-1}(\phi)$, the set of $\sigma$-relational structures
$\tsign$.
$A$ such that $T(A)\models\phi$, is $MSO[\sigma]$-definable.
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}
\end{theorem}
\begin{proof}
\begin{proof}
Idea: any first-order quantification $\exists x$ in $\phi$ is
Idea: any first-order quantification $\exists x$ in $\phi$ is
replaced by the monadic quantification $\exists\overline{X}$,
replaced by the monadic quantification $\exists\overline{X}$,
the atoms $x\leq y$ are replaced by
the atoms $R(x_1,\dots,x_k)$ are replaced by
$\phi_\leq(\overline{X},\overline{Y})$, etc.
$\phi_R(\bar X_1,\ldots,\bar X_k)$, etc.
\end{proof}
\end{proof}
\begin{corollary}
\begin{corollary}
The inverse image of any regular language by an exporegular
The inverse image of any \fo-definable language by an \expreg
function is regular. In particular, exporegular functions have
function is regular. In particular, \expreg functions have
regular domains.
regular domains.
\end{corollary}
\end{corollary}
\subsection{NMSOMI relations}
\subsection{\nmsomi relations}
Just as MSOT can be extended with parameters to add some form of
Just as \msot can be extended with parameters to add some form of
non-determinism, MSOMI can also be extended with parameters to define
non-determinism, \msomi can also be extended with parameters to define
exporegular relations, and the backward translation theorem still
\expreg relations, and the backward translation theorem still
holds. We denote by $\textsf{ExpR}$ the class of exporegular
holds. We denote by $\textsf{ExpR}$ the class of \expreg
relations.
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}
\begin{theorem}
NSST$_{copy}$$\subseteq$\textsf{ExpR}.
NSST$_{copy}$$\subseteq$\textsf{ExpR}.
...
@@ -145,7 +186,7 @@ are of the form $x\not\in X_i \wedge x\in Y_i$.
...
@@ -145,7 +186,7 @@ are of the form $x\not\in X_i \wedge x\in Y_i$.
\end{proof}
\end{proof}
\subsection{ExpoRegular functions of polynomial growth}
\subsection{\Expreg functions of polynomial growth}
\begin{theorem}
\begin{theorem}
$\textsf{ExpF}\cap O(n^d)=\textsf{PolyF}$
$\textsf{ExpF}\cap O(n^d)=\textsf{PolyF}$
...
@@ -159,17 +200,7 @@ are of the form $x\not\in X_i \wedge x\in Y_i$.
...
@@ -159,17 +200,7 @@ are of the form $x\not\in X_i \wedge x\in Y_i$.
Moreover, it is decidable whether an
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}
\section{FO model-checking of transductions with origins: SSTs and pebble transducers}