### structuring

parent 6b983337
tex/intro.tex 0 → 100644
 Semantic Search'' -- a very suggestive term, which is alas seriously under-defined -- has often been touted as the killer application'' of semantic technologies. With a view finder, we can add another possible interpretation: searching mathematical ontologies (here modular theorem prover libraries) at the level of theories -- we call this \defemph{theory classification}. The basic use case is the following: Jane, a mathematician, becomes interested in a class of mathematical objects, say -- as a didactic example -- something she initially calls beautiful subsets'' of a base set $\cB$ (or just beautiful over $\cB$''). These have the following properties: \begin{compactenum} \item the empty set is beautiful over $\cB$ \item every subset of a beautiful set is beautiful over $\cB$ \item If $A$ and $B$ are beautiful over $\cB$and $A$ has more elements than $B$, then there is an $x\in A\backslash B$, such that $B\cup\{x\}$ is beautiful over $\cB$. \end{compactenum} To see what is known about beautiful sets, she types these three conditions into a theory classifier, which interprets them as a \MMT theory $Q$, computes all (total) views from $Q$ into a library $\cL$, and returns presentations of target theories and the assignments made by the views. In our example we have the situation in Figure~\ref{fig:theory-classification-ex}\ednote{MK: Maybe better do this in jEdit and use screenshots} \ednote{Interesting: the $\emptyset$ clause could also be $\exists S$ beautiful'', which is equivalent via the subset clause, maybe use that in the example: define Matroid with that and have a theorem that says that $\emptyset$ is beautiful. Just see what the viewfinder catches.} In the base use case, Jane learns that her beautiful sets'' correspond to the well-known structure of matroids~\cite{wikipedia:matroid}, so she can directly apply matroid theory to her problems. \begin{newpart}{MK: this could/should be extended} In extended use cases, we could \begin{compactitem} \item use partial views to find theories that share significant structure with $Q$, so that we can formalize $Q$ with modularly with minimal effort. Say Jane was interested in dazzling subsets'', i.e. beautiful subsets that obey a fourth condition, then she could just contribute a theory that extends \textsf{matroid} by a formalization of the fourth condition -- and maybe rethink the name. \item use existing views in $\cL$ (compositions of views are vivews) to give Jane more information about her beautiful subsets, e.g. that matroids (and thus beautiful sets) form a generalization of the notion of linear independence from linear algebra. \end{compactitem} \end{newpart} \begin{figure}[ht]\centering\lstset{aboveskip=0pt,belowskip=0pt} \begin{tabular}{|p{5cm}|p{5cm}|}\hline $Q$ & Result \\\hline \begin{lstlisting}[mathescape] theory query : F?MitM include ?set Base : {A} set A beautiful {A} set A $\rightarrow$ prop empty : $\vdash$ beautiful $\emptyset$ subset-closed: $\vdash$ {} \end{lstlisting} & \begin{lstlisting}[mathescape] theory matroid : F?MitM include ?baseset independent {A} set A $\rightarrow$ empty: $\vdash$ hereditary : $\vdash$ augmentation: $\vdash$ \end{lstlisting} \\\hline \end{tabular} \caption{Theory Classification for beautiful sets}\label{fig:theory-classification-ex} \end{figure}
 ... ... @@ -72,78 +72,14 @@ \end{abstract} \setcounter{tocdepth}{2}\tableofcontents\newpage \section{Introduction}\label{sec:intro} \section{Applications}\label{sec:appl} \subsection{Theory Classification}\label{sec:classifier} Semantic Search'' -- a very suggestive term, which is alas seriously under-defined -- has often been touted as the killer application'' of semantic technologies. With a view finder, we can add another possible interpretation: searching mathematical ontologies (here modular theorem prover libraries) at the level of theories -- we call this \defemph{theory classification}. The basic use case is the following: Jane, a mathematician, becomes interested in a class of mathematical objects, say -- as a didactic example -- something she initially calls beautiful subsets'' of a base set $\cB$ (or just beautiful over $\cB$''). These have the following properties: \begin{compactenum} \item the empty set is beautiful over $\cB$ \item every subset of a beautiful set is beautiful over $\cB$ \item If $A$ and $B$ are beautiful over $\cB$and $A$ has more elements than $B$, then there is an $x\in A\backslash B$, such that $B\cup\{x\}$ is beautiful over $\cB$. \end{compactenum} To see what is known about beautiful sets, she types these three conditions into a theory classifier, which interprets them as a \MMT theory $Q$, computes all (total) views from $Q$ into a library $\cL$, and returns presentations of target theories and the assignments made by the views. In our example we have the situation in Figure~\ref{fig:theory-classification-ex}\ednote{MK: Maybe better do this in jEdit and use screenshots} \ednote{Interesting: the $\emptyset$ clause could also be $\exists S$ beautiful'', which is equivalent via the subset clause, maybe use that in the example: define Matroid with that and have a theorem that says that $\emptyset$ is beautiful. Just see what the viewfinder catches.} In the base use case, Jane learns that her beautiful sets'' correspond to the well-known structure of matroids~\cite{wikipedia:matroid}, so she can directly apply matroid theory to her problems. \begin{newpart}{MK: this could/should be extended} In extended use cases, we could \begin{compactitem} \item use partial views to find theories that share significant structure with $Q$, so that we can formalize $Q$ with modularly with minimal effort. Say Jane was interested in dazzling subsets'', i.e. beautiful subsets that obey a fourth condition, then she could just contribute a theory that extends \textsf{matroid} by a formalization of the fourth condition -- and maybe rethink the name. \item use existing views in $\cL$ (compositions of views are vivews) to give Jane more information about her beautiful subsets, e.g. that matroids (and thus beautiful sets) form a generalization of the notion of linear independence from linear algebra. \end{compactitem} \end{newpart} \begin{figure}[ht]\centering\lstset{aboveskip=0pt,belowskip=0pt} \begin{tabular}{|p{5cm}|p{5cm}|}\hline $Q$ & Result \\\hline \begin{lstlisting}[mathescape] theory query : F?MitM include ?set Base : {A} set A beautiful {A} set A $\rightarrow$ prop empty : $\vdash$ beautiful $\emptyset$ subset-closed: $\vdash$ {} \end{lstlisting} & \begin{lstlisting}[mathescape] theory matroid : F?MitM include ?baseset independent {A} set A $\rightarrow$ empty: $\vdash$ hereditary : $\vdash$ augmentation: $\vdash$ \end{lstlisting} \\\hline \end{tabular} \caption{Theory Classification for beautiful sets}\label{fig:theory-classification-ex} \end{figure} \section{Introduction}\label{sec:intro}\input{intro} \section{Preliminaries}\label{sec:prelim}\input{prelim} \section{Viewfinder}\label{sec:viewfinder}\input{viewfinder} \section{Extended Use Case}\label{sec:usecase}\input{usecase} \section{Conclusion}\label{sec:concl} ... ...
tex/prelim.tex 0 → 100644
 MMT theories, flat, bla \ No newline at end of file
tex/usecase.tex 0 → 100644
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