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OAF
alignment-finder
Commits
fafbac5d
Commit
fafbac5d
authored
Apr 12, 2018
by
Dennis Müller
Browse files
structuring
parent
6b983337
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5
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tex/intro.tex
0 → 100644
View file @
fafbac5d
``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}
tex/paper.tex
View file @
fafbac5d
...
...
@@ -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
View file @
fafbac5d
MMT theories, flat, bla
\ No newline at end of file
tex/usecase.tex
0 → 100644
View file @
fafbac5d
tex/viewfinder.tex
0 → 100644
View file @
fafbac5d
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