From 8425416f69c38d6b83ef5c73fd3ca8bf69151407 Mon Sep 17 00:00:00 2001 From: m-iancu <mihnea.iancu88@gmail.com> Date: Sat, 28 Feb 2015 22:44:43 +0100 Subject: [PATCH] more --- adoptions/oldparts.tex | 124 ++++++++++++++++++++++++++++++++++++++++ adoptions/paper.pdf | Bin 474245 -> 448230 bytes adoptions/paper.tex | 3 + adoptions/patterns.tex | 125 +---------------------------------------- 4 files changed, 128 insertions(+), 124 deletions(-) create mode 100644 adoptions/oldparts.tex diff --git a/adoptions/oldparts.tex b/adoptions/oldparts.tex new file mode 100644 index 0000000..a90f5dd --- /dev/null +++ b/adoptions/oldparts.tex @@ -0,0 +1,124 @@ +\begin{oldpart}{from bluenote} + +We will progress by going through the examples from the introduction. \ednote{MK@MK: say + something about what digital libraries should look like: a theory for every variant + highly modularized, narratively enhanced overview for salient theories that are often + used, see Carette/Farmer high-level theories, \cite{CarFarKoh:tr13}} + +\begin{example}[A Course grounded in a Formal Library]\label{sec:course} +Take for instance a course which introduces (naive) set theory informally, but grounds +itself in a formal, modular definition. Then we have the situation in +Figure~\ref{fig:slides-library}. On the right hand side, we have a careful introduction in +the form of a modular theory graph starting at a theory \cn{ZFset} that introduces +membership relation and the axioms of existence, extensionality, and separation and +defines the set constructor $\{\cdot |\cdot\}$ from these axioms. On the left we have a +theory \cn{SET} that ``adopts'' the symbols $\in$ and $\{\cdot |\cdot\}$ via a partial +inclusion $\cn{a_1}$ from \cn{ZFset} to \cn{SET} but ``defines'' them by alluding the +intuitions of the students. Note that a partial inclusion always gives rise to a view in +the opposite direction, here the view \cn{v_1} from \cn{SET} to \cn{set}. Note that we +cannot discharge the proof obligations in \cn{v_1}, since the definition of the set +constructor $\{\cdot |\cdot\}$ is opaque -- i.e. given as natural language, which is not +subject to formal methods; see~\cite{Kohlhase:fsgo13} for a discussion. We should think +of \cn{v_1} as a ``definitional view'' that gives meaning to the opaque parts in \cn{SET}: +the proof obgligations have to be met in order for the diagram to commute (which is an +invariant we want to maintain). + +We call this situation where the symbols of a theory are imported via a partial inclusion +and their meaning is specified via a view that is a partial inverse an \textbf{adoption} +and the morphism pair an \textbf{adoption} morphism. + +\begin{figure}[ht]\centering +\begin{tikzpicture}[yscale=1.7] + \node at (1,2.6) {\large Library}; + \node at (-4.5,2.6) {\large Course}; + \node[thy] (zfset) at (1,-.2) {\mthy{ZFset}{$\in$, \cn{Ex}, \cn{Ext}, \cn{Sep}; $\{\cdot |\cdot\}$}}; + \node[thy] (zfunion) at (-.2,1) {\mthy{ZFunion}{\cn{Un}, $\cup$}}; + \node[thy] (zfpow) at (2.2,1) {\mthy{ZFpow}{\cn{Pow}, $\mathcal{P}$}}; + \node at (1.1,1) {\ldots}; + \node[thy] (zfop) at (1,2) {ZFops}; + \draw[include] (zfset) -- (zfunion); + \draw[include] (zfset) -- (zfpow); + \draw[include] (zfunion) -- (zfop); + \draw[include] (zfpow) -- (zfop); + \node[thy] (set) at (-4.5,-.2) {\mthy{SET}{$\in$, $\{\cdot |\cdot\}$}}; + \node[thy] (sop) at (-4.5,2) {\mthy{SETOPS}{$\cup$,\ldots,$\mathcal{P}$}}; + \draw[view] (set) to[out=-5,in=185] node[below] {$\cn{v_1}$} (zfset); + \draw[pinclude] (zfset) to[out=175,in=5] node[above] {$\cn{a_1}\colon\in,\{\cdot |\cdot\}$}(set); + \draw[view] (sop) to[out=-5,in=185] node[below] {$\cn{v_2}: \text{incl}\; \cn{v_1}$} (zfop); + \draw[pinclude] (zfop) to[out=175,in=5] node[above] {$\cn{a_2}\colon\cup,\ldots,\mathcal{P},\text{incl}\; \cn{a_1}$} (sop); + \draw[include] (set) -- (sop); +\end{tikzpicture} +\caption{A course grounded in a modular Library}\label{fig:slides-library} +\end{figure} + +Let us now continue looking at the example in Figure~\ref{fig:slides-library}: The +informal course materials go on in style, introducing the set operations ranging from set +union to the power set in one go in the theory \cn{SETOPS}. On the library side, we +introduce the set theory axioms one by one, derive the respective operators from them, and +at the end collect all the material in the theory \cn{ZFops}. A this point, we can justify +the theory \cn{SETOPS} via a partial inclusion of the symbols $\cup$, \ldots, +$\mathcal{P}$ from \cn{ZFops}, which gives rise to another definitional view \cn{v2} to +\cn{ZFops}. As \mmt allows assignments to structures, we can simply assign the inclusion +from \cn{SET} to \cn{SETOPS} to the (unique) inclusion from \cn{ZFset} to \cn{ZFops} (as +indicated by the ``assignment'' include $\cn{v_1}$ on $\cn{v_1}$). + +We observe that the two theory graphs are self-contained: the course materials can be +understood without knowing about the library; in particular, the membership relation used +in the definition of the union operator in \cn{SETOPS} is from theory \cn{SET}. + +This self-containedness is important intra-course didactics. But it also has the problem +that the courses become insular; how are students going to communicate with mathematicians +who have learned their maths from other courses? Here is where the views \cn{v_i} come +in. Say the other mathematicians have course theories \cn{\overline{SET}} and +\cn{\overline{SETOPS}} with views \cn{\overline{v_1}} and \cn{\overline{v_2}} into the +same library, then the views \cn{v_1} and \cn{\overline{v_1}} induce a partial +isomorphisms between \cn{SET} and \cn{\overline{SET}} in the sense +of~\cite{KohRabSac:fvip11} (and correspondingly between \cn{SETOPS} and +\cn{\overline{SETOPS}}) that justify communication. +\end{example} + + +\subsection{Accelerated Turing Machines} + +The case of the mathematical paper case study mentioned above is more difficult, since we +do not have a direct generalization relation as in the case above. In this case, the +definition of the accelerated Turing machine involves a concrete step size ($2^{-n}$), +whereas the definition it recaps allows arbitrary sequences of step sizes as long as their +sum remains finite. Thus we have the situation in Figure~\ref{fig:atm}. Theory \cn{ATM} +contains the (opaque) sentence (\ref{lq:atm}), but there cannot be a view from \cn{ATM} to +\cn{atm} as that is more general. But we do have a view to \cn{atm(2^{-n})}, which +naturally arises in treatments of accelerated Turing machines as an example. + +\begin{figure}[ht]\centering + \begin{tikzpicture}[scale=1.3] + \node at (-.5,2.7) {\large Paper}; + \node at (2.5,2.7) {\large Literature}; + \node[thy] (ATMp) at (-.5,2) {\cn{ATMhalt}}; + \node[thy] (catmp) at (2,2) {\cn{atm(2^{-n})halt}}; + \node[thy] (ATM) at (-.5,1) {\cn{ATM}}; + \node[thy] (atm) at (3.5,0) {\cn{atm}}; + \node[thy] (catm) at (2,1) {\cn{atm(2^{-n})}}; + \node[thy,dotted] (atmp) at (3.5,1) {\cn{atmhalt}}; + \draw[view] (ATM) -- node[above] {\cn{v_1}} (catm); + \draw[view] (ATMp) -- node[above] {\cn{v_2}} (catmp); + \draw[include] (atm) -- (catm); + \draw[include] (ATM) -- (ATMp); + \draw[include] (catm) -- (catmp); + \draw[include,dotted] (atm) -- (atmp); + \draw[include,dotted] (atmp) -- (catmp); + \end{tikzpicture} + \caption{Definitional View for ATM}\label{fig:atm} +\end{figure} + +The more important aspect is that the contribution of~\cite{CalStai:natm09} (depicted here +as theory \cn{ATMhalt} as it concerns undecidability of the accelerated halting problem +by accelerated Turing machines) can be justified via a view into a theory +\cn{atm(2^{-n})halt}, which can be systematically constructed from \cn{ATMhalt} modulo +\cn{v_1} as pushouts along inclusions always exist in \mmt. In the particular example, we +can do even better: as the proof in theory \cn{ATMhalt} does not use any properties of the +step size sequence $2^{-n}$, there is a theory \cn{atmhalt} that generalizes +\cn{atm(2^{-n})halt} (shown dotted in Figure~\ref{fig:atm}). Note that this construction +is automatic and needs human intervention -- but one that the authors expect their readers +will readily do, otherwise they would not have restricted themselves to the concrete +sequence. +\end{oldpart} \ No newline at end of file diff --git a/adoptions/paper.pdf b/adoptions/paper.pdf index 23d0ce08f5ed99348eb6158e80a0eb4c030a4bf8..bc4176567d85df1070fd0b953e9da399a5be4dd5 100644 GIT binary patch delta 295320 zcmZquDf4Wtw4iE$i<_>Yo++2=g)2vIaG9EI6cl3>GgL58$WO}Rva#XP_esr5FUe3a zGc=qY@PJXX{%yE#{w)WAy`Qze?XbJ?X1e(ASpfm77A?9G#2?5ny_rS(pTn_rSL*La zUQ@fVvB7z<3D@$C<<qA=-)o-F)-+GR@v~s&BVQA#^DA6JDq=j^7MUg;`pz-Y!|SCX zlehMl24!`T2-{`13V$9se)QI{+e+mYN2hK}eEz3vUtc;S*WQ47qs+WwO;tHzk)3v5 znRb-b%a+D?aU3k)d6+3?YU^XCn5%_T5A^Ci6{_BF<VL^)^>6Fk^=z4ZWJ4O{Vx$hJ zy<aDEc%JyLeX6~tOLV>Gszpv(ZFJH4LZe9jzq|9NZN9F<sk3W!z^;V?wyXBkwjLDa z4$UoDwn`;Tv*0M}yje-JAJp$Sw;<~w<Em?4Q!9>^IQ-nyI@z?KF5YKt_)&&5hjn3} z!ra?DKKb9Cd|Z0cyRBZv0j8<{L?4zv@GJ|ztJwa?*SOPcm&SjI^hZBt|I6j^T>h<e 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\input{patterns} +%\input{oldparts} + \section{Conclusion and Future Work}\label{sec:conc} \input{conc} diff --git a/adoptions/patterns.tex b/adoptions/patterns.tex index 2f17ae3..3959a17 100644 --- a/adoptions/patterns.tex +++ b/adoptions/patterns.tex @@ -8,7 +8,7 @@ First, each paper starts with establishing a common ground on which the results \item Firstly, concepts from the literature are used to conveniently build up the local definitions. From the theory graphs perspective this functions as a (possibly partial) import. \item Secondly, properties of locally introduced concepts are \emph{adopted} from the literature. Mathematically, this is justified by - and (implicit or explicit) equivalence \ednote{``equivalence'' is too strong here} between the local definition and that used by the referenced theorem. + and (implicit or explicit) subsumption between the local definition and that used by the referenced theorem. From the theory graph perspective this function as a theory morphism that induces the properties locally due to its truth-preserving semantics. \end{itemize} Therefore, a paper corresponds, not to a single theory, but to a theory pattern that leads to a theory of the main contxribution of the paper. @@ -391,130 +391,7 @@ given in the literature (which we represent as a realm). \end{figure} -\begin{oldpart}{from bluenote} - -We will progress by going through the examples from the introduction. \ednote{MK@MK: say - something about what digital libraries should look like: a theory for every variant - highly modularized, narratively enhanced overview for salient theories that are often - used, see Carette/Farmer high-level theories, \cite{CarFarKoh:tr13}} - -\begin{example}[A Course grounded in a Formal Library]\label{sec:course} -Take for instance a course which introduces (naive) set theory informally, but grounds -itself in a formal, modular definition. Then we have the situation in -Figure~\ref{fig:slides-library}. On the right hand side, we have a careful introduction in -the form of a modular theory graph starting at a theory \cn{ZFset} that introduces -membership relation and the axioms of existence, extensionality, and separation and -defines the set constructor $\{\cdot |\cdot\}$ from these axioms. On the left we have a -theory \cn{SET} that ``adopts'' the symbols $\in$ and $\{\cdot |\cdot\}$ via a partial -inclusion $\cn{a_1}$ from \cn{ZFset} to \cn{SET} but ``defines'' them by alluding the -intuitions of the students. Note that a partial inclusion always gives rise to a view in -the opposite direction, here the view \cn{v_1} from \cn{SET} to \cn{set}. Note that we -cannot discharge the proof obligations in \cn{v_1}, since the definition of the set -constructor $\{\cdot |\cdot\}$ is opaque -- i.e. given as natural language, which is not -subject to formal methods; see~\cite{Kohlhase:fsgo13} for a discussion. We should think -of \cn{v_1} as a ``definitional view'' that gives meaning to the opaque parts in \cn{SET}: -the proof obgligations have to be met in order for the diagram to commute (which is an -invariant we want to maintain). - -We call this situation where the symbols of a theory are imported via a partial inclusion -and their meaning is specified via a view that is a partial inverse an \textbf{adoption} -and the morphism pair an \textbf{adoption} morphism. -\begin{figure}[ht]\centering -\begin{tikzpicture}[yscale=1.7] - \node at (1,2.6) {\large Library}; - \node at (-4.5,2.6) {\large Course}; - \node[thy] (zfset) at (1,-.2) {\mthy{ZFset}{$\in$, \cn{Ex}, \cn{Ext}, \cn{Sep}; $\{\cdot |\cdot\}$}}; - \node[thy] (zfunion) at (-.2,1) {\mthy{ZFunion}{\cn{Un}, $\cup$}}; - \node[thy] (zfpow) at (2.2,1) {\mthy{ZFpow}{\cn{Pow}, $\mathcal{P}$}}; - \node at (1.1,1) {\ldots}; - \node[thy] (zfop) at (1,2) {ZFops}; - \draw[include] (zfset) -- (zfunion); - \draw[include] (zfset) -- (zfpow); - \draw[include] (zfunion) -- (zfop); - \draw[include] (zfpow) -- (zfop); - \node[thy] (set) at (-4.5,-.2) {\mthy{SET}{$\in$, $\{\cdot |\cdot\}$}}; - \node[thy] (sop) at (-4.5,2) {\mthy{SETOPS}{$\cup$,\ldots,$\mathcal{P}$}}; - \draw[view] (set) to[out=-5,in=185] node[below] {$\cn{v_1}$} (zfset); - \draw[pinclude] (zfset) to[out=175,in=5] node[above] {$\cn{a_1}\colon\in,\{\cdot |\cdot\}$}(set); - \draw[view] (sop) to[out=-5,in=185] node[below] {$\cn{v_2}: \text{incl}\; \cn{v_1}$} (zfop); - \draw[pinclude] (zfop) to[out=175,in=5] node[above] {$\cn{a_2}\colon\cup,\ldots,\mathcal{P},\text{incl}\; \cn{a_1}$} (sop); - \draw[include] (set) -- (sop); -\end{tikzpicture} -\caption{A course grounded in a modular Library}\label{fig:slides-library} -\end{figure} - -Let us now continue looking at the example in Figure~\ref{fig:slides-library}: The -informal course materials go on in style, introducing the set operations ranging from set -union to the power set in one go in the theory \cn{SETOPS}. On the library side, we -introduce the set theory axioms one by one, derive the respective operators from them, and -at the end collect all the material in the theory \cn{ZFops}. A this point, we can justify -the theory \cn{SETOPS} via a partial inclusion of the symbols $\cup$, \ldots, -$\mathcal{P}$ from \cn{ZFops}, which gives rise to another definitional view \cn{v2} to -\cn{ZFops}. As \mmt allows assignments to structures, we can simply assign the inclusion -from \cn{SET} to \cn{SETOPS} to the (unique) inclusion from \cn{ZFset} to \cn{ZFops} (as -indicated by the ``assignment'' include $\cn{v_1}$ on $\cn{v_1}$). - -We observe that the two theory graphs are self-contained: the course materials can be -understood without knowing about the library; in particular, the membership relation used -in the definition of the union operator in \cn{SETOPS} is from theory \cn{SET}. - -This self-containedness is important intra-course didactics. But it also has the problem -that the courses become insular; how are students going to communicate with mathematicians -who have learned their maths from other courses? Here is where the views \cn{v_i} come -in. Say the other mathematicians have course theories \cn{\overline{SET}} and -\cn{\overline{SETOPS}} with views \cn{\overline{v_1}} and \cn{\overline{v_2}} into the -same library, then the views \cn{v_1} and \cn{\overline{v_1}} induce a partial -isomorphisms between \cn{SET} and \cn{\overline{SET}} in the sense -of~\cite{KohRabSac:fvip11} (and correspondingly between \cn{SETOPS} and -\cn{\overline{SETOPS}}) that justify communication. -\end{example} - - -\subsection{Accelerated Turing Machines} - -The case of the mathematical paper case study mentioned above is more difficult, since we -do not have a direct generalization relation as in the case above. In this case, the -definition of the accelerated Turing machine involves a concrete step size ($2^{-n}$), -whereas the definition it recaps allows arbitrary sequences of step sizes as long as their -sum remains finite. Thus we have the situation in Figure~\ref{fig:atm}. Theory \cn{ATM} -contains the (opaque) sentence (\ref{lq:atm}), but there cannot be a view from \cn{ATM} to -\cn{atm} as that is more general. But we do have a view to \cn{atm(2^{-n})}, which -naturally arises in treatments of accelerated Turing machines as an example. - -\begin{figure}[ht]\centering - \begin{tikzpicture}[scale=1.3] - \node at (-.5,2.7) {\large Paper}; - \node at (2.5,2.7) {\large Literature}; - \node[thy] (ATMp) at (-.5,2) {\cn{ATMhalt}}; - \node[thy] (catmp) at (2,2) {\cn{atm(2^{-n})halt}}; - \node[thy] (ATM) at (-.5,1) {\cn{ATM}}; - \node[thy] (atm) at (3.5,0) {\cn{atm}}; - \node[thy] (catm) at (2,1) {\cn{atm(2^{-n})}}; - \node[thy,dotted] (atmp) at (3.5,1) {\cn{atmhalt}}; - \draw[view] (ATM) -- node[above] {\cn{v_1}} (catm); - \draw[view] (ATMp) -- node[above] {\cn{v_2}} (catmp); - \draw[include] (atm) -- (catm); - \draw[include] (ATM) -- (ATMp); - \draw[include] (catm) -- (catmp); - \draw[include,dotted] (atm) -- (atmp); - \draw[include,dotted] (atmp) -- (catmp); - \end{tikzpicture} - \caption{Definitional View for ATM}\label{fig:atm} -\end{figure} - -The more important aspect is that the contribution of~\cite{CalStai:natm09} (depicted here -as theory \cn{ATMhalt} as it concerns undecidability of the accelerated halting problem -by accelerated Turing machines) can be justified via a view into a theory -\cn{atm(2^{-n})halt}, which can be systematically constructed from \cn{ATMhalt} modulo -\cn{v_1} as pushouts along inclusions always exist in \mmt. In the particular example, we -can do even better: as the proof in theory \cn{ATMhalt} does not use any properties of the -step size sequence $2^{-n}$, there is a theory \cn{atmhalt} that generalizes -\cn{atm(2^{-n})halt} (shown dotted in Figure~\ref{fig:atm}). Note that this construction -is automatic and needs human intervention -- but one that the authors expect their readers -will readily do, otherwise they would not have restricted themselves to the concrete -sequence. -\end{oldpart} %%% Local Variables: %%% mode: latex -- GitLab