1.1 --- a/doc-src/isac/jrocnik/eJMT-paper/jrocnik_eJMT.tex Fri Aug 31 20:02:40 2012 +0200
1.2 +++ b/doc-src/isac/jrocnik/eJMT-paper/jrocnik_eJMT.tex Mon Sep 03 20:51:44 2012 +0200
1.3 @@ -69,8 +69,6 @@
1.4 %
1.5 % Please place your own definitions here
1.6 %
1.7 -
1.8 -% isac logos
1.9 \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
1.10 \def\sisac{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
1.11
1.12 @@ -96,7 +94,9 @@
1.13 %
1.14 % document title
1.15 %
1.16 -\title{Interactive Course Material by TP-based Programming}%
1.17 +\title{Trials with TP-based Programming
1.18 +\\
1.19 +for Interactive Course Material}%
1.20 %
1.21 % Single author. Please supply at least your name,
1.22 % email address, and affiliation here.
1.23 @@ -105,7 +105,7 @@
1.24 \textit{Jan Ro\v{c}nik} \\
1.25 jan.rocnik@student.tugraz.at \\
1.26 IST, SPSC\\
1.27 -University of Technologie Graz\\
1.28 +Graz University of Technologie\\
1.29 Austria\end{tabular}
1.30 }%
1.31 %
1.32 @@ -124,7 +124,7 @@
1.33
1.34 Traditional course material in engineering disciplines lacks an
1.35 important component, interactive support for step-wise problem
1.36 -solving. \textbf{TP} (Theorem-Proving) technology is appropriate for one part
1.37 +solving. Theorem-Proving (TP) technology is appropriate for one part
1.38 of such support, in checking user-input. For the other part of such
1.39 support, guiding the learner towards a solution, another kind of
1.40 technology is required. %TODO ... connect to prototype ...
1.41 @@ -143,7 +143,7 @@
1.42 course in Signal Processing: implementation of definitions and
1.43 theorems in TP, formal specification of a problem and step-wise
1.44 development of the program solving the problem. Experiences with the
1.45 -work flow in iterative development with testing and identifying errors
1.46 +ork flow in iterative development with testing and identifying errors
1.47 are described, too. The description clarifies the components missing
1.48 in the prototype's language as well as deficiencies experienced during
1.49 programming.
1.50 @@ -167,25 +167,131 @@
1.51 % and \subparagraph{...} as necessary.
1.52 %
1.53
1.54 -\section{Introduction}
1.55 +\section{Introduction}\label{intro}
1.56
1.57 -\paragraph{Didactics of mathematics} faces a specific issue, a gap between (1) introduction of math concepts and skills and (2) application of these concepts and skills, which usually are separated into different units in curricula (for good reasons). For instance, (1) teaching partial fraction decomposition is separated from (2) application for inverse Z-transform in signal processing.
1.58 -\par
1.59 -This gap is an obstacle for applying math as an fundamental thinking technology in engineering: In (1) motivation is lacking because the question ``What is this stuff good for?'' cannot be treated sufficiently, and in (2) the ``stuff'' is not available to students in higher semesters as widespread experience shows.
1.60 +% \paragraph{Didactics of mathematics}
1.61 +%WN: wenn man in einem high-quality paper von 'didactics' spricht,
1.62 +%WN muss man am state-of-the-art ankn"upfen -- siehe
1.63 +%WN W.Neuper, On the Emergence of TP-based Educational Math Assistants
1.64 +% faces a specific issue, a gap
1.65 +% between (1) introduction of math concepts and skills and (2)
1.66 +% application of these concepts and skills, which usually are separated
1.67 +% into different units in curricula (for good reasons). For instance,
1.68 +% (1) teaching partial fraction decomposition is separated from (2)
1.69 +% application for inverse Z-transform in signal processing.
1.70 +%
1.71 +% \par This gap is an obstacle for applying math as an fundamental
1.72 +% thinking technology in engineering: In (1) motivation is lacking
1.73 +% because the question ``What is this stuff good for?'' cannot be
1.74 +% treated sufficiently, and in (2) the ``stuff'' is not available to
1.75 +% students in higher semesters as widespread experience shows.
1.76 +%
1.77 +% \paragraph{Motivation} taken by this didactic issue on the one hand,
1.78 +% and ongoing research and development on a novel kind of educational
1.79 +% mathematics assistant at Graz University of
1.80 +% Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to
1.81 +% scope with this issue on the other hand, several institutes are
1.82 +% planning to join their expertise: the Institute for Information
1.83 +% Systems and Computer Media (IICM), the Institute for Software
1.84 +% Technology (IST), the Institutes for Mathematics, the Institute for
1.85 +% Signal Processing and Speech Communication (SPSC), the Institute for
1.86 +% Structural Analysis and the Institute of Electrical Measurement and
1.87 +% Measurement Signal Processing.
1.88 +%WN diese Information ist f"ur das Paper zu spezielle, zu aktuell
1.89 +%WN und damit zu verg"anglich.
1.90 +% \par This thesis is the first attempt to tackle the above mentioned
1.91 +% issue, it focuses on Telematics, because these specific studies focus
1.92 +% on mathematics in \emph{STEOP}, the introductory orientation phase in
1.93 +% Austria. \emph{STEOP} is considered an opportunity to investigate the
1.94 +% impact of {\sisac}'s prototype on the issue and others.
1.95 +%
1.96 +
1.97 +Traditional course material in engineering disciplines lacks an
1.98 +important component, interactive support for step-wise problem
1.99 +solving. Theorem-Proving (TP) technology can provide such support by
1.100 +specific services. An important part of such services is called
1.101 +``next-step-guidance'', generated by a specific kind of ``TP-based
1.102 +programming language''. In the
1.103 +{\sisac}-project~\footnote{http://www.ist.tugraz.at/projects/isac/} such
1.104 +a language is prototyped in line with~\cite{plmms10} and built upon
1.105 +the theorem prover
1.106 +Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}.
1.107 +The TP services are coordinated by a specific interpreter for the
1.108 +programming language, called
1.109 +Lucas-Interpreter~\cite{wn:lucas-interp-12}. The language and the
1.110 +interpreter will be briefly re-introduced in order to make the paper
1.111 +self-contained.
1.112
1.113 -\paragraph{Motivation} taken by this didactic issue on the one hand, and ongoing research and development on a novel kind of educational mathematics assistant at Graz University of Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to scope with this issue on the other hand, several institutes are planning to join their expertise: the Institute for Information Systems and Computer Media (IICM), the Institute for Software Technology (IST), the Institutes for Mathematics, the Institute for Signal Processing and Speech Communication (SPSC), the Institute for Structural Analysis and the Institute of Electrical Measurement and Measurement Signal Processing.
1.114 -\par This thesis is the first attempt to tackle the above mentioned issue, it focuses on Telematics, because these specific studies focus on mathematics in \emph{STEOP}, the introductory orientation phase in Austria. \emph{STEOP} is considered an opportunity to investigate the impact of {\sisac}'s prototype on the issue and others.
1.115 +\medskip The main part of the paper is an account of first experiences
1.116 +with programming in this TP-based language. The experience was gained
1.117 +in a case study by the author. The author was considered an ideal
1.118 +candidate for this study for the following reasons: as a student in
1.119 +Telematics (computer science with focus on Signal Processing) he had
1.120 +general knowledge in programming as well as specific domain knowledge
1.121 +in Signal Processing; and he was not involved in the development of
1.122 +{\sisac}'s programming language and interpeter, thus a novice to the
1.123 +language.
1.124
1.125 -The paper will use the problem in Fig.\ref{fig-gclc} as a
1.126 +The goal of the case study was (1) some TP-based programs for
1.127 +interactive course material for a specific ``Adavanced Signal
1.128 +Processing Lab'' in a higher semester, (2) respective program
1.129 +development with as little advice from the {\sisac}-team and (3) records
1.130 +and comments for the main steps of development in an Isabelle theory;
1.131 +this theory should provide guidelines for future programmers. An
1.132 +excerpt from this theory is the main part of this paper.
1.133 +
1.134 +\medskip The paper will use the problem in Fig.\ref{fig-interactive} as a
1.135 running example:
1.136 \begin{figure} [htb]
1.137 \begin{center}
1.138 -%\includegraphics[width=120mm]{fig/newgen/isac-Ztrans-math.png}
1.139 +\includegraphics[width=120mm]{fig/isac-Ztrans-math.png}
1.140 \caption{Step-wise problem solving guided by the TP-based program}
1.141 \label{fig-interactive}
1.142 \end{center}
1.143 -\label{fig-gclc}
1.144 \end{figure}
1.145 +The problem is from the domain of Signal Processing and requests to
1.146 +determine the inverse Z-transform for a given term. Fig.\ref{fig-interactive}
1.147 +also shows the beginning of the interactive construction of a solution
1.148 +for the problem. This construction is done in the right window named
1.149 +``Worksheet''.
1.150 +
1.151 +User-interaction on the Worksheet is {\em checked} and {\em guided} by
1.152 +TP services:
1.153 +\begin{enumerate}
1.154 +\item Formulas input by the user are {\em checked} by TP: such a
1.155 +formula establishes a proof situation --- the prover has to derive the
1.156 +formula from the logical context. The context is built up from the
1.157 +formal specification of the problem (here hidden from the user) by the
1.158 +Lucas-Interpreter.
1.159 +\item If the user gets stuck, the program developed below in this
1.160 +paper ``knows the next step'' from behind the scenes. How the latter
1.161 +TP-service is exploited by dialogue authoring is out of scope of this
1.162 +paper and can be studied in~\cite{gdaroczy-EP-13}.
1.163 +\end{enumerate} It should be noted that the programmer using the
1.164 +TP-based language is not concerned with interaction at all; we will
1.165 +see that the program contains neither input-statements nor
1.166 +output-statements. Rather, interaction is handled by services
1.167 +generated automatically.
1.168 +
1.169 +\medskip So there is a clear separation of concerns: Dialogues are
1.170 +adapted by dialogue authors (in Java-based tools), using automatically
1.171 +generated TP services, while the TP-based program is written by
1.172 +mathematics experts (in Isabelle/ML). The latter is concern of this
1.173 +paper.
1.174 +
1.175 +\medskip The paper is structed as follows: The introduction
1.176 +\S\ref{intro} is followed by a brief re-introduction of the TP-based
1.177 +programming language in \S\ref{PL}, which extends the executable
1.178 +fragment of Isabelle's language (\S\ref{PL-isab}) by tactics which
1.179 +play a specific role in Lucas-Interpretation and in providing the TP
1.180 +services (\S\ref{PL-tacs}). The main part in \S\ref{trial} describes
1.181 +the main steps in developing the program for the running example:
1.182 +prepare domain knowledge, implement the formal specification of the
1.183 +problem, prepare the environment for the program, implement the
1.184 +program. The workflow of programming, debugging and testing is
1.185 +described in \S\ref{workflow}. The conclusion \S\ref{conclusion} will
1.186 +give directions identified for future development.
1.187 +
1.188
1.189 \section{\isac's Prototype for a Programming Language}\label{PL}
1.190 The prototype's language extends the executable fragment in the
1.191 @@ -239,12 +345,14 @@
1.192 % Terms may also contain $\lambda$-abstractions. For example, $\lambda
1.193 % x. \; x$ is the identity function.
1.194
1.195 -\textbf{Formula} are terms of type \textit{bool}. There are the basic
1.196 +\textbf{Formulae} are terms of type \textit{bool}. There are the basic
1.197 constants \textit{True} and \textit{False} and the usual logical
1.198 connectives (in decreasing order of precedence): $\neg, \land, \lor,
1.199 \rightarrow$.
1.200
1.201 -\textbf{Equality} is available in the form of the infix function $=$ of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for formulas, where it means ``if and only if''.
1.202 +\textbf{Equality} is available in the form of the infix function $=$
1.203 +of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for
1.204 +formulas, where it means ``if and only if''.
1.205
1.206 \textbf{Quantifiers} are written $\forall x. \; P$ and $\exists x. \;
1.207 P$. Quantifiers lead to non-executable functions, so functions do not
1.208 @@ -254,7 +362,7 @@
1.209
1.210 \subsection{\isac's Tactics for Lucas-Interpretation}\label{PL-tacs}
1.211 The prototype extends Isabelle's language by specific statements
1.212 -called tactics~\footnote{\sisac's tactics are different from
1.213 +called tactics~\footnote{{\sisac}'s tactics are different from
1.214 Isabelle's tactics: the former concern steps in a calculation, the
1.215 latter concern proof steps.} and tacticals. For the programmer these
1.216 statements are functions with the following signatures:
1.217 @@ -341,70 +449,250 @@
1.218
1.219 .\\.\\.\\TODO\\.\\.\\
1.220
1.221 -\section{Development of a Program on Trial}\label{trial}
1.222 -
1.223 -\subsection{Mechanization of Math in Isabelle/ISAC\label{isabisac}}
1.224 -
1.225 -\paragraph{As mentioned in the introduction,} a prototype of an educational math assistant called {\sisac}\footnote{{\sisac}=\textbf{Isa}belle for \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges the gap between (1) introducation and (2) application of mathematics: {\sisac} is based on Computer Theorem Proving (TP), a technology which requires each fact and each action justified by formal logic, so {{\sisac{}}} makes justifications transparent to students in interactive step-wise problem solving. By that way {\sisac} already can serve both:
1.226 -\begin{enumerate}
1.227 - \item Introduction of math stuff (in e.g. partial fraction decomposition) by stepwise explaining and exercising respective symbolic calculations with ``next step guidance (NSG)'' and rigorously checking steps freely input by students --- this also in context with advanced applications (where the stuff to be taught in higher semesters can be skimmed through by NSG), and
1.228 - \item Application of math stuff in advanced engineering courses (e.g. problems to be solved by inverse Z-transform in a Signal Processing Lab) and now without much ado about basic math techniques (like partial fraction decomposition): ``next step guidance'' supports students in independently (re-)adopting such techniques.
1.229 -\end{enumerate}
1.230 -Before the question is answers, how {\sisac} accomplishes this task from a technical point of view, some remarks on the state-of-the-art is given, therefor follow up Section~\ref{emas}.
1.231 -
1.232 -\subsection{Educational Mathematics Assistants (EMAs)}\label{emas}
1.233 -
1.234 -\paragraph{Educational software in mathematics} is, if at all, based on Computer Algebra Systems (CAS, for instance), Dynamic Geometry Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org} \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These base technologies are used to program math lessons and sometimes even exercises. The latter are cumbersome: the steps towards a solution of such an interactive exercise need to be provided with feedback, where at each step a wide variety of possible input has to be foreseen by the programmer - so such interactive exercises either require high development efforts or the exercises constrain possible inputs.
1.235 -
1.236 -\subparagraph{A new generation} of educational math assistants (EMAs) is emerging presently, which is based on Theorem Proving (TP). TP, for instance Isabelle and Coq, is a technology which requires each fact and each action justified by formal logic. Pushed by demands for \textit{proven} correctness of safety-critical software TP advances into software engineering; from these advancements computer mathematics benefits in general, and math education in particular. Two features of TP are immediately beneficial for learning:
1.237 -
1.238 -\paragraph{TP have knowledge in human readable format,} that is in standard predicate calculus. TP following the LCF-tradition have that knowledge down to the basic definitions of set, equality, etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html}; following the typical deductive development of math, natural numbers are defined and their properties proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html}, etc. Present knowledge mechanized in TP exceeds high-school mathematics by far, however by knowledge required in software technology, and not in other engineering sciences.
1.239 -
1.240 -\paragraph{TP can model the whole problem solving process} in mathematical problem solving {\em within} a coherent logical framework. This is already being done by three projects, by Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.
1.241 -\par
1.242 -Having the whole problem solving process within a logical coherent system, such a design guarantees correctness of intermediate steps and of the result (which seems essential for math software); and the second advantage is that TP provides a wealth of theories which can be exploited for mechanizing other features essential for educational software.
1.243 -
1.244 -\subsubsection{Generation of User Guidance in EMAs}\label{user-guid}
1.245 -
1.246 -One essential feature for educational software is feedback to user input and assistance in coming to a solution.
1.247 -
1.248 -\paragraph{Checking user input} by ATP during stepwise problem solving is being accomplished by the three projects mentioned above exclusively. They model the whole problem solving process as mentioned above, so all what happens between formalized assumptions (or formal specification) and goal (or fulfilled postcondition) can be mechanized. Such mechanization promises to greatly extend the scope of educational software in stepwise problem solving.
1.249 -
1.250 -\paragraph{NSG (Next step guidance)} comprises the system's ability to propose a next step; this is a challenge for TP: either a radical restriction of the search space by restriction to very specific problem classes is required, or much care and effort is required in designing possible variants in the process of problem solving \cite{proof-strategies-11}.
1.251 -\par
1.252 -Another approach is restricted to problem solving in engineering domains, where a problem is specified by input, precondition, output and postcondition, and where the postcondition is proven by ATP behind the scenes: Here the possible variants in the process of problem solving are provided with feedback {\em automatically}, if the problem is described in a TP-based programing language:
1.253 -\cite{plmms10} the programmer only describes the math algorithm without caring about interaction (the respective program is functional and even has no input or output statements!); interaction is generated as a side-effect by the interpreter --- an efficient separation of concern between math programmers and dialog designers promising application all over engineering disciplines.
1.254 -
1.255 -
1.256 -\subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}
1.257 -Authoring new mathematics knowledge in {\sisac} can be compared with ``application programing'' of engineering problems; most of such programing uses CAS-based programing languages (CAS = Computer Algebra Systems; e.g. Mathematica's or Maple's programing language).
1.258 -
1.259 -\paragraph{A novel type of TP-based language} is used by {\sisac{}} \cite{plmms10} for describing how to construct a solution to an engineering problem and for calling equation solvers, integration, etc~\footnote{Implementation of CAS-like functionality in TP is not primarily concerned with efficiency, but with a didactic question: What to decide for: for high-brow algorithms at the state-of-the-art or for elementary algorithms comprehensible for students?} within TP; TP can ensure ``systems that never make a mistake'' \cite{casproto} - are impossible for CAS which have no logics underlying.
1.260 -
1.261 -\subparagraph{Authoring is perfect} by writing such TP based programs; the application programmer is not concerned with interaction or with user guidance: this is concern of a novel kind of program interpreter called Lucas-Interpreter. This interpreter hands over control to a dialog component at each step of calculation (like a debugger at breakpoints) and calls automated TP to check user input following personalized strategies according to a feedback module.
1.262 -\par
1.263 -However ``application programing with TP'' is not done with writing a program: according to the principles of TP, each step must be justified. Such justifications are given by theorems. So all steps must be related to some theorem, if there is no such theorem it must be added to the existing knowledge, which is organized in so-called \textbf{theories} in Isabelle. A theorem must be proven; fortunately Isabelle comprises a mechanism (called ``axiomatization''), which allows to omit proofs. Such a theorem is shown in Example~\ref{eg:neuper1}.
1.264 -
1.265 -\vbox{
1.266 - \begin{example}
1.267 +\section{Development of a Program on Trial}\label{trial}
1.268 +As mentioned above, {\sisac} is an experimental system for a proof of
1.269 +concept for Lucas-Interpretation~\cite{wn:lucas-interp-12}. The
1.270 +latter interprets a specific kind of TP-based programming language,
1.271 +which is as experimental as the whole prototype --- so programming in
1.272 +this language can be only ``on trial'', presently. However, as a
1.273 +prototype, the language addresses essentials described below.
1.274 +
1.275 +\subsection{Mechanization of Math --- Domain Engineering}\label{isabisac}
1.276 +
1.277 +%WN was Fachleute unter obigem Titel interessiert findet
1.278 +%WN unterhalb des auskommentierten Textes.
1.279 +
1.280 +%WN der Text unten spricht Benutzer-Aspekte anund ist nicht speziell
1.281 +%WN auf Computer-Mathematiker fokussiert.
1.282 +% \paragraph{As mentioned in the introduction,} a prototype of an
1.283 +% educational math assistant called
1.284 +% {{\sisac}}\footnote{{{\sisac}}=\textbf{Isa}belle for
1.285 +% \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges
1.286 +% the gap between (1) introducation and (2) application of mathematics:
1.287 +% {{\sisac}} is based on Computer Theorem Proving (TP), a technology which
1.288 +% requires each fact and each action justified by formal logic, so
1.289 +% {{{\sisac}{}}} makes justifications transparent to students in
1.290 +% interactive step-wise problem solving. By that way {{\sisac}} already
1.291 +% can serve both:
1.292 +% \begin{enumerate}
1.293 +% \item Introduction of math stuff (in e.g. partial fraction
1.294 +% decomposition) by stepwise explaining and exercising respective
1.295 +% symbolic calculations with ``next step guidance (NSG)'' and rigorously
1.296 +% checking steps freely input by students --- this also in context with
1.297 +% advanced applications (where the stuff to be taught in higher
1.298 +% semesters can be skimmed through by NSG), and
1.299 +% \item Application of math stuff in advanced engineering courses
1.300 +% (e.g. problems to be solved by inverse Z-transform in a Signal
1.301 +% Processing Lab) and now without much ado about basic math techniques
1.302 +% (like partial fraction decomposition): ``next step guidance'' supports
1.303 +% students in independently (re-)adopting such techniques.
1.304 +% \end{enumerate}
1.305 +% Before the question is answers, how {{\sisac}}
1.306 +% accomplishes this task from a technical point of view, some remarks on
1.307 +% the state-of-the-art is given, therefor follow up Section~\ref{emas}.
1.308 +%
1.309 +% \subsection{Educational Mathematics Assistants (EMAs)}\label{emas}
1.310 +%
1.311 +% \paragraph{Educational software in mathematics} is, if at all, based
1.312 +% on Computer Algebra Systems (CAS, for instance), Dynamic Geometry
1.313 +% Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org}
1.314 +% \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC
1.315 +% http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These
1.316 +% base technologies are used to program math lessons and sometimes even
1.317 +% exercises. The latter are cumbersome: the steps towards a solution of
1.318 +% such an interactive exercise need to be provided with feedback, where
1.319 +% at each step a wide variety of possible input has to be foreseen by
1.320 +% the programmer - so such interactive exercises either require high
1.321 +% development efforts or the exercises constrain possible inputs.
1.322 +%
1.323 +% \subparagraph{A new generation} of educational math assistants (EMAs)
1.324 +% is emerging presently, which is based on Theorem Proving (TP). TP, for
1.325 +% instance Isabelle and Coq, is a technology which requires each fact
1.326 +% and each action justified by formal logic. Pushed by demands for
1.327 +% \textit{proven} correctness of safety-critical software TP advances
1.328 +% into software engineering; from these advancements computer
1.329 +% mathematics benefits in general, and math education in particular. Two
1.330 +% features of TP are immediately beneficial for learning:
1.331 +%
1.332 +% \paragraph{TP have knowledge in human readable format,} that is in
1.333 +% standard predicate calculus. TP following the LCF-tradition have that
1.334 +% knowledge down to the basic definitions of set, equality,
1.335 +% etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html};
1.336 +% following the typical deductive development of math, natural numbers
1.337 +% are defined and their properties
1.338 +% proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html},
1.339 +% etc. Present knowledge mechanized in TP exceeds high-school
1.340 +% mathematics by far, however by knowledge required in software
1.341 +% technology, and not in other engineering sciences.
1.342 +%
1.343 +% \paragraph{TP can model the whole problem solving process} in
1.344 +% mathematical problem solving {\em within} a coherent logical
1.345 +% framework. This is already being done by three projects, by
1.346 +% Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.
1.347 +% \par
1.348 +% Having the whole problem solving process within a logical coherent
1.349 +% system, such a design guarantees correctness of intermediate steps and
1.350 +% of the result (which seems essential for math software); and the
1.351 +% second advantage is that TP provides a wealth of theories which can be
1.352 +% exploited for mechanizing other features essential for educational
1.353 +% software.
1.354 +%
1.355 +% \subsubsection{Generation of User Guidance in EMAs}\label{user-guid}
1.356 +%
1.357 +% One essential feature for educational software is feedback to user
1.358 +% input and assistance in coming to a solution.
1.359 +%
1.360 +% \paragraph{Checking user input} by ATP during stepwise problem solving
1.361 +% is being accomplished by the three projects mentioned above
1.362 +% exclusively. They model the whole problem solving process as mentioned
1.363 +% above, so all what happens between formalized assumptions (or formal
1.364 +% specification) and goal (or fulfilled postcondition) can be
1.365 +% mechanized. Such mechanization promises to greatly extend the scope of
1.366 +% educational software in stepwise problem solving.
1.367 +%
1.368 +% \paragraph{NSG (Next step guidance)} comprises the system's ability to
1.369 +% propose a next step; this is a challenge for TP: either a radical
1.370 +% restriction of the search space by restriction to very specific
1.371 +% problem classes is required, or much care and effort is required in
1.372 +% designing possible variants in the process of problem solving
1.373 +% \cite{proof-strategies-11}.
1.374 +% \par
1.375 +% Another approach is restricted to problem solving in engineering
1.376 +% domains, where a problem is specified by input, precondition, output
1.377 +% and postcondition, and where the postcondition is proven by ATP behind
1.378 +% the scenes: Here the possible variants in the process of problem
1.379 +% solving are provided with feedback {\em automatically}, if the problem
1.380 +% is described in a TP-based programing language: \cite{plmms10} the
1.381 +% programmer only describes the math algorithm without caring about
1.382 +% interaction (the respective program is functional and even has no
1.383 +% input or output statements!); interaction is generated as a
1.384 +% side-effect by the interpreter --- an efficient separation of concern
1.385 +% between math programmers and dialog designers promising application
1.386 +% all over engineering disciplines.
1.387 +%
1.388 +%
1.389 +% \subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}
1.390 +% Authoring new mathematics knowledge in {{\sisac}} can be compared with
1.391 +% ``application programing'' of engineering problems; most of such
1.392 +% programing uses CAS-based programing languages (CAS = Computer Algebra
1.393 +% Systems; e.g. Mathematica's or Maple's programing language).
1.394 +%
1.395 +% \paragraph{A novel type of TP-based language} is used by {{\sisac}{}}
1.396 +% \cite{plmms10} for describing how to construct a solution to an
1.397 +% engineering problem and for calling equation solvers, integration,
1.398 +% etc~\footnote{Implementation of CAS-like functionality in TP is not
1.399 +% primarily concerned with efficiency, but with a didactic question:
1.400 +% What to decide for: for high-brow algorithms at the state-of-the-art
1.401 +% or for elementary algorithms comprehensible for students?} within TP;
1.402 +% TP can ensure ``systems that never make a mistake'' \cite{casproto} -
1.403 +% are impossible for CAS which have no logics underlying.
1.404 +%
1.405 +% \subparagraph{Authoring is perfect} by writing such TP based programs;
1.406 +% the application programmer is not concerned with interaction or with
1.407 +% user guidance: this is concern of a novel kind of program interpreter
1.408 +% called Lucas-Interpreter. This interpreter hands over control to a
1.409 +% dialog component at each step of calculation (like a debugger at
1.410 +% breakpoints) and calls automated TP to check user input following
1.411 +% personalized strategies according to a feedback module.
1.412 +% \par
1.413 +% However ``application programing with TP'' is not done with writing a
1.414 +% program: according to the principles of TP, each step must be
1.415 +% justified. Such justifications are given by theorems. So all steps
1.416 +% must be related to some theorem, if there is no such theorem it must
1.417 +% be added to the existing knowledge, which is organized in so-called
1.418 +% \textbf{theories} in Isabelle. A theorem must be proven; fortunately
1.419 +% Isabelle comprises a mechanism (called ``axiomatization''), which
1.420 +% allows to omit proofs. Such a theorem is shown in
1.421 +% Example~\ref{eg:neuper1}.
1.422 +
1.423 +The running example, introduced by Fig.\ref{fig-interactive} on
1.424 +p.\pageref{fig-interactive}, requires to determine the inverse $\cal
1.425 +Z$-transform for a class of functions. The domain of Signal Processing
1.426 +is accustomed to specific notation for the resulting functions, which
1.427 +are absolutely summable and are called TODO: $u[n]$, where $u$ is the
1.428 +function, $n$ is the argument and the brackets indicate that the
1.429 +arguments are TODO. Surprisingly, Isabelle accepts the rules for
1.430 +${\cal Z}^{-1}$ in this traditional notation~\footnote{Isabelle
1.431 +experts might be particularly surprised, that the brackets do not
1.432 +cause errors in typing (as lists).}:
1.433 +%\vbox{
1.434 +% \begin{example}
1.435 \label{eg:neuper1}
1.436 {\small\begin{tabbing}
1.437 123\=123\=123\=123\=\kill
1.438 \hfill \\
1.439 \>axiomatization where \\
1.440 - \>\> rule1: ``1 = $\delta$ [n]'' and\\
1.441 - \>\> rule2: ``$\vert\vert$ z $\vert\vert$ > 1 ==> z / (z - 1) = u [n]'' and\\
1.442 - \>\> rule3: ``$\vert\vert$ z $\vert\vert$ < 1 ==> z / (z - 1) = -u [-n - 1]'' and \\
1.443 - \>\> rule4: ``$\vert\vert$ z $\vert\vert$ > $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = $\alpha^n$ $\cdot$ u [n]'' and\\
1.444 - \>\> rule5: ``$\vert\vert$ z $\vert\vert$ < $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = -($\alpha^n$) $\cdot$ u [-n - 1]'' and\\
1.445 - \>\> rule6: ``$\vert\vert$ z $\vert\vert$ > 1 ==> z/(z - 1)$^2$ = n $\cdot$ u [n]''
1.446 + \>\> rule1: ``${\cal Z}^{-1}\;1 = \delta [n]$'' and\\
1.447 + \>\> rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow {\cal Z}^{-1}\;z / (z - 1) = u [n]$'' and\\
1.448 + \>\> rule3: ``$\vert\vert$ z $\vert\vert$ < 1 ==> z / (z - 1) = -u [-n - 1]'' and \\
1.449 %TODO
1.450 + \>\> rule4: ``$\vert\vert$ z $\vert\vert$ > $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = $\alpha^n$ $\cdot$ u [n]'' and\\
1.451 %TODO
1.452 + \>\> rule5: ``$\vert\vert$ z $\vert\vert$ < $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = -($\alpha^n$) $\cdot$ u [-n - 1]'' and\\
1.453 %TODO
1.454 + \>\> rule6: ``$\vert\vert$ z $\vert\vert$ > 1 ==> z/(z - 1)$^2$ = n $\cdot$ u [n]''\\
1.455 %TODO
1.456 \end{tabbing}
1.457 }
1.458 - \end{example}
1.459 -}
1.460 -
1.461 -In order to provide TP with logical facts for checking user input, the Lucas-Interpreter requires a \textbf{specification}. Such a specification is shown in Example~\ref{eg:neuper2}.
1.462 -
1.463 +% \end{example}
1.464 +%}
1.465 +These 6 rules can be used as conditional rewrite rules, depending on
1.466 +the respective convergence radius. Satisfaction from notation
1.467 +contrasts with the word {\em axiomatization}: As TP-based, the
1.468 +programming language expects these rules as {\em proved} theorems, and
1.469 +not as axioms implemented in the above brute force manner; otherwise
1.470 +all the verification efforts envisaged (like proof of the
1.471 +post-condition, see below) would be meaningless.
1.472 +
1.473 +Isabelle provides a large body of knowledge, rigorously proven from
1.474 +the basic axioms of mathematics~\footnote{This way of rigorously
1.475 +deriving all knowledge from first principles is called the
1.476 +LCF-paradigm in TP.}. In the case of the Z-Transform the most advanced
1.477 +knowledge can be found in the theoris on Multivariate
1.478 +Analysis~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate\_Analysis}. However,
1.479 +building up knowledge such that a proof for the above rules would be
1.480 +reasonably short and easily comprehensible, still requires lots of
1.481 +work (and is definitely out of scope of our case study).
1.482 +
1.483 +\medskip At the state-of-the-art in mechanization of knowledge in
1.484 +engineering, the process does not stop with the mechanization of
1.485 +mathematics. Rather, ``Formal Methods''~\cite{TODO-formal-methods}
1.486 +proceed to formal description of physical items. Signal Processing,
1.487 +for instance is concerned with physical devices for signal acquisition
1.488 +and reconstruction, which involve measuring a physical signal, storing
1.489 +it, and possibly later rebuilding the original signal or an
1.490 +approximation thereof. For digital systems, this typically includes
1.491 +sampling and quantization; devices for signal compression, including
1.492 +audio compression, image compression, and video compression, etc.
1.493 +``Domain engineering''\cite{db-domain-engineering} is concerned with
1.494 +{\em specification} of these devices' components and features; this
1.495 +part in the process of mechanization is only at the beginning.
1.496 +
1.497 +\medskip TP-based programming, concern of this paper, adds a third
1.498 +part of mechanisation, providing a third axis of ``algorithmic
1.499 +knowledge'' in Fig.\ref{fig:mathuni} on p.\pageref{fig:mathuni}.
1.500 +
1.501 + \begin{figure}
1.502 + \hfill \\
1.503 + \begin{center}
1.504 + \includegraphics{fig/universe}
1.505 + \caption{Didactic ``Math-Universe''\label{fig:mathuni}}
1.506 + \end{center}
1.507 + \end{figure}
1.508 +%WN Deine aktuelle Benennung oben wird Dir kein Fachmann abnehmen;
1.509 +%WN bitte folgende Bezeichnungen nehmen:
1.510 +%WN
1.511 +%WN axis 1: Algorithmic Knowledge (Programs)
1.512 +%WN axis 2: Application-oriented Knowledge (Specifications)
1.513 +%WN axis 3: Deductive Knowledge (Axioms, Definitions, Theorems)
1.514 +
1.515 +\subsection{Specification of the Problem}\label{spec}
1.516 +%WN <--> \chapter 7 der Thesis
1.517 +%WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..
1.518 +In order to provide TP with logical facts for checking user input, the
1.519 +Lucas-Interpreter requires a \textbf{specification}. Such a
1.520 +specification is shown in Example~\ref{eg:neuper2}.
1.521 +
1.522 +-------------------------------------------------------------------
1.523 +
1.524 +Hier brauchen wir die Spezifikation des 'running example' ...
1.525 +
1.526 \vbox{
1.527 \begin{example}
1.528 \label{eg:neuper2}
1.529 @@ -424,8 +712,13 @@
1.530 }
1.531 \end{example}
1.532 }
1.533 -
1.534 -Such a specification is checked before the execution of a program is started, the same applies for sub-programs. In the following example (Example~\ref{eg:subprob}) shows the call of such a subproblem:
1.535 +
1.536 +... und die Implementation in Isac (mit Kommentaren aus dem datatype)
1.537 +
1.538 +%WN das w"urde ich in \sec\label{}
1.539 +Such a specification is checked before the execution of a program is
1.540 +started, the same applies for sub-programs. In the following example
1.541 +(Example~\ref{eg:subprob}) shows the call of such a subproblem:
1.542
1.543 \vbox{
1.544 \begin{example}
1.545 @@ -439,16 +732,52 @@
1.546 \end{tabbing}
1.547 }
1.548 {\small\textit{
1.549 - \noindent If a program requires a result which has to be calculated first we can use a subproblem to do so. In our specific case we wanted to calculate the zeros of a fraction and used a subproblem to calculate the zeros of the denominator polynom.
1.550 + \noindent If a program requires a result which has to be
1.551 +calculated first we can use a subproblem to do so. In our specific
1.552 +case we wanted to calculate the zeros of a fraction and used a
1.553 +subproblem to calculate the zeros of the denominator polynom.
1.554 }}
1.555 \end{example}
1.556 }
1.557 -
1.558 +
1.559 +\subsection{Implementation of the Method}\label{meth}
1.560 +%WN <--> \chapter 7 der Thesis
1.561 +TODO
1.562 +\subsection{Preparation of ML-Functions for the Program}\label{funs}
1.563 +%WN <--> Thesis 6.1 -- 6.3: jene ausw"ahlen, die Du f"ur \label{progr}
1.564 +%WN brauchst
1.565 +TODO
1.566 +\subsection{Implementation of the TP-based Program}\label{progr}
1.567 +%WN <--> \chapter 8 der Thesis
1.568 +TODO
1.569 +
1.570 \section{Workflow of Programming in the Prototype}\label{workflow}
1.571 -
1.572 +-------------------------------------------------------------------
1.573 +
1.574 +``workflow'' heisst: das mache ich zuerst, dann das ...
1.575 +
1.576 +\subsection{Preparations and Trials}\label{flow-prep}
1.577 +TODO ... Build\_Inverse\_Z\_Transform.thy !!!
1.578 +
1.579 +\subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}
1.580 +TODO Build\_Inverse\_Z\_Transform.thy ... ``imports Isac''
1.581 +
1.582 +\subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}
1.583 +TODO http://www.ist.tugraz.at/isac/index.php/Extend\_ISAC\_Knowledge\#Add\_an\_example ?
1.584 +
1.585 +-------------------------------------------------------------------
1.586 +
1.587 +Das unterhalb hab' ich noch nicht durchgearbeitet; einiges w\"are
1.588 +vermutlich auf andere sections aufzuteilen.
1.589 +
1.590 +-------------------------------------------------------------------
1.591 +
1.592 \subsection{Formalization of missing knowledge in Isabelle}
1.593
1.594 -\paragraph{A problem} behind is the mechanization of mathematic theories in TP-bases languages. There is still a huge gap between these algorithms and this what we want as a solution - in Example Signal Processing.
1.595 +\paragraph{A problem} behind is the mechanization of mathematic
1.596 +theories in TP-bases languages. There is still a huge gap between
1.597 +these algorithms and this what we want as a solution - in Example
1.598 +Signal Processing.
1.599
1.600 \vbox{
1.601 \begin{example}
1.602 @@ -457,12 +786,26 @@
1.603 X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY
1.604 \]
1.605 {\small\textit{
1.606 - \noindent A very simple example on this what we call gap is the simplification above. It is needles to say that it is correct and also Isabelle for fills it correct - \emph{always}. But sometimes we don't want expand such terms, sometimes we want another structure of them. Think of a problem were we now would need only the coefficients of $X$ and $Y$. This is what we call the gap between mechanical simplification and the solution.
1.607 + \noindent A very simple example on this what we call gap is the
1.608 +simplification above. It is needles to say that it is correct and also
1.609 +Isabelle for fills it correct - \emph{always}. But sometimes we don't
1.610 +want expand such terms, sometimes we want another structure of
1.611 +them. Think of a problem were we now would need only the coefficients
1.612 +of $X$ and $Y$. This is what we call the gap between mechanical
1.613 +simplification and the solution.
1.614 }}
1.615 \end{example}
1.616 }
1.617
1.618 -\paragraph{We are not able to fill this gap,} until we have to live with it but first have a look on the meaning of this statement: Mechanized math starts from mathematical models and \emph{hopefully} proceeds to match physics. Academic engineering starts from physics (experimentation, measurement) and then proceeds to mathematical modeling and formalization. The process from a physical observance to a mathematical theory is unavoidable bound of setting up a big collection of standards, rules, definition but also exceptions. These are the things making mechanization that difficult.
1.619 +\paragraph{We are not able to fill this gap,} until we have to live
1.620 +with it but first have a look on the meaning of this statement:
1.621 +Mechanized math starts from mathematical models and \emph{hopefully}
1.622 +proceeds to match physics. Academic engineering starts from physics
1.623 +(experimentation, measurement) and then proceeds to mathematical
1.624 +modeling and formalization. The process from a physical observance to
1.625 +a mathematical theory is unavoidable bound of setting up a big
1.626 +collection of standards, rules, definition but also exceptions. These
1.627 +are the things making mechanization that difficult.
1.628
1.629 \vbox{
1.630 \begin{example}
1.631 @@ -471,12 +814,20 @@
1.632 m,\ kg,\ s,\ldots
1.633 \]
1.634 {\small\textit{
1.635 - \noindent Think about some units like that one's above. Behind each unit there is a discerning and very accurate definition: One Meter is the distance the light travels, in a vacuum, through the time of 1 / 299.792.458 second; one kilogram is the weight of a platinum-iridium cylinder in paris; and so on. But are these definitions usable in a computer mechanized world?!
1.636 + \noindent Think about some units like that one's above. Behind
1.637 +each unit there is a discerning and very accurate definition: One
1.638 +Meter is the distance the light travels, in a vacuum, through the time
1.639 +of 1 / 299.792.458 second; one kilogram is the weight of a
1.640 +platinum-iridium cylinder in paris; and so on. But are these
1.641 +definitions usable in a computer mechanized world?!
1.642 }}
1.643 \end{example}
1.644 }
1.645
1.646 -\paragraph{A computer} or a TP-System builds on programs with predefined logical rules and does not know any mathematical trick (follow up example \ref{eg:trick}) or recipe to walk around difficult expressions.
1.647 +\paragraph{A computer} or a TP-System builds on programs with
1.648 +predefined logical rules and does not know any mathematical trick
1.649 +(follow up example \ref{eg:trick}) or recipe to walk around difficult
1.650 +expressions.
1.651
1.652 \vbox{
1.653 \begin{example}
1.654 @@ -486,14 +837,31 @@
1.655 \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]
1.656 \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]
1.657 {\small\textit{
1.658 - \noindent Sometimes it is also useful to be able to apply some \emph{tricks} to get a beautiful and particularly meaningful result, which we are able to interpret. But as seen in this example it can be hard to find out what operations have to be done to transform a result into a meaningful one.
1.659 + \noindent Sometimes it is also useful to be able to apply some
1.660 +\emph{tricks} to get a beautiful and particularly meaningful result,
1.661 +which we are able to interpret. But as seen in this example it can be
1.662 +hard to find out what operations have to be done to transform a result
1.663 +into a meaningful one.
1.664 }}
1.665 \end{example}
1.666 }
1.667
1.668 -\paragraph{The only possibility,} for such a system, is to work through its known definitions and stops if none of these fits. Specified on Signal Processing or any other application it is often possible to walk through by doing simple creases. This creases are in general based on simple math operational but the challenge is to teach the machine \emph{all}\footnote{Its pride to call it \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to reach a high level of \emph{all} but it in real it will still be a survey of knowledge which links to other knowledge and {\sisac{}} a trainer and helper but no human compensating calculator.
1.669 +\paragraph{The only possibility,} for such a system, is to work
1.670 +through its known definitions and stops if none of these
1.671 +fits. Specified on Signal Processing or any other application it is
1.672 +often possible to walk through by doing simple creases. This creases
1.673 +are in general based on simple math operational but the challenge is
1.674 +to teach the machine \emph{all}\footnote{Its pride to call it
1.675 +\emph{all}.} of them. Unfortunately the goal of TP Isabelle is to
1.676 +reach a high level of \emph{all} but it in real it will still be a
1.677 +survey of knowledge which links to other knowledge and {{\sisac}{}} a
1.678 +trainer and helper but no human compensating calculator.
1.679 \par
1.680 -{{\sisac{}}} itself aims to adds an \emph{application} axis (formal specifications of problems out of topics from Signal Processing, etc.) and an \emph{algorithmic} axis to the \emph{deductive} axis of physical knowledge. The result is a three-dimensional universe of mathematics seen in Figure~\ref{fig:mathuni}.
1.681 +{{{\sisac}{}}} itself aims to adds an \emph{application} axis (formal
1.682 +specifications of problems out of topics from Signal Processing, etc.)
1.683 +and an \emph{algorithmic} axis to the \emph{deductive} axis of
1.684 +physical knowledge. The result is a three-dimensional universe of
1.685 +mathematics seen in Figure~\ref{fig:mathuni}.
1.686
1.687 \begin{figure}
1.688 \hfill \\
1.689 @@ -505,7 +873,12 @@
1.690
1.691 \subsection{Notes on Problems with Traditional Notation}
1.692
1.693 -\paragraph{During research} on these topic severely problems on traditional notations have been discovered. Some of them have been known in computer science for many years now and are still unsolved, one of them aggregates with the so called \emph{Lambda Calculus}, Example~\ref{eg:lamda} provides a look on the problem that embarrassed us.
1.694 +\paragraph{During research} on these topic severely problems on
1.695 +traditional notations have been discovered. Some of them have been
1.696 +known in computer science for many years now and are still unsolved,
1.697 +one of them aggregates with the so called \emph{Lambda Calculus},
1.698 +Example~\ref{eg:lamda} provides a look on the problem that embarrassed
1.699 +us.
1.700
1.701 \vbox{
1.702 \begin{example}
1.703 @@ -517,13 +890,31 @@
1.704 \[ f(p)=\ldots\; p \in \quad R \]
1.705
1.706 {\small\textit{
1.707 - \noindent Above we see two equations. The first equation aims to be a mapping of an function from the reel range to the reel one, but when we change only one letter we get the second equation which usually aims to insert a reel point $p$ into the reel function. In computer science now we have the problem to tell the machine (TP) the difference between this two notations. This Problem is called \emph{Lambda Calculus}.
1.708 + \noindent Above we see two equations. The first equation aims to
1.709 +be a mapping of an function from the reel range to the reel one, but
1.710 +when we change only one letter we get the second equation which
1.711 +usually aims to insert a reel point $p$ into the reel function. In
1.712 +computer science now we have the problem to tell the machine (TP) the
1.713 +difference between this two notations. This Problem is called
1.714 +\emph{Lambda Calculus}.
1.715 }}
1.716 \end{example}
1.717 }
1.718
1.719 -\paragraph{An other problem} is that terms are not full simplified in traditional notations, in {\sisac} we have to simplify them complete to check weather results are compatible or not. in e.g. the solutions of an second order linear equation is an rational in {\sisac} but in tradition we keep fractions as long as possible and as long as they aim to be \textit{beautiful} (1/8, 5/16,...).
1.720 -\subparagraph{The math} which should be mechanized in Computer Theorem Provers (\emph{TP}) has (almost) a problem with traditional notations (predicate calculus) for axioms, definitions, lemmas, theorems as a computer program or script is not able to interpret every Greek or Latin letter and every Greek, Latin or whatever calculations symbol. Also if we would be able to handle these symbols we still have a problem to interpret them at all. (Follow up \hbox{Example \ref{eg:symbint1}})
1.721 +\paragraph{An other problem} is that terms are not full simplified in
1.722 +traditional notations, in {{\sisac}} we have to simplify them complete
1.723 +to check weather results are compatible or not. in e.g. the solutions
1.724 +of an second order linear equation is an rational in {{\sisac}} but in
1.725 +tradition we keep fractions as long as possible and as long as they
1.726 +aim to be \textit{beautiful} (1/8, 5/16,...).
1.727 +\subparagraph{The math} which should be mechanized in Computer Theorem
1.728 +Provers (\emph{TP}) has (almost) a problem with traditional notations
1.729 +(predicate calculus) for axioms, definitions, lemmas, theorems as a
1.730 +computer program or script is not able to interpret every Greek or
1.731 +Latin letter and every Greek, Latin or whatever calculations
1.732 +symbol. Also if we would be able to handle these symbols we still have
1.733 +a problem to interpret them at all. (Follow up \hbox{Example
1.734 +\ref{eg:symbint1}})
1.735
1.736 \vbox{
1.737 \begin{example}
1.738 @@ -532,12 +923,18 @@
1.739 u\left[n\right] \ \ldots \ unitstep
1.740 \]
1.741 {\small\textit{
1.742 - \noindent The unitstep is something we need to solve Signal Processing problem classes. But in {{\sisac{}}} the rectangular brackets have a different meaning. So we abuse them for our requirements. We get something which is not defined, but usable. The Result is syntax only without semantic.
1.743 + \noindent The unitstep is something we need to solve Signal
1.744 +Processing problem classes. But in {{{\sisac}{}}} the rectangular
1.745 +brackets have a different meaning. So we abuse them for our
1.746 +requirements. We get something which is not defined, but usable. The
1.747 +Result is syntax only without semantic.
1.748 }}
1.749 \end{example}
1.750 }
1.751
1.752 -In different problems, symbols and letters have different meanings and ask for different ways to get through. (Follow up \hbox{Example \ref{eg:symbint2}})
1.753 +In different problems, symbols and letters have different meanings and
1.754 +ask for different ways to get through. (Follow up \hbox{Example
1.755 +\ref{eg:symbint2}})
1.756
1.757 \vbox{
1.758 \begin{example}
1.759 @@ -546,18 +943,28 @@
1.760 \widehat{\ }\ \widehat{\ }\ \widehat{\ } \ \ldots \ exponent
1.761 \]
1.762 {\small\textit{
1.763 - \noindent For using exponents the three \texttt{widehat} symbols are required. The reason for that is due the development of {{\sisac{}}} the single \texttt{widehat} and also the double were already in use for different operations.
1.764 + \noindent For using exponents the three \texttt{widehat} symbols
1.765 +are required. The reason for that is due the development of
1.766 +{{{\sisac}{}}} the single \texttt{widehat} and also the double were
1.767 +already in use for different operations.
1.768 }}
1.769 \end{example}
1.770 }
1.771
1.772 -\paragraph{Also the output} can be a problem. We are familiar with a specified notations and style taught in university but a computer program has no knowledge of the form proved by a professor and the machines themselves also have not yet the possibilities to print every symbol (correct) Recent developments provide proofs in a human readable format but according to the fact that there is no money for good working formal editors yet, the style is one thing we have to live with.
1.773 +\paragraph{Also the output} can be a problem. We are familiar with a
1.774 +specified notations and style taught in university but a computer
1.775 +program has no knowledge of the form proved by a professor and the
1.776 +machines themselves also have not yet the possibilities to print every
1.777 +symbol (correct) Recent developments provide proofs in a human
1.778 +readable format but according to the fact that there is no money for
1.779 +good working formal editors yet, the style is one thing we have to
1.780 +live with.
1.781
1.782 \section{Problems rising out of the Development Environment}
1.783
1.784 fehlermeldungen! TODO
1.785
1.786 -\section{Conclusion}
1.787 +\section{Conclusion}\label{conclusion}
1.788
1.789 TODO
1.790