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1.63 -\fancyhead[c]{\small The Electronic Journal of Mathematics%
1.64 -\ and Technology, Volume 1, Number 1, ISSN 1933-2823} %
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1.75 -\def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
1.76 -\def\sisac{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
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1.98 -\begin{document}
1.99 -%
1.100 -% document title
1.101 -%
1.102 -\title{Trials with TP-based Programming
1.103 -\\
1.104 -for Interactive Course Material}%
1.105 -%
1.106 -% Single author. Please supply at least your name,
1.107 -% email address, and affiliation here.
1.108 -%
1.109 -\author{\begin{tabular}{c}
1.110 -\textit{Jan Ro\v{c}nik} \\
1.111 -jan.rocnik@student.tugraz.at \\
1.112 -IST, SPSC\\
1.113 -Graz University of Technology\\
1.114 -Austria\end{tabular}
1.115 -}%
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1.125 -%
1.126 -% abstract
1.127 -%
1.128 -\begin{abstract}
1.129 -
1.130 -Traditional course material in engineering disciplines lacks an
1.131 -important component, interactive support for step-wise problem
1.132 -solving. Theorem-Proving (TP) technology is appropriate for one part
1.133 -of such support, in checking user-input. For the other part of such
1.134 -support, guiding the learner towards a solution, another kind of
1.135 -technology is required.
1.136 -
1.137 -Both kinds of support can be achieved by so-called
1.138 -Lucas-Interpretation which combines deduction and computation and, for
1.139 -the latter, uses a novel kind of programming language. This language
1.140 -is based on (Computer) Theorem Proving (TP), thus called a ``TP-based
1.141 -programming language''.
1.142 -
1.143 -This paper is the experience report of the first ``application
1.144 -programmer'' using this language for creating exercises in step-wise
1.145 -problem solving for an advanced lab in Signal Processing. The tasks
1.146 -involved in TP-based programming are described together with the
1.147 -experience gained from a prototype of the programming language and of
1.148 -it's interpreter.
1.149 -
1.150 -The report concludes with a positive proof of concept, states
1.151 -insufficiency usability of the prototype and captures the requirements
1.152 -for further development of both, the programming language and the
1.153 -interpreter.
1.154 -%
1.155 -\end{abstract}%
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1.166 -% etc. Please also use \subsubsection{...}, \paragraph{...}
1.167 -% and \subparagraph{...} as necessary.
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1.169 -
1.170 -\section{Introduction}\label{intro}
1.171 -
1.172 -% \paragraph{Didactics of mathematics}
1.173 -%WN: wenn man in einem high-quality paper von 'didactics' spricht,
1.174 -%WN muss man am state-of-the-art ankn"upfen -- siehe
1.175 -%WN W.Neuper, On the Emergence of TP-based Educational Math Assistants
1.176 -% faces a specific issue, a gap
1.177 -% between (1) introduction of math concepts and skills and (2)
1.178 -% application of these concepts and skills, which usually are separated
1.179 -% into different units in curricula (for good reasons). For instance,
1.180 -% (1) teaching partial fraction decomposition is separated from (2)
1.181 -% application for inverse Z-transform in signal processing.
1.182 -%
1.183 -% \par This gap is an obstacle for applying math as an fundamental
1.184 -% thinking technology in engineering: In (1) motivation is lacking
1.185 -% because the question ``What is this stuff good for?'' cannot be
1.186 -% treated sufficiently, and in (2) the ``stuff'' is not available to
1.187 -% students in higher semesters as widespread experience shows.
1.188 -%
1.189 -% \paragraph{Motivation} taken by this didactic issue on the one hand,
1.190 -% and ongoing research and development on a novel kind of educational
1.191 -% mathematics assistant at Graz University of
1.192 -% Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to
1.193 -% scope with this issue on the other hand, several institutes are
1.194 -% planning to join their expertise: the Institute for Information
1.195 -% Systems and Computer Media (IICM), the Institute for Software
1.196 -% Technology (IST), the Institutes for Mathematics, the Institute for
1.197 -% Signal Processing and Speech Communication (SPSC), the Institute for
1.198 -% Structural Analysis and the Institute of Electrical Measurement and
1.199 -% Measurement Signal Processing.
1.200 -%WN diese Information ist f"ur das Paper zu spezielle, zu aktuell
1.201 -%WN und damit zu verg"anglich.
1.202 -% \par This thesis is the first attempt to tackle the above mentioned
1.203 -% issue, it focuses on Telematics, because these specific studies focus
1.204 -% on mathematics in \emph{STEOP}, the introductory orientation phase in
1.205 -% Austria. \emph{STEOP} is considered an opportunity to investigate the
1.206 -% impact of {\sisac}'s prototype on the issue and others.
1.207 -%
1.208 -
1.209 -Traditional course material in engineering disciplines lacks an
1.210 -important component, interactive support for step-wise problem
1.211 -solving. The lack becomes evident by comparing existing course
1.212 -material with the sheets collected from written exams (in case solving
1.213 -engineering problems is {\em not} deteriorated to multiple choice
1.214 -tests) on the topics addressed by the materials.
1.215 -Theorem-Proving (TP) technology can provide such support by
1.216 -specific services. An important part of such services is called
1.217 -``next-step-guidance'', generated by a specific kind of ``TP-based
1.218 -programming language''. In the
1.219 -{\sisac}-project~\footnote{http://www.ist.tugraz.at/projects/isac/} such
1.220 -a language is prototyped in line with~\cite{plmms10} and built upon
1.221 -the theorem prover Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}
1.222 -\footnote{http://isabelle.in.tum.de/}.
1.223 -The TP services are coordinated by a specific interpreter for the
1.224 -programming language, called
1.225 -Lucas-Interpreter~\cite{wn:lucas-interp-12}. The language
1.226 - will be briefly re-introduced in order to make the paper
1.227 -self-contained.
1.228 -
1.229 -The main part of the paper is an account of first experiences
1.230 -with programming in this TP-based language. The experience was gained
1.231 -in a case study by the author. The author was considered an ideal
1.232 -candidate for this study for the following reasons: as a student in
1.233 -Telematics (computer science with focus on Signal Processing) he had
1.234 -general knowledge in programming as well as specific domain knowledge
1.235 -in Signal Processing; and he was {\em not} involved in the development of
1.236 -{\sisac}'s programming language and interpreter, thus being a novice to the
1.237 -language.
1.238 -
1.239 -The goals of the case study were: (1) to identify some TP-based programs for
1.240 -interactive course material for a specific ``Advanced Signal
1.241 -Processing Lab'' in a higher semester, (2) respective program
1.242 -development with as little advice as possible from the {\sisac}-team and (3)
1.243 -to document records and comments for the main steps of development in an
1.244 -Isabelle theory; this theory should provide guidelines for future programmers.
1.245 -An excerpt from this theory is the main part of this paper.
1.246 -\par
1.247 -
1.248 -\medskip The major example resulting from the case study will be used
1.249 -as running example throughout this paper. This example requires a
1.250 -program resembling the size of real-world applications in engineering;
1.251 -such a size was considered essential for the case study, since there
1.252 -are many small programs for a long time (mainly concerned with
1.253 -elementary Computer Algebra like simplification, equation solving,
1.254 -calculus, etc.~\footnote{The programs existing in the {\sisac}
1.255 -prototype are found at
1.256 -http://www.ist.tugraz.at/projects/isac/www/kbase/met/index\_met.html})
1.257 -
1.258 -\paragraph{The mathematical background of the running example} is the
1.259 -following: In Signal Processing, ``the ${\cal Z}$-transform for
1.260 -discrete-time signals is the counterpart of the Laplace transform for
1.261 -continuous-time signals, and they each have a similar relationship to
1.262 -the corresponding Fourier transform. One motivation for introducing
1.263 -this generalization is that the Fourier transform does not converge
1.264 -for all sequences, and it is useful to have a generalization of the
1.265 -Fourier transform that encompasses a broader class of signals. A
1.266 -second advantage is that in analytic problems, the ${\cal Z}$-transform
1.267 -notation is often more convenient than the Fourier transform
1.268 -notation.'' ~\cite[p. 128]{oppenheim2010discrete}. The ${\cal Z}$-transform
1.269 -is defined as
1.270 -\begin{equation*}
1.271 -X(z)=\sum_{n=-\infty }^{\infty }x[n]z^{-n}
1.272 -\end{equation*}
1.273 -where a discrete time sequence $x[n]$ is transformed into the function
1.274 -$X(z)$ where $z$ is a continuous complex variable. The inverse
1.275 -function is addressed in the running example and can be determined by
1.276 -the integral
1.277 -\begin{equation*}
1.278 -x[n]=\frac{1}{2\pi j} \oint_{C} X(z)\cdot z^{n-1} dz
1.279 -\end{equation*}
1.280 -where the letter $C$ represents a contour within the range of
1.281 -convergence of the ${\cal Z}$-transform. The unit circle can be a special
1.282 -case of this contour. Remember that $j$ is the complex number in the
1.283 -domain of engineering. As this transform requires high effort to
1.284 -be solved, tables of commonly used transform pairs are used in
1.285 -education as well as in engineering practice; such tables can be found
1.286 -at~\cite{wiki:1} or~\cite[Table~3.1]{oppenheim2010discrete} as well.
1.287 -A completely solved and more detailed example can be found at
1.288 -~\cite[p. 149f]{oppenheim2010discrete}.
1.289 -
1.290 -Following conventions in engineering education and in practice, the
1.291 -running example solves the problem by use of a table.
1.292 -
1.293 -\paragraph{Support for interactive stepwise problem solving} in the
1.294 -{\sisac} prototype is shown in Fig.\ref{fig-interactive}~\footnote{ Fig.\ref{fig-interactive} also shows the prototype status of {\sisac}; for instance,
1.295 -the lack of 2-dimensional presentation and input of formulas is the major obstacle for field-tests in standard classes.}:
1.296 -A student inputs formulas line by line on the \textit{``Worksheet''},
1.297 -and each step (i.e. each formula on completion) is immediately checked
1.298 -by the system, such that at most {\em one inconsistent} formula can reside on
1.299 -the Worksheet (on the input line, marked by the red $\otimes$).
1.300 -\begin{figure} [htb]
1.301 -\begin{center}
1.302 -\includegraphics[width=140mm]{fig/isac-Ztrans-math-3}
1.303 -%\includegraphics[width=140mm]{fig/isac-Ztrans-math}
1.304 -\caption{Step-wise problem solving guided by the TP-based program
1.305 -\label{fig-interactive}}
1.306 -\end{center}
1.307 -\end{figure}
1.308 -If the student gets stuck and does not know the formula to proceed
1.309 -with, there is the button \framebox{NEXT} presenting the next formula
1.310 -on the Worksheet; this feature is called ``next-step-guidance''~\cite{wn:lucas-interp-12}. The button \framebox{AUTO} immediately delivers the
1.311 -final result in case the student is not interested in intermediate
1.312 -steps.
1.313 -
1.314 -Adaptive dialogue guidance is already under
1.315 -construction~\cite{gdaroczy-EP-13} and the two buttons will disappear,
1.316 -since their presence is not wanted in many learning scenarios (in
1.317 -particular, {\em not} in written exams).
1.318 -
1.319 -The buttons \framebox{Theories}, \framebox{Problems} and
1.320 -\framebox{Methods} are the entry points for interactive lookup of the
1.321 -underlying knowledge. For instance, pushing \framebox{Theories} in
1.322 -the configuration shown in Fig.\ref{fig-interactive}, pops up a
1.323 -``Theory browser'' displaying the theorem(s) justifying the current
1.324 -step. The browser allows to lookup all other theories, thus
1.325 -supporting indepentend investigation of underlying definitions,
1.326 -theorems, proofs --- where the HTML representation of the browsers is
1.327 -ready for arbitrary multimedia add-ons. Likewise, the browsers for
1.328 -\framebox{Problems} and \framebox{Methods} support context sensitive
1.329 -as well as interactive access to specifications and programs
1.330 -respectively.
1.331 -
1.332 -There is also a simple web-based representation of knowledge items;
1.333 -the items under consideration in this paper can be looked up as
1.334 -well
1.335 -~\footnote{\href{http://www.ist.tugraz.at/projects/isac/www/kbase/thy/browser\_info/HOL/HOL-Real/Isac/Inverse\_Z\_Transform.thy}{http://www.ist.tugraz.at/projects/isac/www/kbase/thy/browser\_info/HOL/HOL-Real/Isac/\textbf{Inverse\_Z\_Transform.thy}}}
1.336 -~\footnote{\href{http://www.ist.tugraz.at/projects/isac/www/kbase/thy/browser\_info/HOL/HOL-Real/Isac/Partial\_Fractions.thy}{http://www.ist.tugraz.at/projects/isac/www/kbase/thy/browser\_info/HOL/HOL-Real/Isac/\textbf{Partial\_Fractions.thy}}}
1.337 -~\footnote{\href{http://www.ist.tugraz.at/projects/isac/www/kbase/thy/browser\_info/HOL/HOL-Real/Isac/Build\_Inverse\_Z\_Transform.thy}{http://www.ist.tugraz.at/projects/isac/www/kbase/thy/browser\_info/HOL/HOL-Real/Isac/\textbf{Build\_Inverse\_Z\_Transform.thy}}}.
1.338 -
1.339 -% can be explained by having a look at
1.340 -% Fig.\ref{fig-interactive} which shows the beginning of the interactive
1.341 -% construction of a solution for the problem. This construction is done in the
1.342 -% right window named ``Worksheet''.
1.343 -% \par
1.344 -% User-interaction on the Worksheet is {\em checked} and {\em guided} by
1.345 -% TP services:
1.346 -% \begin{enumerate}
1.347 -% \item Formulas input by the user are {\em checked} by TP: such a
1.348 -% formula establishes a proof situation --- the prover has to derive the
1.349 -% formula from the logical context. The context is built up from the
1.350 -% formal specification of the problem (here hidden from the user) by the
1.351 -% Lucas-Interpreter.
1.352 -% \item If the user gets stuck, the program developed below in this
1.353 -% paper ``knows the next step'' and Lucas-Interpretation provides services
1.354 -% featuring so-called ``next-step-guidance''; this is out of scope of this
1.355 -% paper and can be studied in~\cite{gdaroczy-EP-13}.
1.356 -% \end{enumerate} It should be noted that the programmer using the
1.357 -% TP-based language is not concerned with interaction at all; we will
1.358 -% see that the program contains neither input-statements nor
1.359 -% output-statements. Rather, interaction is handled by the interpreter
1.360 -% of the language.
1.361 -%
1.362 -% So there is a clear separation of concerns: Dialogues are adapted by
1.363 -% dialogue authors (in Java-based tools), using TP services provided by
1.364 -% Lucas-Interpretation. The latter acts on programs developed by
1.365 -% mathematics-authors (in Isabelle/ML); their task is concern of this
1.366 -% paper.
1.367 -
1.368 -\bigskip The paper is structured as follows: The introduction
1.369 -\S\ref{intro} is followed by a brief re-introduction of the TP-based
1.370 -programming language in \S\ref{PL}, which extends the executable
1.371 -fragment of Isabelle's language (\S\ref{PL-isab}) by tactics which
1.372 -play a specific role in Lucas-Interpretation and in providing the TP
1.373 -services (\S\ref{PL-tacs}). The main part \S\ref{trial} describes
1.374 -the main steps in developing the program for the running example:
1.375 -prepare domain knowledge, implement the formal specification of the
1.376 -problem, prepare the environment for the interpreter, implement the
1.377 -program in \S\ref{isabisac} to \S\ref{progr} respectively.
1.378 -The work-flow of programming, debugging and testing is
1.379 -described in \S\ref{workflow}. The conclusion \S\ref{conclusion} will
1.380 -give directions identified for future development.
1.381 -
1.382 -
1.383 -\section{\isac's Prototype for a Programming Language}\label{PL}
1.384 -The prototype of the language and of the Lucas-Interpreter is briefly
1.385 -described from the point of view of a programmer. The language extends
1.386 -the executable fragment of Higher-Order Logic (HOL) in the theorem prover
1.387 -Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}.
1.388 -
1.389 -\subsection{The Executable Fragment of Isabelle's Language}\label{PL-isab}
1.390 -The executable fragment consists of data-type and function
1.391 -definitions. It's usability even suggests that fragment for
1.392 -introductory courses \cite{nipkow-prog-prove}. HOL is a typed logic whose type system resembles that of functional programming
1.393 -languages. Thus there are
1.394 -\begin{description}
1.395 -\item[base types,] in particular \textit{bool}, the type of truth
1.396 -values, \textit{nat}, \textit{int}, \textit{complex}, and the types of
1.397 -natural, integer and complex numbers respectively in mathematics.
1.398 -\item[type constructors] allow to define arbitrary types, from
1.399 -\textit{set}, \textit{list} to advanced data-structures like
1.400 -\textit{trees}, red-black-trees etc.
1.401 -\item[function types,] denoted by $\Rightarrow$.
1.402 -\item[type variables,] denoted by $^\prime a, ^\prime b$ etc, provide
1.403 -type polymorphism. Isabelle automatically computes the type of each
1.404 -variable in a term by use of Hindley-Milner type inference
1.405 -\cite{pl:hind97,Milner-78}.
1.406 -\end{description}
1.407 -
1.408 -\textbf{Terms} are formed as in functional programming by applying
1.409 -functions to arguments. If $f$ is a function of type
1.410 -$\tau_1\Rightarrow \tau_2$ and $t$ is a term of type $\tau_1$ then
1.411 -$f\;t$ is a term of type~$\tau_2$. $t\;::\;\tau$ means that term $t$
1.412 -has type $\tau$. There are many predefined infix symbols like $+$ and
1.413 -$\leq$ most of which are overloaded for various types.
1.414 -
1.415 -HOL also supports some basic constructs from functional programming:
1.416 -{\footnotesize\it\label{isabelle-stmts}
1.417 -\begin{tabbing} 123\=\kill
1.418 -01\>$( \; {\tt if} \; b \; {\tt then} \; t_1 \; {\tt else} \; t_2 \;)$\\
1.419 -02\>$( \; {\tt let} \; x=t \; {\tt in} \; u \; )$\\
1.420 -03\>$( \; {\tt case} \; t \; {\tt of} \; {\it pat}_1
1.421 - \Rightarrow t_1 \; |\dots| \; {\it pat}_n\Rightarrow t_n \; )$
1.422 -\end{tabbing}}
1.423 -\noindent The running example's program uses some of these elements
1.424 -(marked by {\tt tt-font} on p.\pageref{s:impl}): for instance {\tt
1.425 -let}\dots{\tt in} in lines {\rm 02} \dots {\rm 13}. In fact, the whole program
1.426 -is an Isabelle term with specific function constants like {\tt
1.427 -program}, {\tt Take}, {\tt Rewrite}, {\tt Subproblem} and {\tt
1.428 -Rewrite\_Set} in lines {\rm 01, 03. 04, 07, 10} and {\rm 11, 12}
1.429 -respectively.
1.430 -
1.431 -% Terms may also contain $\lambda$-abstractions. For example, $\lambda
1.432 -% x. \; x$ is the identity function.
1.433 -
1.434 -%JR warum auskommentiert? WN2...
1.435 -%WN2 weil ein Punkt wie dieser in weiteren Zusammenh"angen innerhalb
1.436 -%WN2 des Papers auftauchen m"usste; nachdem ich einen solchen
1.437 -%WN2 Zusammenhang _noch_ nicht sehe, habe ich den Punkt _noch_ nicht
1.438 -%WN2 gel"oscht.
1.439 -%WN2 Wenn der Punkt nicht weiter gebraucht wird, nimmt er nur wertvollen
1.440 -%WN2 Platz f"ur Anderes weg.
1.441 -
1.442 -\textbf{Formulae} are terms of type \textit{bool}. There are the basic
1.443 -constants \textit{True} and \textit{False} and the usual logical
1.444 -connectives (in decreasing order of precedence): $\neg, \land, \lor,
1.445 -\rightarrow$.
1.446 -
1.447 -\textbf{Equality} is available in the form of the infix function $=$
1.448 -of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for
1.449 -formulas, where it means ``if and only if''.
1.450 -
1.451 -\textbf{Quantifiers} are written $\forall x. \; P$ and $\exists x. \;
1.452 -P$. Quantifiers lead to non-executable functions, so functions do not
1.453 -always correspond to programs, for instance, if comprising \\$(
1.454 -\;{\it if} \; \exists x.\;P \; {\it then} \; e_1 \; {\it else} \; e_2
1.455 -\;)$.
1.456 -
1.457 -\subsection{\isac's Tactics for Lucas-Interpretation}\label{PL-tacs}
1.458 -The prototype extends Isabelle's language by specific statements
1.459 -called tactics~\footnote{{\sisac}'s. These tactics are different from
1.460 -Isabelle's tactics: the former concern steps in a calculation, the
1.461 -latter concern proofs.}. For the programmer these
1.462 -statements are functions with the following signatures:
1.463 -
1.464 -\begin{description}
1.465 -\item[Rewrite:] ${\it theorem}\Rightarrow{\it term}\Rightarrow{\it
1.466 -term} * {\it term}\;{\it list}$:
1.467 -this tactic applies {\it theorem} to a {\it term} yielding a {\it
1.468 -term} and a {\it term list}, the list are assumptions generated by
1.469 -conditional rewriting. For instance, the {\it theorem}
1.470 -$b\not=0\land c\not=0\Rightarrow\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$
1.471 -applied to the {\it term} $\frac{2\cdot x}{3\cdot x}$ yields
1.472 -$(\frac{2}{3}, [x\not=0])$.
1.473 -
1.474 -\item[Rewrite\_Set:] ${\it ruleset}\Rightarrow{\it
1.475 -term}\Rightarrow{\it term} * {\it term}\;{\it list}$:
1.476 -this tactic applies {\it ruleset} to a {\it term}; {\it ruleset} is
1.477 -a confluent and terminating term rewrite system, in general. If
1.478 -none of the rules ({\it theorem}s) is applicable on interpretation
1.479 -of this tactic, an exception is thrown.
1.480 -
1.481 -% \item[Rewrite\_Inst:] ${\it substitution}\Rightarrow{\it
1.482 -% theorem}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
1.483 -% list}$:
1.484 -%
1.485 -% \item[Rewrite\_Set\_Inst:] ${\it substitution}\Rightarrow{\it
1.486 -% ruleset}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
1.487 -% list}$:
1.488 -
1.489 -%SPACEvvv
1.490 -\item[Substitute:] ${\it substitution}\Rightarrow{\it
1.491 -term}\Rightarrow{\it term}$: allows to access sub-terms.
1.492 -%SPACE^^^
1.493 -
1.494 -\item[Take:] ${\it term}\Rightarrow{\it term}$:
1.495 -this tactic has no effect in the program; but it creates a side-effect
1.496 -by Lucas-Interpretation (see below) and writes {\it term} to the
1.497 -Worksheet.
1.498 -
1.499 -\item[Subproblem:] ${\it theory} * {\it specification} * {\it
1.500 -method}\Rightarrow{\it argument}\;{\it list}\Rightarrow{\it term}$:
1.501 -this tactic is a generalisation of a function call: it takes an
1.502 -\textit{argument list} as usual, and additionally a triple consisting
1.503 -of an Isabelle \textit{theory}, an implicit \textit{specification} of the
1.504 -program and a \textit{method} containing data for Lucas-Interpretation,
1.505 -last not least a program (as an explicit specification)~\footnote{In
1.506 -interactive tutoring these three items can be determined explicitly
1.507 -by the user.}.
1.508 -\end{description}
1.509 -The tactics play a specific role in
1.510 -Lucas-Interpretation~\cite{wn:lucas-interp-12}: they are treated as
1.511 -break-points where, as a side-effect, a line is added to a calculation
1.512 -as a protocol for proceeding towards a solution in step-wise problem
1.513 -solving. At the same points Lucas-Interpretation serves interactive
1.514 -tutoring and hands over control to the user. The user is free to
1.515 -investigate underlying knowledge, applicable theorems, etc. And the
1.516 -user can proceed constructing a solution by input of a tactic to be
1.517 -applied or by input of a formula; in the latter case the
1.518 -Lucas-Interpreter has built up a logical context (initialised with the
1.519 -precondition of the formal specification) such that Isabelle can
1.520 -derive the formula from this context --- or give feedback, that no
1.521 -derivation can be found.
1.522 -
1.523 -\subsection{Tactics as Control Flow Statements}
1.524 -The flow of control in a program can be determined by {\tt if then else}
1.525 -and {\tt case of} as mentioned on p.\pageref{isabelle-stmts} and also
1.526 -by additional tactics:
1.527 -\begin{description}
1.528 -\item[Repeat:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it
1.529 -term}$: iterates over tactics which take a {\it term} as argument as
1.530 -long as a tactic is applicable (for instance, {\tt Rewrite\_Set} might
1.531 -not be applicable).
1.532 -
1.533 -\item[Try:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it term}$:
1.534 -if {\it tactic} is applicable, then it is applied to {\it term},
1.535 -otherwise {\it term} is passed on without changes.
1.536 -
1.537 -\item[Or:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
1.538 -term}\Rightarrow{\it term}$: If the first {\it tactic} is applicable,
1.539 -it is applied to the first {\it term} yielding another {\it term},
1.540 -otherwise the second {\it tactic} is applied; if none is applicable an
1.541 -exception is raised.
1.542 -
1.543 -\item[@@:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
1.544 -term}\Rightarrow{\it term}$: applies the first {\it tactic} to the
1.545 -first {\it term} yielding an intermediate term (not appearing in the
1.546 -signature) to which the second {\it tactic} is applied.
1.547 -
1.548 -\item[While:] ${\it term::bool}\Rightarrow{\it tactic}\Rightarrow{\it
1.549 -term}\Rightarrow{\it term}$: if the first {\it term} is true, then the
1.550 -{\it tactic} is applied to the first {\it term} yielding an
1.551 -intermediate term (not appearing in the signature); the intermediate
1.552 -term is added to the environment the first {\it term} is evaluated in
1.553 -etc. as long as the first {\it term} is true.
1.554 -\end{description}
1.555 -The tactics are not treated as break-points by Lucas-Interpretation
1.556 -and thus do neither contribute to the calculation nor to interaction.
1.557 -
1.558 -\section{Concepts and Tasks in TP-based Programming}\label{trial}
1.559 -%\section{Development of a Program on Trial}
1.560 -
1.561 -This section presents all the concepts involved in TP-based
1.562 -programming and all the tasks to be accomplished by programmers. The
1.563 -presentation uses the running example from
1.564 -Fig.\ref{fig-interactive} on p.\pageref{fig-interactive}.
1.565 -
1.566 -\subsection{Mechanization of Math --- Domain Engineering}\label{isabisac}
1.567 -
1.568 -%WN was Fachleute unter obigem Titel interessiert findet sich
1.569 -%WN unterhalb des auskommentierten Textes.
1.570 -
1.571 -%WN der Text unten spricht Benutzer-Aspekte anund ist nicht speziell
1.572 -%WN auf Computer-Mathematiker fokussiert.
1.573 -% \paragraph{As mentioned in the introduction,} a prototype of an
1.574 -% educational math assistant called
1.575 -% {{\sisac}}\footnote{{{\sisac}}=\textbf{Isa}belle for
1.576 -% \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges
1.577 -% the gap between (1) introducation and (2) application of mathematics:
1.578 -% {{\sisac}} is based on Computer Theorem Proving (TP), a technology which
1.579 -% requires each fact and each action justified by formal logic, so
1.580 -% {{{\sisac}{}}} makes justifications transparent to students in
1.581 -% interactive step-wise problem solving. By that way {{\sisac}} already
1.582 -% can serve both:
1.583 -% \begin{enumerate}
1.584 -% \item Introduction of math stuff (in e.g. partial fraction
1.585 -% decomposition) by stepwise explaining and exercising respective
1.586 -% symbolic calculations with ``next step guidance (NSG)'' and rigorously
1.587 -% checking steps freely input by students --- this also in context with
1.588 -% advanced applications (where the stuff to be taught in higher
1.589 -% semesters can be skimmed through by NSG), and
1.590 -% \item Application of math stuff in advanced engineering courses
1.591 -% (e.g. problems to be solved by inverse Z-transform in a Signal
1.592 -% Processing Lab) and now without much ado about basic math techniques
1.593 -% (like partial fraction decomposition): ``next step guidance'' supports
1.594 -% students in independently (re-)adopting such techniques.
1.595 -% \end{enumerate}
1.596 -% Before the question is answers, how {{\sisac}}
1.597 -% accomplishes this task from a technical point of view, some remarks on
1.598 -% the state-of-the-art is given, therefor follow up Section~\ref{emas}.
1.599 -%
1.600 -% \subsection{Educational Mathematics Assistants (EMAs)}\label{emas}
1.601 -%
1.602 -% \paragraph{Educational software in mathematics} is, if at all, based
1.603 -% on Computer Algebra Systems (CAS, for instance), Dynamic Geometry
1.604 -% Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org}
1.605 -% \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC
1.606 -% http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These
1.607 -% base technologies are used to program math lessons and sometimes even
1.608 -% exercises. The latter are cumbersome: the steps towards a solution of
1.609 -% such an interactive exercise need to be provided with feedback, where
1.610 -% at each step a wide variety of possible input has to be foreseen by
1.611 -% the programmer - so such interactive exercises either require high
1.612 -% development efforts or the exercises constrain possible inputs.
1.613 -%
1.614 -% \subparagraph{A new generation} of educational math assistants (EMAs)
1.615 -% is emerging presently, which is based on Theorem Proving (TP). TP, for
1.616 -% instance Isabelle and Coq, is a technology which requires each fact
1.617 -% and each action justified by formal logic. Pushed by demands for
1.618 -% \textit{proven} correctness of safety-critical software TP advances
1.619 -% into software engineering; from these advancements computer
1.620 -% mathematics benefits in general, and math education in particular. Two
1.621 -% features of TP are immediately beneficial for learning:
1.622 -%
1.623 -% \paragraph{TP have knowledge in human readable format,} that is in
1.624 -% standard predicate calculus. TP following the LCF-tradition have that
1.625 -% knowledge down to the basic definitions of set, equality,
1.626 -% etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html};
1.627 -% following the typical deductive development of math, natural numbers
1.628 -% are defined and their properties
1.629 -% proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html},
1.630 -% etc. Present knowledge mechanized in TP exceeds high-school
1.631 -% mathematics by far, however by knowledge required in software
1.632 -% technology, and not in other engineering sciences.
1.633 -%
1.634 -% \paragraph{TP can model the whole problem solving process} in
1.635 -% mathematical problem solving {\em within} a coherent logical
1.636 -% framework. This is already being done by three projects, by
1.637 -% Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.
1.638 -% \par
1.639 -% Having the whole problem solving process within a logical coherent
1.640 -% system, such a design guarantees correctness of intermediate steps and
1.641 -% of the result (which seems essential for math software); and the
1.642 -% second advantage is that TP provides a wealth of theories which can be
1.643 -% exploited for mechanizing other features essential for educational
1.644 -% software.
1.645 -%
1.646 -% \subsubsection{Generation of User Guidance in EMAs}\label{user-guid}
1.647 -%
1.648 -% One essential feature for educational software is feedback to user
1.649 -% input and assistance in coming to a solution.
1.650 -%
1.651 -% \paragraph{Checking user input} by ATP during stepwise problem solving
1.652 -% is being accomplished by the three projects mentioned above
1.653 -% exclusively. They model the whole problem solving process as mentioned
1.654 -% above, so all what happens between formalized assumptions (or formal
1.655 -% specification) and goal (or fulfilled postcondition) can be
1.656 -% mechanized. Such mechanization promises to greatly extend the scope of
1.657 -% educational software in stepwise problem solving.
1.658 -%
1.659 -% \paragraph{NSG (Next step guidance)} comprises the system's ability to
1.660 -% propose a next step; this is a challenge for TP: either a radical
1.661 -% restriction of the search space by restriction to very specific
1.662 -% problem classes is required, or much care and effort is required in
1.663 -% designing possible variants in the process of problem solving
1.664 -% \cite{proof-strategies-11}.
1.665 -% \par
1.666 -% Another approach is restricted to problem solving in engineering
1.667 -% domains, where a problem is specified by input, precondition, output
1.668 -% and postcondition, and where the postcondition is proven by ATP behind
1.669 -% the scenes: Here the possible variants in the process of problem
1.670 -% solving are provided with feedback {\em automatically}, if the problem
1.671 -% is described in a TP-based programing language: \cite{plmms10} the
1.672 -% programmer only describes the math algorithm without caring about
1.673 -% interaction (the respective program is functional and even has no
1.674 -% input or output statements!); interaction is generated as a
1.675 -% side-effect by the interpreter --- an efficient separation of concern
1.676 -% between math programmers and dialog designers promising application
1.677 -% all over engineering disciplines.
1.678 -%
1.679 -%
1.680 -% \subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}
1.681 -% Authoring new mathematics knowledge in {{\sisac}} can be compared with
1.682 -% ``application programing'' of engineering problems; most of such
1.683 -% programing uses CAS-based programing languages (CAS = Computer Algebra
1.684 -% Systems; e.g. Mathematica's or Maple's programing language).
1.685 -%
1.686 -% \paragraph{A novel type of TP-based language} is used by {{\sisac}{}}
1.687 -% \cite{plmms10} for describing how to construct a solution to an
1.688 -% engineering problem and for calling equation solvers, integration,
1.689 -% etc~\footnote{Implementation of CAS-like functionality in TP is not
1.690 -% primarily concerned with efficiency, but with a didactic question:
1.691 -% What to decide for: for high-brow algorithms at the state-of-the-art
1.692 -% or for elementary algorithms comprehensible for students?} within TP;
1.693 -% TP can ensure ``systems that never make a mistake'' \cite{casproto} -
1.694 -% are impossible for CAS which have no logics underlying.
1.695 -%
1.696 -% \subparagraph{Authoring is perfect} by writing such TP based programs;
1.697 -% the application programmer is not concerned with interaction or with
1.698 -% user guidance: this is concern of a novel kind of program interpreter
1.699 -% called Lucas-Interpreter. This interpreter hands over control to a
1.700 -% dialog component at each step of calculation (like a debugger at
1.701 -% breakpoints) and calls automated TP to check user input following
1.702 -% personalized strategies according to a feedback module.
1.703 -% \par
1.704 -% However ``application programing with TP'' is not done with writing a
1.705 -% program: according to the principles of TP, each step must be
1.706 -% justified. Such justifications are given by theorems. So all steps
1.707 -% must be related to some theorem, if there is no such theorem it must
1.708 -% be added to the existing knowledge, which is organized in so-called
1.709 -% \textbf{theories} in Isabelle. A theorem must be proven; fortunately
1.710 -% Isabelle comprises a mechanism (called ``axiomatization''), which
1.711 -% allows to omit proofs. Such a theorem is shown in
1.712 -% Example~\ref{eg:neuper1}.
1.713 -
1.714 -The running example requires to determine the inverse ${\cal Z}$-transform
1.715 -for a class of functions. The domain of Signal Processing
1.716 -is accustomed to specific notation for the resulting functions, which
1.717 -are absolutely capable of being totalled and are called step-response: $u[n]$, where $u$ is the
1.718 -function, $n$ is the argument and the brackets indicate that the
1.719 -arguments are discrete. Surprisingly, Isabelle accepts the rules for
1.720 -$z^{-1}$ in this traditional notation~\footnote{Isabelle
1.721 -experts might be particularly surprised, that the brackets do not
1.722 -cause errors in typing (as lists).}:
1.723 -%\vbox{
1.724 -% \begin{example}
1.725 - \label{eg:neuper1}
1.726 - {\footnotesize\begin{tabbing}
1.727 - 123\=123\=123\=123\=\kill
1.728 -
1.729 - 01\>axiomatization where \\
1.730 - 02\>\> rule1: ``$z^{-1}\;1 = \delta [n]$'' and\\
1.731 - 03\>\> rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow z^{-1}\;z / (z - 1) = u [n]$'' and\\
1.732 - 04\>\> rule3: ``$\vert\vert z \vert\vert < 1 \Rightarrow z / (z - 1) = -u [-n - 1]$'' and \\
1.733 - 05\>\> rule4: ``$\vert\vert z \vert\vert > \vert\vert$ $\alpha$ $\vert\vert \Rightarrow z / (z - \alpha) = \alpha^n \cdot u [n]$'' and\\
1.734 - 06\>\> rule5: ``$\vert\vert z \vert\vert < \vert\vert \alpha \vert\vert \Rightarrow z / (z - \alpha) = -(\alpha^n) \cdot u [-n - 1]$'' and\\
1.735 - 07\>\> rule6: ``$\vert\vert z \vert\vert > 1 \Rightarrow z/(z - 1)^2 = n \cdot u [n]$''
1.736 - \end{tabbing}}
1.737 -% \end{example}
1.738 -%}
1.739 -These 6 rules can be used as conditional rewrite rules, depending on
1.740 -the respective convergence radius. Satisfaction from accordance with traditional
1.741 -notation contrasts with the above word {\em axiomatization}: As TP-based, the
1.742 -programming language expects these rules as {\em proved} theorems, and
1.743 -not as axioms implemented in the above brute force manner; otherwise
1.744 -all the verification efforts envisaged (like proof of the
1.745 -post-condition, see below) would be meaningless.
1.746 -
1.747 -Isabelle provides a large body of knowledge, rigorously proved from
1.748 -the basic axioms of mathematics~\footnote{This way of rigorously
1.749 -deriving all knowledge from first principles is called the
1.750 -LCF-paradigm in TP.}. In the case of the ${\cal Z}$-transform the most advanced
1.751 -knowledge can be found in the theories on Multivariate
1.752 -Analysis~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate\_Analysis}. However,
1.753 -building up knowledge such that a proof for the above rules would be
1.754 -reasonably short and easily comprehensible, still requires lots of
1.755 -work (and is definitely out of scope of our case study).
1.756 -
1.757 -%REMOVED DUE TO SPACE CONSTRAINTS
1.758 -%At the state-of-the-art in mechanization of knowledge in engineering
1.759 -%sciences, the process does not stop with the mechanization of
1.760 -%mathematics traditionally used in these sciences. Rather, ``Formal
1.761 -%Methods''~\cite{ fm-03} are expected to proceed to formal and explicit
1.762 -%description of physical items. Signal Processing, for instance is
1.763 -%concerned with physical devices for signal acquisition and
1.764 -%reconstruction, which involve measuring a physical signal, storing it,
1.765 -%and possibly later rebuilding the original signal or an approximation
1.766 -%thereof. For digital systems, this typically includes sampling and
1.767 -%quantization; devices for signal compression, including audio
1.768 -%compression, image compression, and video compression, etc. ``Domain
1.769 -%engineering''\cite{db:dom-eng} is concerned with {\em specification}
1.770 -%of these devices' components and features; this part in the process of
1.771 -%mechanization is only at the beginning in domains like Signal
1.772 -%Processing.
1.773 -%
1.774 -%TP-based programming, concern of this paper, is determined to
1.775 -%add ``algorithmic knowledge'' to the mechanised body of knowledge.
1.776 -%% in Fig.\ref{fig:mathuni} on
1.777 -%% p.\pageref{fig:mathuni}. As we shall see below, TP-based programming
1.778 -%% starts with a formal {\em specification} of the problem to be solved.
1.779 -%% \begin{figure}
1.780 -%% \begin{center}
1.781 -%% \includegraphics[width=110mm]{../../fig/jrocnik/math-universe-small}
1.782 -%% \caption{The three-dimensional universe of mathematics knowledge}
1.783 -%% \label{fig:mathuni}
1.784 -%% \end{center}
1.785 -%% \end{figure}
1.786 -%% The language for both axes is defined in the axis at the bottom, deductive
1.787 -%% knowledge, in {\sisac} represented by Isabelle's theories.
1.788 -
1.789 -\subsection{Preparation of Simplifiers for the Program}\label{simp}
1.790 -
1.791 -All evaluation in the prototype's Lucas-Interpreter is done by term rewriting on
1.792 -Isabelle's terms, see \S\ref{meth} below; in this section some of respective
1.793 -preparations are described. In order to work reliably with term rewriting, the
1.794 -respective rule-sets must be confluent and terminating~\cite{nipk:rew-all-that},
1.795 -then they are called (canonical) simplifiers. These properties do not go without
1.796 -saying, their establishment is a difficult task for the programmer; this task is
1.797 -not yet supported in the prototype.
1.798 -
1.799 -The prototype rewrites using theorems only. Axioms which are theorems as well
1.800 -have been already shown in \S\ref{eg:neuper1} on p.\pageref{eg:neuper1} , we
1.801 -assemble them in a rule-set and apply them in ML as follows:
1.802 -
1.803 -{\footnotesize
1.804 -\begin{verbatim}
1.805 - 01 val inverse_z = Rls
1.806 - 02 {id = "inverse_z",
1.807 - 03 rew_ord = dummy_ord,
1.808 - 04 erls = Erls,
1.809 - 05 rules = [Thm ("rule1", @{thm rule1}), Thm ("rule2", @{thm rule1}),
1.810 - 06 Thm ("rule3", @{thm rule3}), Thm ("rule4", @{thm rule4}),
1.811 - 07 Thm ("rule5", @{thm rule5}), Thm ("rule6", @{thm rule6})],
1.812 - 08 errpatts = [],
1.813 - 09 scr = ""}
1.814 -\end{verbatim}}
1.815 -
1.816 -\noindent The items, line by line, in the above record have the following purpose:
1.817 -\begin{description}
1.818 -\item[01..02] the ML-value \textit{inverse\_z} stores it's identifier
1.819 -as a string for ``reflection'' when switching between the language
1.820 -layers of Isabelle/ML (like in the Lucas-Interpreter) and
1.821 -Isabelle/Isar (like in the example program on p.\pageref{s:impl} on
1.822 -line {\rm 12}).
1.823 -
1.824 -\item[03..04] both, (a) the rewrite-order~\cite{nipk:rew-all-that}
1.825 -\textit{rew\_ord} and (b) the rule-set \textit{erls} are trivial here:
1.826 -(a) the \textit{rules} in {\rm 07..12} don't need ordered rewriting
1.827 -and (b) the assumptions of the \textit{rules} need not be evaluated
1.828 -(they just go into the context during rewriting).
1.829 -
1.830 -\item[05..07] the \textit{rules} are the axioms from p.\pageref{eg:neuper1};
1.831 -also ML-functions (\S\ref{funs}) can come into this list as shown in
1.832 -\S\ref{flow-prep}; so they are distinguished by type-constructors \textit{Thm}
1.833 -and \textit{Calc} respectively; for the purpose of reflection both
1.834 -contain their identifiers.
1.835 -
1.836 -\item[08..09] are error-patterns not discussed here and \textit{scr}
1.837 -is prepared to get a program, automatically generated by {\sisac} for
1.838 -producing intermediate rewrites when requested by the user.
1.839 -
1.840 -\end{description}
1.841 -
1.842 -%OUTCOMMENTED DUE TO SPACE RESTRICTIONS
1.843 -% \noindent It is advisable to immediately test rule-sets; for that
1.844 -% purpose an appropriate term has to be created; \textit{parse} takes a
1.845 -% context \textit{ctxt} and a string (with \textit{ZZ\_1} denoting ${\cal
1.846 -% Z}^{-1}$) and creates a term:
1.847 -%
1.848 -% {\footnotesize
1.849 -% \begin{verbatim}
1.850 -% 01 ML {*
1.851 -% 02 val t = parse ctxt "ZZ_1 (z / (z - 1) + z / (z - </alpha>) + 1)";
1.852 -% 03 *}
1.853 -% 04 val t = Const ("Build_Inverse_Z_Transform.ZZ_1",
1.854 -% 05 "RealDef.real => RealDef.real => RealDef.real") $
1.855 -% 06 (Const (...) $ (Const (...) $ Free (...) $ (Const (...) $ Free (...)
1.856 -% \end{verbatim}}
1.857 -%
1.858 -% \noindent The internal representation of the term, as required for
1.859 -% rewriting, consists of \textit{Const}ants, a pair of a string
1.860 -% \textit{"Groups.plus\_class.plus"} for $+$ and a type, variables
1.861 -% \textit{Free} and the respective constructor \textit{\$}. Now the
1.862 -% term can be rewritten by the rule-set \textit{inverse\_z}:
1.863 -%
1.864 -% {\footnotesize
1.865 -% \begin{verbatim}
1.866 -% 01 ML {*
1.867 -% 02 val SOME (t', asm) = rewrite_set_ @{theory} inverse\_z t;
1.868 -% 03 term2str t';
1.869 -% 04 terms2str asm;
1.870 -% 05 *}
1.871 -% 06 val it = "u[n] + </alpha> ^ n * u[n] + </delta>[n]" : string
1.872 -% 07 val it = "|| z || > 1 & || z || > </alpha>" : string
1.873 -% \end{verbatim}}
1.874 -%
1.875 -% \noindent The resulting term \textit{t} and the assumptions
1.876 -% \textit{asm} are converted to readable strings by \textit{term2str}
1.877 -% and \textit{terms2str}.
1.878 -
1.879 -\subsection{Preparation of ML-Functions}\label{funs}
1.880 -Some functionality required in programming, cannot be accomplished by
1.881 -rewriting. So the prototype has a mechanism to call functions within
1.882 -the rewrite-engine: certain redexes in Isabelle terms call these
1.883 -functions written in SML~\cite{pl:milner97}, the implementation {\em
1.884 -and} meta-language of Isabelle. The programmer has to use this
1.885 -mechanism.
1.886 -
1.887 -In the running example's program on p.\pageref{s:impl} the lines {\rm
1.888 -05} and {\rm 06} contain such functions; we go into the details with
1.889 -\textit{argument\_in X\_z;}. This function fetches the argument from a
1.890 -function application: Line {\rm 03} in the example calculation on
1.891 -p.\pageref{exp-calc} is created by line {\rm 06} of the example
1.892 -program on p.\pageref{s:impl} where the program's environment assigns
1.893 -the value \textit{X z} to the variable \textit{X\_z}; so the function
1.894 -shall extract the argument \textit{z}.
1.895 -
1.896 -\medskip In order to be recognised as a function constant in the
1.897 -program source the constant needs to be declared in a theory, here in
1.898 -\textit{Build\_Inverse\_Z\_Transform.thy}; then it can be parsed in
1.899 -the context \textit{ctxt} of that theory:
1.900 -
1.901 -{\footnotesize
1.902 -\begin{verbatim}
1.903 -01 consts
1.904 -02 argument'_in :: "real => real" ("argument'_in _" 10)
1.905 -\end{verbatim}}
1.906 -
1.907 -%^3.2^ ML {* val SOME t = parse ctxt "argument_in (X z)"; *}
1.908 -%^3.2^ val t = Const ("Build_Inverse_Z_Transform.argument'_in", "RealDef.real ⇒ RealDef.real")
1.909 -%^3.2^ $ (Free ("X", "RealDef.real ⇒ RealDef.real") $ Free ("z", "RealDef.real")): term
1.910 -%^3.2^ \end{verbatim}}
1.911 -%^3.2^
1.912 -%^3.2^ \noindent Parsing produces a term \texttt{t} in internal
1.913 -%^3.2^ representation~\footnote{The attentive reader realizes the
1.914 -%^3.2^ differences between interal and extermal representation even in the
1.915 -%^3.2^ strings, i.e \texttt{'\_}}, consisting of \texttt{Const
1.916 -%^3.2^ ("argument'\_in", type)} and the two variables \texttt{Free ("X",
1.917 -%^3.2^ type)} and \texttt{Free ("z", type)}, \texttt{\$} is the term
1.918 -%^3.2^ constructor.
1.919 -The function body below is implemented directly in SML,
1.920 -i.e in an \texttt{ML \{* *\}} block; the function definition provides
1.921 -a unique prefix \texttt{eval\_} to the function name:
1.922 -
1.923 -{\footnotesize
1.924 -\begin{verbatim}
1.925 -01 ML {*
1.926 -02 fun eval_argument_in _
1.927 -03 "Build_Inverse_Z_Transform.argument'_in"
1.928 -04 (t as (Const ("Build_Inverse_Z_Transform.argument'_in", _) $(f $arg))) _ =
1.929 -05 if is_Free arg (*could be something to be simplified before*)
1.930 -06 then SOME (term2str t ^"="^ term2str arg, Trueprop $(mk_equality (t, arg)))
1.931 -07 else NONE
1.932 -08 | eval_argument_in _ _ _ _ = NONE;
1.933 -09 *}
1.934 -\end{verbatim}}
1.935 -
1.936 -\noindent The function body creates either \texttt{NONE}
1.937 -telling the rewrite-engine to search for the next redex, or creates an
1.938 -ad-hoc theorem for rewriting, thus the programmer needs to adopt many
1.939 -technicalities of Isabelle, for instance, the \textit{Trueprop}
1.940 -constant.
1.941 -
1.942 -\bigskip This sub-task particularly sheds light on basic issues in the
1.943 -design of a programming language, the integration of differential language
1.944 -layers, the layer of Isabelle/Isar and Isabelle/ML.
1.945 -
1.946 -Another point of improvement for the prototype is the rewrite-engine: The
1.947 -program on p.\pageref{s:impl} would not allow to contract the two lines {\rm 05}
1.948 -and {\rm 06} to
1.949 -
1.950 -{\small\it\label{s:impl}
1.951 -\begin{tabbing}
1.952 -123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
1.953 -\>{\rm 05/06}\>\>\> (z::real) = argument\_in (lhs X\_eq) ;
1.954 -\end{tabbing}}
1.955 -
1.956 -\noindent because nested function calls would require creating redexes
1.957 -inside-out; however, the prototype's rewrite-engine only works top down
1.958 -from the root of a term down to the leaves.
1.959 -
1.960 -How all these technicalities are to be checked in the prototype is
1.961 -shown in \S\ref{flow-prep} below.
1.962 -
1.963 -% \paragraph{Explicit Problems} require explicit methods to solve them, and within
1.964 -% this methods we have some explicit steps to do. This steps can be unique for
1.965 -% a special problem or refindable in other problems. No mather what case, such
1.966 -% steps often require some technical functions behind. For the solving process
1.967 -% of the Inverse Z Transformation and the corresponding partial fraction it was
1.968 -% neccessary to build helping functions like \texttt{get\_denominator},
1.969 -% \texttt{get\_numerator} or \texttt{argument\_in}. First two functions help us
1.970 -% to filter the denominator or numerator out of a fraction, last one helps us to
1.971 -% get to know the bound variable in a equation.
1.972 -% \par
1.973 -% By taking \texttt{get\_denominator} as an example, we want to explain how to
1.974 -% implement new functions into the existing system and how we can later use them
1.975 -% in our program.
1.976 -%
1.977 -% \subsubsection{Find a place to Store the Function}
1.978 -%
1.979 -% The whole system builds up on a well defined structure of Knowledge. This
1.980 -% Knowledge sets up at the Path:
1.981 -% \begin{center}\ttfamily src/Tools/isac/Knowledge\normalfont\end{center}
1.982 -% For implementing the Function \texttt{get\_denominator} (which let us extract
1.983 -% the denominator out of a fraction) we have choosen the Theory (file)
1.984 -% \texttt{Rational.thy}.
1.985 -%
1.986 -% \subsubsection{Write down the new Function}
1.987 -%
1.988 -% In upper Theory we now define the new function and its purpose:
1.989 -% \begin{verbatim}
1.990 -% get_denominator :: "real => real"
1.991 -% \end{verbatim}
1.992 -% This command tells the machine that a function with the name
1.993 -% \texttt{get\_denominator} exists which gets a real expression as argument and
1.994 -% returns once again a real expression. Now we are able to implement the function
1.995 -% itself, upcoming example now shows the implementation of
1.996 -% \texttt{get\_denominator}.
1.997 -%
1.998 -% %\begin{example}
1.999 -% \label{eg:getdenom}
1.1000 -% \begin{verbatim}
1.1001 -%
1.1002 -% 01 (*
1.1003 -% 02 *("get_denominator",
1.1004 -% 03 * ("Rational.get_denominator", eval_get_denominator ""))
1.1005 -% 04 *)
1.1006 -% 05 fun eval_get_denominator (thmid:string) _
1.1007 -% 06 (t as Const ("Rational.get_denominator", _) $
1.1008 -% 07 (Const ("Rings.inverse_class.divide", _) $num
1.1009 -% 08 $denom)) thy =
1.1010 -% 09 SOME (mk_thmid thmid ""
1.1011 -% 10 (Print_Mode.setmp []
1.1012 -% 11 (Syntax.string_of_term (thy2ctxt thy)) denom) "",
1.1013 -% 12 Trueprop $ (mk_equality (t, denom)))
1.1014 -% 13 | eval_get_denominator _ _ _ _ = NONE;\end{verbatim}
1.1015 -% %\end{example}
1.1016 -%
1.1017 -% Line \texttt{07} and \texttt{08} are describing the mode of operation the best -
1.1018 -% there is a fraction\\ (\ttfamily Rings.inverse\_class.divide\normalfont)
1.1019 -% splittet
1.1020 -% into its two parts (\texttt{\$num \$denom}). The lines before are additionals
1.1021 -% commands for declaring the function and the lines after are modeling and
1.1022 -% returning a real variable out of \texttt{\$denom}.
1.1023 -%
1.1024 -% \subsubsection{Add a test for the new Function}
1.1025 -%
1.1026 -% \paragraph{Everytime when adding} a new function it is essential also to add
1.1027 -% a test for it. Tests for all functions are sorted in the same structure as the
1.1028 -% knowledge it self and can be found up from the path:
1.1029 -% \begin{center}\ttfamily test/Tools/isac/Knowledge\normalfont\end{center}
1.1030 -% This tests are nothing very special, as a first prototype the functionallity
1.1031 -% of a function can be checked by evaluating the result of a simple expression
1.1032 -% passed to the function. Example~\ref{eg:getdenomtest} shows the test for our
1.1033 -% \textit{just} created function \texttt{get\_denominator}.
1.1034 -%
1.1035 -% %\begin{example}
1.1036 -% \label{eg:getdenomtest}
1.1037 -% \begin{verbatim}
1.1038 -%
1.1039 -% 01 val thy = @{theory Isac};
1.1040 -% 02 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
1.1041 -% 03 val SOME (_, t') = eval_get_denominator "" 0 t thy;
1.1042 -% 04 if term2str t' = "get_denominator ((a + x) / b) = b" then ()
1.1043 -% 05 else error "get_denominator ((a + x) / b) = b" \end{verbatim}
1.1044 -% %\end{example}
1.1045 -%
1.1046 -% \begin{description}
1.1047 -% \item[01] checks if the proofer set up on our {\sisac{}} System.
1.1048 -% \item[02] passes a simple expression (fraction) to our suddenly created
1.1049 -% function.
1.1050 -% \item[04] checks if the resulting variable is the correct one (in this case
1.1051 -% ``b'' the denominator) and returns.
1.1052 -% \item[05] handels the error case and reports that the function is not able to
1.1053 -% solve the given problem.
1.1054 -% \end{description}
1.1055 -
1.1056 -\subsection{Specification of the Problem}\label{spec}
1.1057 -%WN <--> \chapter 7 der Thesis
1.1058 -%WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..
1.1059 -
1.1060 -Mechanical treatment requires to translate a textual problem
1.1061 -description like in Fig.\ref{fig-interactive} on
1.1062 -p.\pageref{fig-interactive} into a {\em formal} specification. The
1.1063 -formal specification of the running example could look like is this
1.1064 -~\footnote{The ``TODO'' in the postcondition indicates, that postconditions
1.1065 -are not yet handled in the prototype; in particular, the postcondition, i.e.
1.1066 -the correctness of the result is not yet automatically proved.}:
1.1067 -
1.1068 -%WN Hier brauchen wir die Spezifikation des 'running example' ...
1.1069 -%JR Habe input, output und precond vom Beispiel eingefügt brauche aber Hilfe bei
1.1070 -%JR der post condition - die existiert für uns ja eigentlich nicht aka
1.1071 -%JR haben sie bis jetzt nicht beachtet WN...
1.1072 -%WN2 Mein Vorschlag ist, das TODO zu lassen und deutlich zu kommentieren.
1.1073 -%JR2 done
1.1074 -
1.1075 -\label{eg:neuper2}
1.1076 -{\small\begin{tabbing}
1.1077 - 123\=123\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill
1.1078 - %\hfill \\
1.1079 - \>Specification:\\
1.1080 - \> \>input \>: ${\it filterExpression} \;\;X\;z=\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}, \;{\it domain}\;\mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$\\
1.1081 - \>\>precond \>: $\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}\;\; {\it continuous\_on}\; \mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$ \\
1.1082 - \>\>output \>: stepResponse $x[n]$ \\
1.1083 - \>\>postcond \>: TODO
1.1084 -\end{tabbing}}
1.1085 -
1.1086 -%JR wie besprochen, kein remark, keine begründung, nur simples "nicht behandelt"
1.1087 -
1.1088 -% \begin{remark}
1.1089 -% Defining the postcondition requires a high amount mathematical
1.1090 -% knowledge, the difficult part in our case is not to set up this condition
1.1091 -% nor it is more to define it in a way the interpreter is able to handle it.
1.1092 -% Due the fact that implementing that mechanisms is quite the same amount as
1.1093 -% creating the programm itself, it is not avaible in our prototype.
1.1094 -% \label{rm:postcond}
1.1095 -% \end{remark}
1.1096 -
1.1097 -The implementation of the formal specification in the present
1.1098 -prototype, still bar-bones without support for authoring, is done
1.1099 -like that:
1.1100 -%WN Kopie von Inverse_Z_Transform.thy, leicht versch"onert:
1.1101 -
1.1102 -{\footnotesize\label{exp-spec}
1.1103 -\begin{verbatim}
1.1104 - 00 ML {*
1.1105 - 01 store_specification
1.1106 - 02 (prepare_specification
1.1107 - 03 "pbl_SP_Ztrans_inv"
1.1108 - 04 ["Jan Rocnik"]
1.1109 - 05 thy
1.1110 - 06 ( ["Inverse", "Z_Transform", "SignalProcessing"],
1.1111 - 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1.1112 - 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1.1113 - 09 ("#Find" , ["stepResponse n_eq"]),
1.1114 - 10 ("#Post" , [" TODO "])])
1.1115 - 11 prls
1.1116 - 12 NONE
1.1117 - 13 [["SignalProcessing","Z_Transform","Inverse"]]);
1.1118 - 14 *}
1.1119 -\end{verbatim}}
1.1120 -
1.1121 -Although the above details are partly very technical, we explain them
1.1122 -in order to document some intricacies of TP-based programming in the
1.1123 -present state of the {\sisac} prototype:
1.1124 -\begin{description}
1.1125 -\item[01..02]\textit{store\_specification:} stores the result of the
1.1126 -function \textit{prep\_specification} in a global reference
1.1127 -\textit{Unsynchronized.ref}, which causes principal conflicts with
1.1128 -Isabelle's asynchronous document model~\cite{Wenzel-11:doc-orient} and
1.1129 -parallel execution~\cite{Makarius-09:parall-proof} and is under
1.1130 -reconstruction already.
1.1131 -
1.1132 -\textit{prep\_specification:} translates the specification to an internal format
1.1133 -which allows efficient processing; see for instance line {\rm 07}
1.1134 -below.
1.1135 -\item[03..04] are a unique identifier for the specification within {\sisac}
1.1136 -and the ``mathematics author'' holding the copy-rights.
1.1137 -\item[05] is the Isabelle \textit{theory} required to parse the
1.1138 -specification in lines {\rm 07..10}.
1.1139 -\item[06] is a key into the tree of all specifications as presented to
1.1140 -the user (where some branches might be hidden by the dialogue
1.1141 -component).
1.1142 -\item[07..10] are the specification with input, pre-condition, output
1.1143 -and post-condition respectively; note that the specification contains
1.1144 -variables to be instantiated with concrete values for a concrete problem ---
1.1145 -thus the specification actually captures a class of problems. The post-condition is not handled in
1.1146 -the prototype presently.
1.1147 -\item[11] is a rule-set (defined elsewhere) for evaluation of the pre-condition: \textit{(rhs X\_eq) is\_continuous\_in D}, instantiated with the values of a concrete problem, evaluates to true or false --- and all evaluation is done by
1.1148 -rewriting determined by rule-sets.
1.1149 -\item[12]\textit{NONE:} could be \textit{SOME ``solve ...''} for a
1.1150 -problem associated to a function from Computer Algebra (like an
1.1151 -equation solver) which is not the case here.
1.1152 -\item[13] is a list of methods solving the specified problem (here
1.1153 -only one list item) represented analogously to {\rm 06}.
1.1154 -\end{description}
1.1155 -
1.1156 -
1.1157 -%WN die folgenden Erkl"arungen finden sich durch "grep -r 'datatype pbt' *"
1.1158 -%WN ...
1.1159 -% type pbt =
1.1160 -% {guh : guh, (*unique within this isac-knowledge*)
1.1161 -% mathauthors: string list, (*copyright*)
1.1162 -% init : pblID, (*to start refinement with*)
1.1163 -% thy : theory, (* which allows to compile that pbt
1.1164 -% TODO: search generalized for subthy (ref.p.69*)
1.1165 -% (*^^^ WN050912 NOT used during application of the problem,
1.1166 -% because applied terms may be from 'subthy' as well as from super;
1.1167 -% thus we take 'maxthy'; see match_ags !*)
1.1168 -% cas : term option,(*'CAS-command'*)
1.1169 -% prls : rls, (* for preds in where_*)
1.1170 -% where_: term list, (* where - predicates*)
1.1171 -% ppc : pat list,
1.1172 -% (*this is the model-pattern;
1.1173 -% it contains "#Given","#Where","#Find","#Relate"-patterns
1.1174 -% for constraints on identifiers see "fun cpy_nam"*)
1.1175 -% met : metID list}; (* methods solving the pbt*)
1.1176 -%
1.1177 -%WN weil dieser Code sehr unaufger"aumt ist, habe ich die Erkl"arungen
1.1178 -%WN oben selbst geschrieben.
1.1179 -
1.1180 -
1.1181 -
1.1182 -
1.1183 -%WN das w"urde ich in \sec\label{progr} verschieben und
1.1184 -%WN das SubProblem partial fractions zum Erkl"aren verwenden.
1.1185 -% Such a specification is checked before the execution of a program is
1.1186 -% started, the same applies for sub-programs. In the following example
1.1187 -% (Example~\ref{eg:subprob}) shows the call of such a subproblem:
1.1188 -%
1.1189 -% \vbox{
1.1190 -% \begin{example}
1.1191 -% \label{eg:subprob}
1.1192 -% \hfill \\
1.1193 -% {\ttfamily \begin{tabbing}
1.1194 -% ``(L\_L::bool list) = (\=SubProblem (\=Test','' \\
1.1195 -% ``\>\>[linear,univariate,equation,test],'' \\
1.1196 -% ``\>\>[Test,solve\_linear])'' \\
1.1197 -% ``\>[BOOL equ, REAL z])'' \\
1.1198 -% \end{tabbing}
1.1199 -% }
1.1200 -% {\small\textit{
1.1201 -% \noindent If a program requires a result which has to be
1.1202 -% calculated first we can use a subproblem to do so. In our specific
1.1203 -% case we wanted to calculate the zeros of a fraction and used a
1.1204 -% subproblem to calculate the zeros of the denominator polynom.
1.1205 -% }}
1.1206 -% \end{example}
1.1207 -% }
1.1208 -
1.1209 -\subsection{Implementation of the Method}\label{meth}
1.1210 -A method collects all data required to interpret a certain program by
1.1211 -Lucas-Interpretation. The \texttt{program} from p.\pageref{s:impl} of
1.1212 -the running example is embedded on the last line in the following method:
1.1213 -%The methods represent the different ways a problem can be solved. This can
1.1214 -%include mathematical tactics as well as tactics taught in different courses.
1.1215 -%Declaring the Method itself gives us the possibilities to describe the way of
1.1216 -%calculation in deep, as well we get the oppertunities to build in different
1.1217 -%rulesets.
1.1218 -
1.1219 -{\footnotesize
1.1220 -\begin{verbatim}
1.1221 - 00 ML {*
1.1222 - 01 store_method
1.1223 - 02 (prep_method
1.1224 - 03 "SP_InverseZTransformation_classic"
1.1225 - 04 ["Jan Rocnik"]
1.1226 - 05 thy
1.1227 - 06 ( ["SignalProcessing", "Z_Transform", "Inverse"],
1.1228 - 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1.1229 - 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1.1230 - 09 ("#Find" , ["stepResponse n_eq"]),
1.1231 - 10 rew_ord erls
1.1232 - 11 srls prls nrls
1.1233 - 12 errpats
1.1234 - 13 program);
1.1235 - 14 *}
1.1236 -\end{verbatim}}
1.1237 -
1.1238 -\noindent The above code stores the whole structure analogously to a
1.1239 -specification as described above:
1.1240 -\begin{description}
1.1241 -\item[01..06] are identical to those for the example specification on
1.1242 -p.\pageref{exp-spec}.
1.1243 -
1.1244 -\item[07..09] show something looking like the specification; this is a
1.1245 -{\em guard}: as long as not all \textit{Given} items are present and
1.1246 -the \textit{Pre}-conditions is not true, interpretation of the program
1.1247 -is not started.
1.1248 -
1.1249 -\item[10..11] all concern rewriting (the respective data are defined elsewhere): \textit{rew\_ord} is the rewrite order~\cite{nipk:rew-all-that} in case
1.1250 -\textit{program} contains a \textit{Rewrite} tactic; and in case the respective rule is a conditional rewrite-rule, \textit{erls} features evaluating the conditions. The rule-sets
1.1251 -\textit{srls, prls, nrls} feature evaluating (a) the ML-functions in the program (e.g.
1.1252 -\textit{lhs, argument\_in, rhs} in the program on p.\pageref{s:impl}, (b) the pre-condition analogous to the specification in line 11 on p.\pageref{exp-spec}
1.1253 -and (c) is required for the derivation-machinery checking user-input formulas.
1.1254 -
1.1255 -\item[12..13] \textit{errpats} are error-patterns~\cite{gdaroczy-EP-13} for this method and \textit{program} is the variable holding the example from p.\pageref {s:impl}.
1.1256 -\end{description}
1.1257 -The many rule-sets above cause considerable efforts for the
1.1258 -programmers, in particular, because there are no tools for checking
1.1259 -essential features of rule-sets.
1.1260 -
1.1261 -% is again very technical and goes hard in detail. Unfortunataly
1.1262 -% most declerations are not essential for a basic programm but leads us to a huge
1.1263 -% range of powerful possibilities.
1.1264 -%
1.1265 -% \begin{description}
1.1266 -% \item[01..02] stores the method with the given name into the system under a global
1.1267 -% reference.
1.1268 -% \item[03] specifies the topic within which context the method can be found.
1.1269 -% \item[04..05] as the requirements for different methods can be deviant we
1.1270 -% declare what is \emph{given} and and what to \emph{find} for this specific method.
1.1271 -% The code again helds on the topic of the case studie, where the inverse
1.1272 -% z-transformation does a switch between a term describing a electrical filter into
1.1273 -% its step response. Also the datatype has to be declared (bool - due the fact that
1.1274 -% we handle equations).
1.1275 -% \item[06] \emph{rewrite order} is the order of this rls (ruleset), where one
1.1276 -% theorem of it is used for rewriting one single step.
1.1277 -% \item[07] \texttt{rls} is the currently used ruleset for this method. This set
1.1278 -% has already been defined before.
1.1279 -% \item[08] we would have the possiblitiy to add this method to a predefined tree of
1.1280 -% calculations, i.eg. if it would be a sub of a bigger problem, here we leave it
1.1281 -% independend.
1.1282 -% \item[09] The \emph{source ruleset}, can be used to evaluate list expressions in
1.1283 -% the source.
1.1284 -% \item[10] \emph{predicates ruleset} can be used to indicates predicates within
1.1285 -% model patterns.
1.1286 -% \item[11] The \emph{check ruleset} summarizes rules for checking formulas
1.1287 -% elementwise.
1.1288 -% \item[12] \emph{error patterns} which are expected in this kind of method can be
1.1289 -% pre-specified to recognize them during the method.
1.1290 -% \item[13] finally the \emph{canonical ruleset}, declares the canonical simplifier
1.1291 -% of the specific method.
1.1292 -% \item[14] for this code snipset we don't specify the programm itself and keep it
1.1293 -% empty. Follow up \S\ref{progr} for informations on how to implement this
1.1294 -% \textit{main} part.
1.1295 -% \end{description}
1.1296 -
1.1297 -\subsection{Implementation of the TP-based Program}\label{progr}
1.1298 -So finally all the prerequisites are described and the final task can
1.1299 -be addressed. The program below comes back to the running example: it
1.1300 -computes a solution for the problem from Fig.\ref{fig-interactive} on
1.1301 -p.\pageref{fig-interactive}. The reader is reminded of
1.1302 -\S\ref{PL-isab}, the introduction of the programming language:
1.1303 -
1.1304 -{\footnotesize\it\label{s:impl}
1.1305 -\begin{tabbing}
1.1306 -123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
1.1307 -\>{\rm 00}\>ML \{*\\
1.1308 -\>{\rm 00}\>val program =\\
1.1309 -\>{\rm 01}\> "{\tt Program} InverseZTransform (X\_eq::bool) = \\
1.1310 -\>{\rm 02}\>\> {\tt let} \\
1.1311 -\>{\rm 03}\>\>\> X\_eq = {\tt Take} X\_eq ; \\
1.1312 -\>{\rm 04}\>\>\> X\_eq = {\tt Rewrite} prep\_for\_part\_frac X\_eq ; \\
1.1313 -\>{\rm 05}\>\>\> (X\_z::real) = lhs X\_eq ; \\ %no inside-out evaluation
1.1314 -\>{\rm 06}\>\>\> (z::real) = argument\_in X\_z; \\
1.1315 -\>{\rm 07}\>\>\> (part\_frac::real) = {\tt SubProblem} \\
1.1316 -\>{\rm 08}\>\>\>\>\>\>\>\> ( Isac, [partial\_fraction, rational, simplification], [] )\\
1.1317 -%\>{\rm 10}\>\>\>\>\>\>\>\>\> [simplification, of\_rationals, to\_partial\_fraction] ) \\
1.1318 -\>{\rm 09}\>\>\>\>\>\>\>\> [ (rhs X\_eq)::real, z::real ]; \\
1.1319 -\>{\rm 10}\>\>\> (X'\_eq::bool) = {\tt Take} ((X'::real =$>$ bool) z = ZZ\_1 part\_frac) ; \\
1.1320 -\>{\rm 11}\>\>\> X'\_eq = (({\tt Rewrite\_Set} prep\_for\_inverse\_z) @@ \\
1.1321 -\>{\rm 12}\>\>\>\>\> $\;\;$ ({\tt Rewrite\_Set} inverse\_z)) X'\_eq \\
1.1322 -\>{\rm 13}\>\> {\tt in } \\
1.1323 -\>{\rm 14}\>\>\> X'\_eq"\\
1.1324 -\>{\rm 15}\>*\}
1.1325 -\end{tabbing}}
1.1326 -% ORIGINAL FROM Inverse_Z_Transform.thy
1.1327 -% "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1.1328 -% "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1.1329 -% " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1.1330 -% " (X'_z::real) = lhs X'; "^(* ?X' z*)
1.1331 -% " (zzz::real) = argument_in X'_z; "^(* z *)
1.1332 -% " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1.1333 -%
1.1334 -% " (pbz::real) = (SubProblem (Isac', "^(**)
1.1335 -% " [partial_fraction,rational,simplification], "^
1.1336 -% " [simplification,of_rationals,to_partial_fraction]) "^
1.1337 -% " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1.1338 -%
1.1339 -% " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1.1340 -% " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1.1341 -% " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1.1342 -% " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1.1343 -% " n_eq = (Rewrite_Set inverse_z False) X_zeq; "^(*([4], Res), X_z = 4 * (1 / 2) ^^^ ?n * ?u [?n] + -4 * (-1 / 4) ^^^ ?n * ?u [?n]*)
1.1344 -% " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1.1345 -% "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1.1346 -The program is represented as a string and part of the method in
1.1347 -\S\ref{meth}. As mentioned in \S\ref{PL} the program is purely
1.1348 -functional and lacks any input statements and output statements. So
1.1349 -the steps of calculation towards a solution (and interactive tutoring
1.1350 -in step-wise problem solving) are created as a side-effect by
1.1351 -Lucas-Interpretation. The side-effects are triggered by the tactics
1.1352 -\texttt{Take}, \texttt{Rewrite}, \texttt{SubProblem} and
1.1353 -\texttt{Rewrite\_Set} in the above lines {\rm 03, 04, 07, 10, 11} and
1.1354 -{\rm 12} respectively. These tactics produce the respective lines in the
1.1355 -calculation on p.\pageref{flow-impl}.
1.1356 -
1.1357 -The above lines {\rm 05, 06} do not contain a tactics, so they do not
1.1358 -immediately contribute to the calculation on p.\pageref{flow-impl};
1.1359 -rather, they compute actual arguments for the \texttt{SubProblem} in
1.1360 -line {\rm 09}~\footnote{The tactics also are break-points for the
1.1361 -interpreter, where control is handed over to the user in interactive
1.1362 -tutoring.}. Line {\rm 11} contains tactical \textit{@@}.
1.1363 -
1.1364 -\medskip The above program also indicates the dominant role of interactive
1.1365 -selection of knowledge in the three-dimensional universe of
1.1366 -mathematics. The \texttt{SubProblem} in the above lines
1.1367 -{\rm 07..09} is more than a function call with the actual arguments
1.1368 -\textit{[ (rhs X\_eq)::real, z::real ]}. The programmer has to determine
1.1369 -three items:
1.1370 -
1.1371 -\begin{enumerate}
1.1372 -\item the theory, in the example \textit{Isac} because different
1.1373 -methods can be selected in Pt.3 below, which are defined in different
1.1374 -theories with \textit{Isac} collecting them.
1.1375 -\item the specification identified by \textit{[partial\_fraction,
1.1376 -rational, simplification]} in the tree of specifications; this
1.1377 -specification is analogous to the specification of the main program
1.1378 -described in \S\ref{spec}; the problem is to find a ``partial fraction
1.1379 -decomposition'' for a univariate rational polynomial.
1.1380 -\item the method in the above example is \textit{[ ]}, i.e. empty,
1.1381 -which supposes the interpreter to select one of the methods predefined
1.1382 -in the specification, for instance in line {\rm 13} in the running
1.1383 -example's specification on p.\pageref{exp-spec}~\footnote{The freedom
1.1384 -(or obligation) for selection carries over to the student in
1.1385 -interactive tutoring.}.
1.1386 -\end{enumerate}
1.1387 -
1.1388 -The program code, above presented as a string, is parsed by Isabelle's
1.1389 -parser --- the program is an Isabelle term. This fact is expected to
1.1390 -simplify verification tasks in the future; on the other hand, this
1.1391 -fact causes troubles in error detection which are discussed as part
1.1392 -of the work-flow in the subsequent section.
1.1393 -
1.1394 -\section{Work-flow of Programming in the Prototype}\label{workflow}
1.1395 -The new prover IDE Isabelle/jEdit~\cite{makar-jedit-12} is a great
1.1396 -step forward for interactive theory and proof development. The
1.1397 -{\sisac}-prototype re-uses this IDE as a programming environment. The
1.1398 -experiences from this re-use show, that the essential components are
1.1399 -available from Isabelle/jEdit. However, additional tools and features
1.1400 -are required to achieve acceptable usability.
1.1401 -
1.1402 -So notable experiences are reported here, also as a requirement
1.1403 -capture for further development of TP-based languages and respective
1.1404 -IDEs.
1.1405 -
1.1406 -\subsection{Preparations and Trials}\label{flow-prep}
1.1407 -The many sub-tasks to be accomplished {\em before} the first line of
1.1408 -program code can be written and tested suggest an approach which
1.1409 -step-wise establishes the prerequisites. The case study underlying
1.1410 -this paper~\cite{jrocnik-bakk} documents the approach in a separate
1.1411 -Isabelle theory,
1.1412 -\textit{Build\_Inverse\_Z\_Transform.thy}~\footnote{http://www.ist.tugraz.at/projects/isac/publ/Build\_Inverse\_Z\_Transform.thy}. Part
1.1413 -II in the study comprises this theory, \LaTeX ed from the theory by
1.1414 -use of Isabelle's document preparation system. This paper resembles
1.1415 -the approach in \S\ref{isabisac} to \S\ref{meth}, which in actual
1.1416 -implementation work involves several iterations.
1.1417 -
1.1418 -\bigskip For instance, only the last step, implementing the program
1.1419 -described in \S\ref{meth}, reveals details required. Let us assume,
1.1420 -this is the ML-function \textit{argument\_in} required in line {\rm 06}
1.1421 -of the example program on p.\pageref{s:impl}; how this function needs
1.1422 -to be implemented in the prototype has been discussed in \S\ref{funs}
1.1423 -already.
1.1424 -
1.1425 -Now let us assume, that calling this function from the program code
1.1426 -does not work; so testing this function is required in order to find out
1.1427 -the reason: type errors, a missing entry of the function somewhere or
1.1428 -even more nasty technicalities \dots
1.1429 -
1.1430 -{\footnotesize
1.1431 -\begin{verbatim}
1.1432 -01 ML {*
1.1433 -02 val SOME t = parseNEW ctxt "argument_in (X (z::real))";
1.1434 -03 val SOME (str, t') = eval_argument_in ""
1.1435 -04 "Build_Inverse_Z_Transform.argument'_in" t 0;
1.1436 -05 term2str t';
1.1437 -06 *}
1.1438 -07 val it = "(argument_in X z) = z": string\end{verbatim}}
1.1439 -
1.1440 -\noindent So, this works: we get an ad-hoc theorem, which used in
1.1441 -rewriting would reduce \texttt{argument\_in X z} to \texttt{z}. Now we check this
1.1442 -reduction and create a rule-set \texttt{rls} for that purpose:
1.1443 -
1.1444 -{\footnotesize
1.1445 -\begin{verbatim}
1.1446 -01 ML {*
1.1447 -02 val rls = append_rls "test" e_rls
1.1448 -03 [Calc ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")]
1.1449 -04 val SOME (t', asm) = rewrite_set_ @{theory} rls t;
1.1450 -05 *}
1.1451 -06 val t' = Free ("z", "RealDef.real"): term
1.1452 -07 val asm = []: term list\end{verbatim}}
1.1453 -
1.1454 -\noindent The resulting term \texttt{t'} is \texttt{Free ("z",
1.1455 -"RealDef.real")}, i.e the variable \texttt{z}, so all is
1.1456 -perfect. Probably we have forgotten to store this function correctly~?
1.1457 -We review the respective \texttt{calclist} (again an
1.1458 -\textit{Unsynchronized.ref} to be removed in order to adjust to
1.1459 -Isabelle/Isar's asynchronous document model):
1.1460 -
1.1461 -{\footnotesize
1.1462 -\begin{verbatim}
1.1463 -01 calclist:= overwritel (! calclist,
1.1464 -02 [("argument_in",
1.1465 -03 ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")),
1.1466 -04 ...
1.1467 -05 ]);\end{verbatim}}
1.1468 -
1.1469 -\noindent The entry is perfect. So what is the reason~? Ah, probably there
1.1470 -is something messed up with the many rule-sets in the method, see \S\ref{meth} ---
1.1471 -right, the function \texttt{argument\_in} is not contained in the respective
1.1472 -rule-set \textit{srls} \dots this just as an example of the intricacies in
1.1473 -debugging a program in the present state of the prototype.
1.1474 -
1.1475 -\subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}
1.1476 -Given all the prerequisites from \S\ref{isabisac} to \S\ref{meth},
1.1477 -usually developed within several iterations, the program can be
1.1478 -assembled; on p.\pageref{s:impl} there is the complete program of the
1.1479 -running example.
1.1480 -
1.1481 -The completion of this program required efforts for several weeks
1.1482 -(after some months of familiarisation with {\sisac}), caused by the
1.1483 -abundance of intricacies indicated above. Also writing the program is
1.1484 -not pleasant, given Isabelle/Isar/ without add-ons for
1.1485 -programming. Already writing and parsing a few lines of program code
1.1486 -is a challenge: the program is an Isabelle term; Isabelle's parser,
1.1487 -however, is not meant for huge terms like the program of the running
1.1488 -example. So reading out the specific error (usually type errors) from
1.1489 -Isabelle's message is difficult.
1.1490 -
1.1491 -\medskip Testing the evaluation of the program has to rely on very
1.1492 -simple tools. Step-wise execution is modeled by a function
1.1493 -\texttt{me}, short for mathematics-engine~\footnote{The interface used
1.1494 -by the front-end which created the calculation on
1.1495 -p.\pageref{fig-interactive} is different from this function}:
1.1496 -%the following is a simplification of the actual function
1.1497 -
1.1498 -{\footnotesize
1.1499 -\begin{verbatim}
1.1500 -01 ML {* me; *}
1.1501 -02 val it = tac -> ctree * pos -> mout * tac * ctree * pos\end{verbatim}}
1.1502 -
1.1503 -\noindent This function takes as arguments a tactic \texttt{tac} which
1.1504 -determines the next step, the step applied to the interpreter-state
1.1505 -\texttt{ctree * pos} as last argument taken. The interpreter-state is
1.1506 -a pair of a tree \texttt{ctree} representing the calculation created
1.1507 -(see the example below) and a position \texttt{pos} in the
1.1508 -calculation. The function delivers a quadruple, beginning with the new
1.1509 -formula \texttt{mout} and the next tactic followed by the new
1.1510 -interpreter-state.
1.1511 -
1.1512 -This function allows to stepwise check the program:
1.1513 -
1.1514 -{\footnotesize\label{ml-check-program}
1.1515 -\begin{verbatim}
1.1516 -01 ML {*
1.1517 -02 val fmz =
1.1518 -03 ["filterExpression (X z = 3 / ((z::real) + 1/10 - 1/50*(1/z)))",
1.1519 -04 "stepResponse (x[n::real]::bool)"];
1.1520 -05 val (dI,pI,mI) =
1.1521 -06 ("Isac",
1.1522 -07 ["Inverse", "Z_Transform", "SignalProcessing"],
1.1523 -08 ["SignalProcessing","Z_Transform","Inverse"]);
1.1524 -09 val (mout, tac, ctree, pos) = CalcTreeTEST [(fmz, (dI, pI, mI))];
1.1525 -10 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1.1526 -11 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1.1527 -12 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1.1528 -13 ...
1.1529 -\end{verbatim}}
1.1530 -
1.1531 -\noindent Several dozens of calls for \texttt{me} are required to
1.1532 -create the lines in the calculation below (including the sub-problems
1.1533 -not shown). When an error occurs, the reason might be located
1.1534 -many steps before: if evaluation by rewriting, as done by the prototype,
1.1535 -fails, then first nothing happens --- the effects come later and
1.1536 -cause unpleasant checks.
1.1537 -
1.1538 -The checks comprise watching the rewrite-engine for many different
1.1539 -kinds of rule-sets (see \S\ref{meth}), the interpreter-state, in
1.1540 -particular the environment and the context at the states position ---
1.1541 -all checks have to rely on simple functions accessing the
1.1542 -\texttt{ctree}. So getting the calculation below (which resembles the
1.1543 -calculation in Fig.\ref{fig-interactive} on p.\pageref{fig-interactive})
1.1544 -is the result of several weeks of development:
1.1545 -
1.1546 -{\small\it\label{exp-calc}
1.1547 -\begin{tabbing}
1.1548 -123l\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=\kill
1.1549 -\>{\rm 01}\> $\bullet$ \> {\tt Problem } (Inverse\_Z\_Transform, [Inverse, Z\_Transform, SignalProcessing]) \`\\
1.1550 -\>{\rm 02}\>\> $\vdash\;\;X z = \frac{3}{z - \frac{1}{4} - \frac{1}{8} \cdot z^{-1}}$ \`{\footnotesize {\tt Take} X\_eq}\\
1.1551 -\>{\rm 03}\>\> $X z = \frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}$ \`{\footnotesize {\tt Rewrite} prep\_for\_part\_frac X\_eq}\\
1.1552 -\>{\rm 04}\>\> $\bullet$\> {\tt Problem } [partial\_fraction,rational,simplification] \`{\footnotesize {\tt SubProblem} \dots}\\
1.1553 -\>{\rm 05}\>\>\> $\vdash\;\;\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=$ \`- - -\\
1.1554 -\>{\rm 06}\>\>\> $\frac{24}{-1 + -2 \cdot z + 8 \cdot z^2}$ \`- - -\\
1.1555 -\>{\rm 07}\>\>\> $\bullet$\> solve ($-1 + -2 \cdot z + 8 \cdot z^2,\;z$ ) \`- - -\\
1.1556 -\>{\rm 08}\>\>\>\> $\vdash$ \> $\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=0$ \`- - -\\
1.1557 -\>{\rm 09}\>\>\>\> $z = \frac{2+\sqrt{-4+8}}{16}\;\lor\;z = \frac{2-\sqrt{-4+8}}{16}$ \`- - -\\
1.1558 -\>{\rm 10}\>\>\>\> $z = \frac{1}{2}\;\lor\;z =$ \_\_\_ \`- - -\\
1.1559 -\> \>\>\>\> \_\_\_ \`- - -\\
1.1560 -\>{\rm 11}\>\> \dots\> $\frac{4}{z - \frac{1}{2}} + \frac{-4}{z - \frac{-1}{4}}$ \`\\
1.1561 -\>{\rm 12}\>\> $X^\prime z = {\cal z}^{-1} (\frac{4}{z - \frac{1}{2}} + \frac{-4}{z - \frac{-1}{4}})$ \`{\footnotesize {\tt Take} ((X'::real =$>$ bool) z = ZZ\_1 part\_frac)}\\
1.1562 -\>{\rm 13}\>\> $X^\prime z = {\cal z}^{-1} (4\cdot\frac{z}{z - \frac{1}{2}} + -4\cdot\frac{z}{z - \frac{-1}{4}})$ \`{\footnotesize{\tt Rewrite\_Set} prep\_for\_inverse\_z X'\_eq }\\
1.1563 -\>{\rm 14}\>\> $X^\prime z = 4\cdot(\frac{1}{2})^n \cdot u [n] + -4\cdot(\frac{-1}{4})^n \cdot u [n]$ \`{\footnotesize {\tt Rewrite\_Set} inverse\_z X'\_eq}\\
1.1564 -\>{\rm 15}\> \dots\> $X^\prime z = 4\cdot(\frac{1}{2})^n \cdot u [n] + -4\cdot(\frac{-1}{4})^n \cdot u [n]$ \`{\footnotesize {\tt Check\_Postcond}}
1.1565 -\end{tabbing}}
1.1566 -The tactics on the right margin of the above calculation are those in
1.1567 -the program on p.\pageref{s:impl} which create the respective formulas
1.1568 -on the left.
1.1569 -% ORIGINAL FROM Inverse_Z_Transform.thy
1.1570 -% "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1.1571 -% "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1.1572 -% " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1.1573 -% " (X'_z::real) = lhs X'; "^(* ?X' z*)
1.1574 -% " (zzz::real) = argument_in X'_z; "^(* z *)
1.1575 -% " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1.1576 -%
1.1577 -% " (pbz::real) = (SubProblem (Isac', "^(**)
1.1578 -% " [partial_fraction,rational,simplification], "^
1.1579 -% " [simplification,of_rationals,to_partial_fraction]) "^
1.1580 -% " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1.1581 -%
1.1582 -% " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1.1583 -% " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1.1584 -% " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1.1585 -% " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1.1586 -% " n_eq = (Rewrite_Set inverse_z False) X_zeq; "^(*([4], Res), X_z = 4 * (1 / 2) ^^^ ?n * ?u [?n] + -4 * (-1 / 4) ^^^ ?n * ?u [?n]*)
1.1587 -% " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1.1588 -% "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1.1589 -
1.1590 -\subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}
1.1591 -Finally \textit{Build\_Inverse\_Z\_Transform.thy} has got the job done
1.1592 -and the knowledge accumulated in it can be distributed to appropriate
1.1593 -theories: the program to \textit{Inverse\_Z\_Transform.thy}, the
1.1594 -sub-problem accomplishing the partial fraction decomposition to
1.1595 -\textit{Partial\_Fractions.thy}. Since there are hacks into Isabelle's
1.1596 -internals, this kind of distribution is not trivial. For instance, the
1.1597 -function \texttt{argument\_in} in \S\ref{funs} explicitly contains a
1.1598 -string with the theory it has been defined in, so this string needs to
1.1599 -be updated from \texttt{Build\_Inverse\_Z\_Transform} to
1.1600 -\texttt{Atools} if that function is transferred to theory
1.1601 -\textit{Atools.thy}.
1.1602 -
1.1603 -In order to obtain the functionality presented in Fig.\ref{fig-interactive} on p.\pageref{fig-interactive} data must be exported from SML-structures to XML.
1.1604 -This process is also rather bare-bones without authoring tools and is
1.1605 -described in detail in the {\sisac} wiki~\footnote{http://www.ist.tugraz.at/isac/index.php/Generate\_representations\_for\_ISAC\_Knowledge}.
1.1606 -
1.1607 -% \newpage
1.1608 -% -------------------------------------------------------------------
1.1609 -%
1.1610 -% Material, falls noch Platz bleibt ...
1.1611 -%
1.1612 -% -------------------------------------------------------------------
1.1613 -%
1.1614 -%
1.1615 -% \subsubsection{Trials on Notation and Termination}
1.1616 -%
1.1617 -% \paragraph{Technical notations} are a big problem for our piece of software,
1.1618 -% but the reason for that isn't a fault of the software itself, one of the
1.1619 -% troubles comes out of the fact that different technical subtopics use different
1.1620 -% symbols and notations for a different purpose. The most famous example for such
1.1621 -% a symbol is the complex number $i$ (in cassique math) or $j$ (in technical
1.1622 -% math). In the specific part of signal processing one of this notation issues is
1.1623 -% the use of brackets --- we use round brackets for analoge signals and squared
1.1624 -% brackets for digital samples. Also if there is no problem for us to handle this
1.1625 -% fact, we have to tell the machine what notation leads to wich meaning and that
1.1626 -% this purpose seperation is only valid for this special topic - signal
1.1627 -% processing.
1.1628 -% \subparagraph{In the programming language} itself it is not possible to declare
1.1629 -% fractions, exponents, absolutes and other operators or remarks in a way to make
1.1630 -% them pretty to read; our only posssiblilty were ASCII characters and a handfull
1.1631 -% greek symbols like: $\alpha, \beta, \gamma, \phi,\ldots$.
1.1632 -% \par
1.1633 -% With the upper collected knowledge it is possible to check if we were able to
1.1634 -% donate all required terms and expressions.
1.1635 -%
1.1636 -% \subsubsection{Definition and Usage of Rules}
1.1637 -%
1.1638 -% \paragraph{The core} of our implemented problem is the Z-Transformation, due
1.1639 -% the fact that the transformation itself would require higher math which isn't
1.1640 -% yet avaible in our system we decided to choose the way like it is applied in
1.1641 -% labratory and problem classes at our university - by applying transformation
1.1642 -% rules (collected in transformation tables).
1.1643 -% \paragraph{Rules,} in {\sisac{}}'s programming language can be designed by the
1.1644 -% use of axiomatizations like shown in Example~\ref{eg:ruledef}
1.1645 -%
1.1646 -% \begin{example}
1.1647 -% \label{eg:ruledef}
1.1648 -% \hfill\\
1.1649 -% \begin{verbatim}
1.1650 -% axiomatization where
1.1651 -% rule1: ``1 = $\delta$[n]'' and
1.1652 -% rule2: ``|| z || > 1 ==> z / (z - 1) = u [n]'' and
1.1653 -% rule3: ``|| z || < 1 ==> z / (z - 1) = -u [-n - 1]''
1.1654 -% \end{verbatim}
1.1655 -% \end{example}
1.1656 -%
1.1657 -% This rules can be collected in a ruleset and applied to a given expression as
1.1658 -% follows in Example~\ref{eg:ruleapp}.
1.1659 -%
1.1660 -% \begin{example}
1.1661 -% \hfill\\
1.1662 -% \label{eg:ruleapp}
1.1663 -% \begin{enumerate}
1.1664 -% \item Store rules in ruleset:
1.1665 -% \begin{verbatim}
1.1666 -% val inverse_Z = append_rls "inverse_Z" e_rls
1.1667 -% [ Thm ("rule1",num_str @{thm rule1}),
1.1668 -% Thm ("rule2",num_str @{thm rule2}),
1.1669 -% Thm ("rule3",num_str @{thm rule3})
1.1670 -% ];\end{verbatim}
1.1671 -% \item Define exression:
1.1672 -% \begin{verbatim}
1.1673 -% val sample_term = str2term "z/(z-1)+z/(z-</delta>)+1";\end{verbatim}
1.1674 -% \item Apply ruleset:
1.1675 -% \begin{verbatim}
1.1676 -% val SOME (sample_term', asm) =
1.1677 -% rewrite_set_ thy true inverse_Z sample_term;\end{verbatim}
1.1678 -% \end{enumerate}
1.1679 -% \end{example}
1.1680 -%
1.1681 -% The use of rulesets makes it much easier to develop our designated applications,
1.1682 -% but the programmer has to be careful and patient. When applying rulesets
1.1683 -% two important issues have to be mentionend:
1.1684 -% \subparagraph{How often} the rules have to be applied? In case of
1.1685 -% transformations it is quite clear that we use them once but other fields
1.1686 -% reuqire to apply rules until a special condition is reached (e.g.
1.1687 -% a simplification is finished when there is nothing to be done left).
1.1688 -% \subparagraph{The order} in which rules are applied often takes a big effect
1.1689 -% and has to be evaluated for each purpose once again.
1.1690 -% \par
1.1691 -% In our special case of Signal Processing and the rules defined in
1.1692 -% Example~\ref{eg:ruledef} we have to apply rule~1 first of all to transform all
1.1693 -% constants. After this step has been done it no mather which rule fit's next.
1.1694 -%
1.1695 -% \subsubsection{Helping Functions}
1.1696 -%
1.1697 -% \paragraph{New Programms require,} often new ways to get through. This new ways
1.1698 -% means that we handle functions that have not been in use yet, they can be
1.1699 -% something special and unique for a programm or something famous but unneeded in
1.1700 -% the system yet. In our dedicated example it was for example neccessary to split
1.1701 -% a fraction into numerator and denominator; the creation of such function and
1.1702 -% even others is described in upper Sections~\ref{simp} and \ref{funs}.
1.1703 -%
1.1704 -% \subsubsection{Trials on equation solving}
1.1705 -% %simple eq and problem with double fractions/negative exponents
1.1706 -% \paragraph{The Inverse Z-Transformation} makes it neccessary to solve
1.1707 -% equations degree one and two. Solving equations in the first degree is no
1.1708 -% problem, wether for a student nor for our machine; but even second degree
1.1709 -% equations can lead to big troubles. The origin of this troubles leads from
1.1710 -% the build up process of our equation solving functions; they have been
1.1711 -% implemented some time ago and of course they are not as good as we want them to
1.1712 -% be. Wether or not following we only want to show how cruel it is to build up new
1.1713 -% work on not well fundamentials.
1.1714 -% \subparagraph{A simple equation solving,} can be set up as shown in the next
1.1715 -% example:
1.1716 -%
1.1717 -% \begin{example}
1.1718 -% \begin{verbatim}
1.1719 -%
1.1720 -% val fmz =
1.1721 -% ["equality (-1 + -2 * z + 8 * z ^^^ 2 = (0::real))",
1.1722 -% "solveFor z",
1.1723 -% "solutions L"];
1.1724 -%
1.1725 -% val (dI',pI',mI') =
1.1726 -% ("Isac",
1.1727 -% ["abcFormula","degree_2","polynomial","univariate","equation"],
1.1728 -% ["no_met"]);\end{verbatim}
1.1729 -% \end{example}
1.1730 -%
1.1731 -% Here we want to solve the equation: $-1+-2\cdot z+8\cdot z^{2}=0$. (To give
1.1732 -% a short overview on the commands; at first we set up the equation and tell the
1.1733 -% machine what's the bound variable and where to store the solution. Second step
1.1734 -% is to define the equation type and determine if we want to use a special method
1.1735 -% to solve this type.) Simple checks tell us that the we will get two results for
1.1736 -% this equation and this results will be real.
1.1737 -% So far it is easy for us and for our machine to solve, but
1.1738 -% mentioned that a unvariate equation second order can have three different types
1.1739 -% of solutions it is getting worth.
1.1740 -% \subparagraph{The solving of} all this types of solutions is not yet supported.
1.1741 -% Luckily it was needed for us; but something which has been needed in this
1.1742 -% context, would have been the solving of an euation looking like:
1.1743 -% $-z^{-2}+-2\cdot z^{-1}+8=0$ which is basically the same equation as mentioned
1.1744 -% before (remember that befor it was no problem to handle for the machine) but
1.1745 -% now, after a simple equivalent transformation, we are not able to solve
1.1746 -% it anymore.
1.1747 -% \subparagraph{Error messages} we get when we try to solve something like upside
1.1748 -% were very confusing and also leads us to no special hint about a problem.
1.1749 -% \par The fault behind is, that we have no well error handling on one side and
1.1750 -% no sufficient formed equation solving on the other side. This two facts are
1.1751 -% making the implemention of new material very difficult.
1.1752 -%
1.1753 -% \subsection{Formalization of missing knowledge in Isabelle}
1.1754 -%
1.1755 -% \paragraph{A problem} behind is the mechanization of mathematic
1.1756 -% theories in TP-bases languages. There is still a huge gap between
1.1757 -% these algorithms and this what we want as a solution - in Example
1.1758 -% Signal Processing.
1.1759 -%
1.1760 -% \vbox{
1.1761 -% \begin{example}
1.1762 -% \label{eg:gap}
1.1763 -% \[
1.1764 -% X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY
1.1765 -% \]
1.1766 -% {\small\textit{
1.1767 -% \noindent A very simple example on this what we call gap is the
1.1768 -% simplification above. It is needles to say that it is correct and also
1.1769 -% Isabelle for fills it correct - \emph{always}. But sometimes we don't
1.1770 -% want expand such terms, sometimes we want another structure of
1.1771 -% them. Think of a problem were we now would need only the coefficients
1.1772 -% of $X$ and $Y$. This is what we call the gap between mechanical
1.1773 -% simplification and the solution.
1.1774 -% }}
1.1775 -% \end{example}
1.1776 -% }
1.1777 -%
1.1778 -% \paragraph{We are not able to fill this gap,} until we have to live
1.1779 -% with it but first have a look on the meaning of this statement:
1.1780 -% Mechanized math starts from mathematical models and \emph{hopefully}
1.1781 -% proceeds to match physics. Academic engineering starts from physics
1.1782 -% (experimentation, measurement) and then proceeds to mathematical
1.1783 -% modeling and formalization. The process from a physical observance to
1.1784 -% a mathematical theory is unavoidable bound of setting up a big
1.1785 -% collection of standards, rules, definition but also exceptions. These
1.1786 -% are the things making mechanization that difficult.
1.1787 -%
1.1788 -% \vbox{
1.1789 -% \begin{example}
1.1790 -% \label{eg:units}
1.1791 -% \[
1.1792 -% m,\ kg,\ s,\ldots
1.1793 -% \]
1.1794 -% {\small\textit{
1.1795 -% \noindent Think about some units like that one's above. Behind
1.1796 -% each unit there is a discerning and very accurate definition: One
1.1797 -% Meter is the distance the light travels, in a vacuum, through the time
1.1798 -% of 1 / 299.792.458 second; one kilogram is the weight of a
1.1799 -% platinum-iridium cylinder in paris; and so on. But are these
1.1800 -% definitions usable in a computer mechanized world?!
1.1801 -% }}
1.1802 -% \end{example}
1.1803 -% }
1.1804 -%
1.1805 -% \paragraph{A computer} or a TP-System builds on programs with
1.1806 -% predefined logical rules and does not know any mathematical trick
1.1807 -% (follow up example \ref{eg:trick}) or recipe to walk around difficult
1.1808 -% expressions.
1.1809 -%
1.1810 -% \vbox{
1.1811 -% \begin{example}
1.1812 -% \label{eg:trick}
1.1813 -% \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]
1.1814 -% \[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)=
1.1815 -% \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]
1.1816 -% \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]
1.1817 -% {\small\textit{
1.1818 -% \noindent Sometimes it is also useful to be able to apply some
1.1819 -% \emph{tricks} to get a beautiful and particularly meaningful result,
1.1820 -% which we are able to interpret. But as seen in this example it can be
1.1821 -% hard to find out what operations have to be done to transform a result
1.1822 -% into a meaningful one.
1.1823 -% }}
1.1824 -% \end{example}
1.1825 -% }
1.1826 -%
1.1827 -% \paragraph{The only possibility,} for such a system, is to work
1.1828 -% through its known definitions and stops if none of these
1.1829 -% fits. Specified on Signal Processing or any other application it is
1.1830 -% often possible to walk through by doing simple creases. This creases
1.1831 -% are in general based on simple math operational but the challenge is
1.1832 -% to teach the machine \emph{all}\footnote{Its pride to call it
1.1833 -% \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to
1.1834 -% reach a high level of \emph{all} but it in real it will still be a
1.1835 -% survey of knowledge which links to other knowledge and {{\sisac}{}} a
1.1836 -% trainer and helper but no human compensating calculator.
1.1837 -% \par
1.1838 -% {{{\sisac}{}}} itself aims to adds \emph{Algorithmic Knowledge} (formal
1.1839 -% specifications of problems out of topics from Signal Processing, etc.)
1.1840 -% and \emph{Application-oriented Knowledge} to the \emph{deductive} axis of
1.1841 -% physical knowledge. The result is a three-dimensional universe of
1.1842 -% mathematics seen in Figure~\ref{fig:mathuni}.
1.1843 -%
1.1844 -% \begin{figure}
1.1845 -% \begin{center}
1.1846 -% \includegraphics{fig/universe}
1.1847 -% \caption{Didactic ``Math-Universe'': Algorithmic Knowledge (Programs) is
1.1848 -% combined with Application-oriented Knowledge (Specifications) and Deductive Knowledge (Axioms, Definitions, Theorems). The Result
1.1849 -% leads to a three dimensional math universe.\label{fig:mathuni}}
1.1850 -% \end{center}
1.1851 -% \end{figure}
1.1852 -%
1.1853 -% %WN Deine aktuelle Benennung oben wird Dir kein Fachmann abnehmen;
1.1854 -% %WN bitte folgende Bezeichnungen nehmen:
1.1855 -% %WN
1.1856 -% %WN axis 1: Algorithmic Knowledge (Programs)
1.1857 -% %WN axis 2: Application-oriented Knowledge (Specifications)
1.1858 -% %WN axis 3: Deductive Knowledge (Axioms, Definitions, Theorems)
1.1859 -% %WN
1.1860 -% %WN und bitte die R"ander von der Grafik wegschneiden (was ich f"ur *.pdf
1.1861 -% %WN nicht hinkriege --- weshalb ich auch die eJMT-Forderung nicht ganz
1.1862 -% %WN verstehe, separierte PDFs zu schicken; ich w"urde *.png schicken)
1.1863 -%
1.1864 -% %JR Ränder und beschriftung geändert. Keine Ahnung warum eJMT sich pdf's
1.1865 -% %JR wünschen, würde ebenfalls png oder ähnliches verwenden, aber wenn pdf's
1.1866 -% %JR gefordert werden WN2...
1.1867 -% %WN2 meiner Meinung nach hat sich eJMT unklar ausgedr"uckt (z.B. kann
1.1868 -% %WN2 man meines Wissens pdf-figures nicht auf eine bestimmte Gr"osse
1.1869 -% %WN2 zusammenschneiden um die R"ander weg zu bekommen)
1.1870 -% %WN2 Mein Vorschlag ist, in umserem tex-file bei *.png zu bleiben und
1.1871 -% %WN2 png + pdf figures mitzuschicken.
1.1872 -%
1.1873 -% \subsection{Notes on Problems with Traditional Notation}
1.1874 -%
1.1875 -% \paragraph{During research} on these topic severely problems on
1.1876 -% traditional notations have been discovered. Some of them have been
1.1877 -% known in computer science for many years now and are still unsolved,
1.1878 -% one of them aggregates with the so called \emph{Lambda Calculus},
1.1879 -% Example~\ref{eg:lamda} provides a look on the problem that embarrassed
1.1880 -% us.
1.1881 -%
1.1882 -% \vbox{
1.1883 -% \begin{example}
1.1884 -% \label{eg:lamda}
1.1885 -%
1.1886 -% \[ f(x)=\ldots\; \quad R \rightarrow \quad R \]
1.1887 -%
1.1888 -%
1.1889 -% \[ f(p)=\ldots\; p \in \quad R \]
1.1890 -%
1.1891 -% {\small\textit{
1.1892 -% \noindent Above we see two equations. The first equation aims to
1.1893 -% be a mapping of an function from the reel range to the reel one, but
1.1894 -% when we change only one letter we get the second equation which
1.1895 -% usually aims to insert a reel point $p$ into the reel function. In
1.1896 -% computer science now we have the problem to tell the machine (TP) the
1.1897 -% difference between this two notations. This Problem is called
1.1898 -% \emph{Lambda Calculus}.
1.1899 -% }}
1.1900 -% \end{example}
1.1901 -% }
1.1902 -%
1.1903 -% \paragraph{An other problem} is that terms are not full simplified in
1.1904 -% traditional notations, in {{\sisac}} we have to simplify them complete
1.1905 -% to check weather results are compatible or not. in e.g. the solutions
1.1906 -% of an second order linear equation is an rational in {{\sisac}} but in
1.1907 -% tradition we keep fractions as long as possible and as long as they
1.1908 -% aim to be \textit{beautiful} (1/8, 5/16,...).
1.1909 -% \subparagraph{The math} which should be mechanized in Computer Theorem
1.1910 -% Provers (\emph{TP}) has (almost) a problem with traditional notations
1.1911 -% (predicate calculus) for axioms, definitions, lemmas, theorems as a
1.1912 -% computer program or script is not able to interpret every Greek or
1.1913 -% Latin letter and every Greek, Latin or whatever calculations
1.1914 -% symbol. Also if we would be able to handle these symbols we still have
1.1915 -% a problem to interpret them at all. (Follow up \hbox{Example
1.1916 -% \ref{eg:symbint1}})
1.1917 -%
1.1918 -% \vbox{
1.1919 -% \begin{example}
1.1920 -% \label{eg:symbint1}
1.1921 -% \[
1.1922 -% u\left[n\right] \ \ldots \ unitstep
1.1923 -% \]
1.1924 -% {\small\textit{
1.1925 -% \noindent The unitstep is something we need to solve Signal
1.1926 -% Processing problem classes. But in {{{\sisac}{}}} the rectangular
1.1927 -% brackets have a different meaning. So we abuse them for our
1.1928 -% requirements. We get something which is not defined, but usable. The
1.1929 -% Result is syntax only without semantic.
1.1930 -% }}
1.1931 -% \end{example}
1.1932 -% }
1.1933 -%
1.1934 -% In different problems, symbols and letters have different meanings and
1.1935 -% ask for different ways to get through. (Follow up \hbox{Example
1.1936 -% \ref{eg:symbint2}})
1.1937 -%
1.1938 -% \vbox{
1.1939 -% \begin{example}
1.1940 -% \label{eg:symbint2}
1.1941 -% \[
1.1942 -% \widehat{\ }\ \widehat{\ }\ \widehat{\ } \ \ldots \ exponent
1.1943 -% \]
1.1944 -% {\small\textit{
1.1945 -% \noindent For using exponents the three \texttt{widehat} symbols
1.1946 -% are required. The reason for that is due the development of
1.1947 -% {{{\sisac}{}}} the single \texttt{widehat} and also the double were
1.1948 -% already in use for different operations.
1.1949 -% }}
1.1950 -% \end{example}
1.1951 -% }
1.1952 -%
1.1953 -% \paragraph{Also the output} can be a problem. We are familiar with a
1.1954 -% specified notations and style taught in university but a computer
1.1955 -% program has no knowledge of the form proved by a professor and the
1.1956 -% machines themselves also have not yet the possibilities to print every
1.1957 -% symbol (correct) Recent developments provide proofs in a human
1.1958 -% readable format but according to the fact that there is no money for
1.1959 -% good working formal editors yet, the style is one thing we have to
1.1960 -% live with.
1.1961 -%
1.1962 -% \section{Problems rising out of the Development Environment}
1.1963 -%
1.1964 -% fehlermeldungen! TODO
1.1965 -
1.1966 -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{verbatim}
1.1967 -
1.1968 -\section{Summary and Conclusions}\label{conclusion}
1.1969 -
1.1970 -%JR obvious
1.1971 -
1.1972 -%This paper gives a first experience report about programming with a
1.1973 -%TP-based programming language.
1.1974 -
1.1975 -A brief re-introduction of the novel kind of programming
1.1976 -language by example of the {\sisac}-prototype makes the paper
1.1977 -self-contained. The main section describes all the main concepts
1.1978 -involved in TP-based programming and all the sub-tasks concerning
1.1979 -respective implementation in the {\sisac} prototype: mechanisation of mathematics and domain
1.1980 -modeling, implementation of term rewriting systems for the
1.1981 -rewriting-engine, formal (implicit) specification of the problem to be
1.1982 -(explicitly) described by the program, implementation of the many components
1.1983 -required for Lucas-Interpretation and finally implementation of the
1.1984 -program itself.
1.1985 -
1.1986 -The many concepts and sub-tasks involved in programming require a
1.1987 -comprehensive work-flow; first experiences with the work-flow as
1.1988 -supported by the present prototype are described as well: Isabelle +
1.1989 -Isar + jEdit provide appropriate components for establishing an
1.1990 -efficient development environment integrating computation and
1.1991 -deduction. However, the present state of the prototype is far off a
1.1992 -state appropriate for wide-spread use: the prototype of the program
1.1993 -language lacks expressiveness and elegance, the prototype of the
1.1994 -development environment is hardly usable: error messages still address
1.1995 -the developer of the prototype's interpreter rather than the
1.1996 -application programmer, implementation of the many settings for the
1.1997 -Lucas-Interpreter is cumbersome.
1.1998 -
1.1999 -\subsection{Conclusions for Future Development}
1.2000 -From the above mentioned experiences a successful proof of concept can be concluded:
1.2001 -programming arbitrary problems from engineering sciences is possible,
1.2002 -in principle even in the prototype. Furthermore the experiences allow
1.2003 -to conclude detailed requirements for further development:
1.2004 -\begin{enumerate}
1.2005 -\item Clarify underlying logics such that programming is smoothly
1.2006 -integrated with verification of the program; the post-condition should
1.2007 -be proved more or less automatically, otherwise working engineers
1.2008 -would not encounter such programming.
1.2009 -\item Combine the prototype's programming language with Isabelle's
1.2010 -powerful function package and probably with more of SML's
1.2011 -pattern-matching features; include parallel execution on multi-core
1.2012 -machines into the language design.
1.2013 -\item Extend the prototype's Lucas-Interpreter such that it also
1.2014 -handles functions defined by use of Isabelle's functions package; and
1.2015 -generalize Isabelle's code generator such that efficient code for the
1.2016 -whole definition of the programming language can be generated (for
1.2017 -multi-core machines).
1.2018 -\item Develop an efficient development environment with
1.2019 -integration of programming and proving, with management not only of
1.2020 -Isabelle theories, but also of large collections of specifications and
1.2021 -of programs.
1.2022 -\item\label{CAS} Extend Isabelle's computational features in direction of
1.2023 -\textit{verfied} Computer Algebra: simplification extended by
1.2024 -algorithms beyond rewriting (cancellation of multivariate rationals,
1.2025 -factorisation, partial fraction decomposition, etc), equation solving
1.2026 -, integration, etc.
1.2027 -\end{enumerate}
1.2028 -Provided successful accomplishment, these points provide distinguished
1.2029 -components for virtual workbenches appealing to practitioners of
1.2030 -engineering in the near future.
1.2031 -
1.2032 -\subsection{Preview to Development of Course Material}
1.2033 -Interactive course material, as addressed by the title,
1.2034 -can comprise step-wise problem solving created as a side-effect of a
1.2035 -TP-based program: The introduction \S\ref{intro} briefly shows that Lucas-Interpretation not only provides an
1.2036 -interactive programming environment, Lucas-Interpretation also can
1.2037 -provide TP-based services for a flexible dialogue component with
1.2038 -adaptive user guidance for independent and inquiry-based learning.
1.2039 -
1.2040 -However, the {\sisac} prototype is not ready for use in field-tests,
1.2041 -not only due to the above five requirements not sufficiently
1.2042 -accomplished, but also due to usability of the fron-end, in particular
1.2043 -the lack of an editor for formulas in 2-dimension representation.
1.2044 -
1.2045 -Nevertheless, the experiences from the case study described in this
1.2046 -paper, allow to give a preview to the development of course material,
1.2047 -if based on Lucas-Interpretation:
1.2048 -
1.2049 -\paragraph{Development of material from scratch} is too much effort
1.2050 -just for e-learning; this has become clear with the case study. For
1.2051 -getting support for stepwise problem solving just in {\em one} example
1.2052 -class, the one presented in this paper, involved the following tasks:
1.2053 -\begin{itemize}
1.2054 -\item Adapt the equation solver; since that was too laborous, the
1.2055 -program has been adapted in an unelegant way.
1.2056 -\item Implement an algorithms for partial fraction decomposition,
1.2057 -which is considered a standard normal form in Computer Algebra.
1.2058 -\item Implement a specification for partial fraction decomposition and
1.2059 -locate it appropriately in the hierarchy of specification.
1.2060 -\item Declare definitions and theorems within the theory of
1.2061 -${\cal Z}$-transform, and prove the theorems (which was not done in the
1.2062 -case study).
1.2063 -\end{itemize}
1.2064 -On the other hand, for the one the class of problems implemented,
1.2065 -adding an arbitrary number of examples within this class requires a
1.2066 -few minutes~\footnote{As shown in Fig.\ref{fig-interactive}, an
1.2067 -example is called from an HTML-file by an URL, which addresses an
1.2068 -XML-structure holding the respective data as shown on
1.2069 -p.\pageref{ml-check-program}.} and the support for individual stepwise
1.2070 -problem solving comes for free.
1.2071 -
1.2072 -\paragraph{E-learning benefits from Formal Domain Engineering} which can be
1.2073 -expected for various domains in the near future. In order to cope with
1.2074 -increasing complexity in domain of technology, specific domain
1.2075 -knowledge is beeing mechanised, not only for software technology
1.2076 -\footnote{For instance, the Archive of Formal Proofs
1.2077 -http://afp.sourceforge.net/} but also for other engineering domains
1.2078 -\cite{Dehbonei&94,Hansen94b,db:dom-eng}. This fairly new part of
1.2079 -engineering sciences is called ``domain engineering'' in
1.2080 -\cite{db:SW-engIII}.
1.2081 -
1.2082 -Given this kind of mechanised knowledge including mathematical
1.2083 -theories, domain specific definitions, specifications and algorithms,
1.2084 -theorems and proofs, then e-learning with support for individual
1.2085 -stepwise problem solving will not be much ado anymore; then e-learning
1.2086 -media in technology education can be derived from this knowledge with
1.2087 -reasonable effort.
1.2088 -
1.2089 -\paragraph{Development differentiates into tasks} more separated than
1.2090 -without Lucas-Interpretation and more challenginging in specific
1.2091 -expertise. These are the kinds of experts expected to cooperate in
1.2092 -development of
1.2093 -\begin{itemize}
1.2094 -\item ``Domain engineers'', who accomplish fairly novel tasks
1.2095 -described in this paper.
1.2096 -\item Course designers, who provide the instructional design according
1.2097 -to curricula, together with usability experts and media designers, are
1.2098 -indispensable in production of e-learning media at the state-of-the
1.2099 -art.
1.2100 -\item ``Dialog designers'', whose part of development is clearly
1.2101 -separated from the part of domain engineers as a consequence of
1.2102 -Lucas-Interpretation: TP-based programs are functional, as mentioned,
1.2103 -and are only concerned with describing mathematics --- and not at all
1.2104 -concerned with interaction, psychology, learning theory and the like,
1.2105 -because there are no in/output statements. Dialog designers can expect
1.2106 -a high-level rule-based language~\cite{gdaroczy-EP-13} for describing
1.2107 -their part.
1.2108 -\end{itemize}
1.2109 -
1.2110 -% response-to-referees:
1.2111 -% (2.1) details of novel technology in order to estimate the impact
1.2112 -% (2.2) which kinds of expertise are required for production of e-learning media (instructional design, math authoring, dialog authoring, media design)
1.2113 -% (2.3) what in particular is required for programming new exercises supported by next-step-guidance (expertise / efforts)
1.2114 -% (2.4) estimation of break-even points for development of next-step-guidance
1.2115 -% (2.5) usability of ISAC prototype at the present state
1.2116 -%
1.2117 -% The points (1.*) seem to be well covered in the paper, the points (2.*) are not. So I decided to address the points (2.*) in a separate section §5.1."".
1.2118 -
1.2119 -\bigskip\noindent For this decade there seems to be a window of opportunity opening from
1.2120 -one side inreasing demand for formal domain engineering and from the
1.2121 -other side from TP more and more gaining industrial relevance. Within
1.2122 -this window, development of TP-based educational software can take
1.2123 -benefit from the fact, that the TPs leading in Europe, Coq~\cite{coq-team-10} and
1.2124 -Isabelle are still open source together with the major part of
1.2125 -mechanised knowledge.%~\footnote{NICTA}.
1.2126 -
1.2127 -\bibliographystyle{alpha}
1.2128 -{\small\bibliography{references}}
1.2129 -
1.2130 -\end{document}
1.2131 -% LocalWords: TP IST SPSC Telematics Dialogues dialogue HOL bool nat Hindley
1.2132 -% LocalWords: Milner tt Subproblem Formulae ruleset generalisation initialised
1.2133 -% LocalWords: axiomatization LCF Simplifiers simplifiers Isar rew Thm Calc SML
1.2134 -% LocalWords: recognised hoc Trueprop redexes Unsynchronized pre rhs ord erls
1.2135 -% LocalWords: srls prls nrls lhs errpats InverseZTransform SubProblem IDE IDEs
1.2136 -% LocalWords: univariate jEdit rls RealDef calclist familiarisation ons pos eq
1.2137 -% LocalWords: mout ctree SignalProcessing frac ZZ Postcond Atools wiki SML's
1.2138 -% LocalWords: mechanisation multi