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jan@42463: \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
jan@42463: \def\sisac{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
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jan@42463: \begin{document}
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neuper@42464: \title{Trials with TP-based Programming
neuper@42464: \\
neuper@42464: for Interactive Course Material}%
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jan@42463: % Single author.  Please supply at least your name,
jan@42463: % email address, and affiliation here.
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jan@42463: \author{\begin{tabular}{c}
jan@42463: \textit{Jan Ro\v{c}nik} \\
jan@42463: jan.rocnik@student.tugraz.at \\
jan@42463: IST, SPSC\\
neuper@42514: Graz University of Technology\\
jan@42463: Austria\end{tabular}
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jan@42463: % abstract
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jan@42463: \begin{abstract}
jan@42463: 
jan@42463: Traditional course material in engineering disciplines lacks an
jan@42463: important component, interactive support for step-wise problem
neuper@42464: solving. Theorem-Proving (TP) technology is appropriate for one part
jan@42463: of such support, in checking user-input. For the other part of such
jan@42463: support, guiding the learner towards a solution, another kind of
neuper@42516: technology is required.
jan@42463: 
jan@42511: Both kinds of support can be achieved by so-called
neuper@42504: Lucas-Interpretation which combines deduction and computation and, for
neuper@42504: the latter, uses a novel kind of programming language. This language
neuper@42504: is based on (Computer) Theorem Proving (TP), thus called a ``TP-based
neuper@42504: programming language''.
jan@42463: 
neuper@42504: This paper is the experience report of the first ``application
neuper@42507: programmer'' using this language for creating exercises in step-wise
neuper@42507: problem solving for an advanced lab in Signal Processing. The tasks
neuper@42507: involved in TP-based programming are described together with the
neuper@42507: experience gained from a prototype of the programming language and of
neuper@42507: it's interpreter.
neuper@42504: 
neuper@42504: The report concludes with a positive proof of concept, states
jan@42512: insufficiency usability of the prototype and captures the requirements
neuper@42504: for further development of both, the programming language and the
neuper@42504: interpreter.
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jan@42463: \end{abstract}%
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jan@42463: 
neuper@42464: \section{Introduction}\label{intro}
jan@42463: 
jan@42466: % \paragraph{Didactics of mathematics} 
jan@42466: %WN: wenn man in einem high-quality paper von 'didactics' spricht, 
jan@42466: %WN muss man am state-of-the-art ankn"upfen -- siehe
jan@42466: %WN W.Neuper, On the Emergence of TP-based Educational Math Assistants
neuper@42464: % faces a specific issue, a gap
neuper@42464: % between (1) introduction of math concepts and skills and (2)
neuper@42464: % application of these concepts and skills, which usually are separated
neuper@42464: % into different units in curricula (for good reasons). For instance,
neuper@42464: % (1) teaching partial fraction decomposition is separated from (2)
neuper@42464: % application for inverse Z-transform in signal processing.
neuper@42464: % 
neuper@42464: % \par This gap is an obstacle for applying math as an fundamental
neuper@42464: % thinking technology in engineering: In (1) motivation is lacking
neuper@42464: % because the question ``What is this stuff good for?'' cannot be
neuper@42464: % treated sufficiently, and in (2) the ``stuff'' is not available to
neuper@42464: % students in higher semesters as widespread experience shows.
neuper@42464: % 
neuper@42464: % \paragraph{Motivation} taken by this didactic issue on the one hand,
neuper@42464: % and ongoing research and development on a novel kind of educational
neuper@42464: % mathematics assistant at Graz University of
neuper@42464: % Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to
neuper@42464: % scope with this issue on the other hand, several institutes are
neuper@42464: % planning to join their expertise: the Institute for Information
neuper@42464: % Systems and Computer Media (IICM), the Institute for Software
neuper@42464: % Technology (IST), the Institutes for Mathematics, the Institute for
neuper@42464: % Signal Processing and Speech Communication (SPSC), the Institute for
neuper@42464: % Structural Analysis and the Institute of Electrical Measurement and
neuper@42464: % Measurement Signal Processing.
jan@42466: %WN diese Information ist f"ur das Paper zu spezielle, zu aktuell 
jan@42466: %WN und damit zu verg"anglich.
neuper@42464: % \par This thesis is the first attempt to tackle the above mentioned
neuper@42464: % issue, it focuses on Telematics, because these specific studies focus
neuper@42464: % on mathematics in \emph{STEOP}, the introductory orientation phase in
neuper@42464: % Austria. \emph{STEOP} is considered an opportunity to investigate the
neuper@42464: % impact of {\sisac}'s prototype on the issue and others.
neuper@42464: % 
jan@42466: 
jan@42502: Traditional course material in engineering disciplines lacks an
neuper@42464: important component, interactive support for step-wise problem
neuper@48772: solving. The lack becomes evident by comparing existing course
neuper@48772: material with the sheets collected from written exams (in case solving
neuper@48772: engineering problems is {\em not} deteriorated to multiple choice
neuper@48772: tests) on the topics addressed by the materials.
neuper@48772: Theorem-Proving (TP) technology can provide such support by
neuper@42464: specific services. An important part of such services is called
neuper@42464: ``next-step-guidance'', generated by a specific kind of ``TP-based
neuper@42464: programming language''. In the
neuper@42464: {\sisac}-project~\footnote{http://www.ist.tugraz.at/projects/isac/} such
neuper@42464: a language is prototyped in line with~\cite{plmms10} and built upon
jan@48766: the theorem prover Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}
jan@48766: \footnote{http://isabelle.in.tum.de/}.
neuper@42464: The TP services are coordinated by a specific interpreter for the
neuper@42464: programming language, called
neuper@48772: Lucas-Interpreter~\cite{wn:lucas-interp-12}. The language 
neuper@48772:  will be briefly re-introduced in order to make the paper
neuper@42464: self-contained.
jan@42463: 
neuper@42504: The main part of the paper is an account of first experiences
neuper@42464: with programming in this TP-based language. The experience was gained
neuper@42464: in a case study by the author. The author was considered an ideal
neuper@42464: candidate for this study for the following reasons: as a student in
neuper@42464: Telematics (computer science with focus on Signal Processing) he had
neuper@42464: general knowledge in programming as well as specific domain knowledge
neuper@42504: in Signal Processing; and he was {\em not} involved in the development of
jan@48766: {\sisac}'s programming language and interpreter, thus being a novice to the
neuper@42464: language.
jan@42463: 
jan@48766: The goals of the case study were: (1) to identify some TP-based programs for
jan@42511: interactive course material for a specific ``Advanced Signal
neuper@42464: Processing Lab'' in a higher semester, (2) respective program
jan@48766: development with as little advice as possible from the {\sisac}-team and (3) 
jan@48766: to document records and comments for the main steps of development in an
jan@48766: Isabelle theory; this theory should provide guidelines for future programmers.
jan@48766: An excerpt from this theory is the main part of this paper.
jan@42466: \par
neuper@48771: 
neuper@48771: \medskip The major example resulting from the case study will be used
neuper@48771: as running example throughout this paper. This example requires a
neuper@48771: program resembling the size of real-world applications in engineering;
neuper@48771: such a size was considered essential for the case study, since there
neuper@48771: are many small programs for a long time (mainly concerned with
neuper@48771: elementary Computer Algebra like simplification, equation solving,
neuper@48771: calculus, etc.~\footnote{The programs existing in the {\sisac}
neuper@48771: prototype are found at
neuper@48771: http://www.ist.tugraz.at/projects/isac/www/kbase/met/index\_met.html})
neuper@48771: 
neuper@48771: \paragraph{The mathematical background of the running example} is the
jan@48774: following: In Signal Processing, ``the ${\cal Z}$-transform for
neuper@48771: discrete-time signals is the counterpart of the Laplace transform for
neuper@48771: continuous-time signals, and they each have a similar relationship to
neuper@48771: the corresponding Fourier transform. One motivation for introducing
neuper@48771: this generalization is that the Fourier transform does not converge
neuper@48771: for all sequences, and it is useful to have a generalization of the
neuper@48771: Fourier transform that encompasses a broader class of signals. A
jan@48774: second advantage is that in analytic problems, the ${\cal Z}$-transform
neuper@48771: notation is often more convenient than the Fourier transform
jan@48774: notation.''  ~\cite[p. 128]{oppenheim2010discrete}.  The ${\cal Z}$-transform
neuper@48771: is defined as
neuper@48771: \begin{equation*}
neuper@48771: X(z)=\sum_{n=-\infty }^{\infty }x[n]z^{-n}
neuper@48771: \end{equation*}
neuper@48771: where a discrete time sequence $x[n]$ is transformed into the function
neuper@48771: $X(z)$ where $z$ is a continuous complex variable. The inverse
neuper@48771: function is addressed in the running example and can be determined by
neuper@48771: the integral
neuper@48771: \begin{equation*}
neuper@48771: x[n]=\frac{1}{2\pi j} \oint_{C} X(z)\cdot z^{n-1} dz
neuper@48771: \end{equation*}
neuper@48771: where the letter $C$ represents a contour within the range of
jan@48774: convergence of the ${\cal Z}$-transform. The unit circle can be a special
neuper@48771: case of this contour. Remember that $j$ is the complex number in the
jan@48774: domain of engineering.  As this transform requires high effort to
neuper@48771: be solved, tables of commonly used transform pairs are used in
neuper@48771: education as well as in engineering practice; such tables can be found
neuper@48771: at~\cite{wiki:1} or~\cite[Table~3.1]{oppenheim2010discrete} as well.
neuper@48771: A completely solved and more detailed example can be found at
neuper@48771: ~\cite[p. 149f]{oppenheim2010discrete}. 
neuper@48771: 
neuper@48771: Following conventions in engineering education and in practice, the
neuper@48771: running example solves the problem by use of a table. 
neuper@48771: 
neuper@48771: \paragraph{Support for interactive stepwise problem solving} in the
neuper@48771: {\sisac} prototype is shown in Fig.\ref{fig-interactive}~\footnote{ Fig.\ref{fig-interactive} also shows the prototype status of {\sisac}; for instance,
neuper@48771: the lack of 2-dimensional presentation and input of formulas is the major obstacle for field-tests in standard classes.}:
neuper@48771: A student inputs formulas line by line on the \textit{``Worksheet''},
neuper@48771: and each step (i.e. each formula on completion) is immediately checked
neuper@48775: by the system, such that at most {\em one inconsistent} formula can reside on
neuper@48771: the Worksheet (on the input line, marked by the red $\otimes$).
jan@42463: \begin{figure} [htb]
jan@42463: \begin{center}
neuper@42468: \includegraphics[width=140mm]{fig/isac-Ztrans-math-3}
neuper@42468: %\includegraphics[width=140mm]{fig/isac-Ztrans-math}
jan@42512: \caption{Step-wise problem solving guided by the TP-based program
jan@42512: \label{fig-interactive}}
jan@42463: \end{center}
jan@42463: \end{figure}
neuper@48772: If the student gets stuck and does not know the formula to proceed
neuper@48772: with, there is the button \framebox{NEXT} presenting the next formula
neuper@48777: on the Worksheet; this feature is called ``next-step-guidance''~\cite{wn:lucas-interp-12}. The button \framebox{AUTO} immediately delivers the
neuper@48772: final result in case the student is not interested in intermediate
neuper@48772: steps.
jan@42466: 
neuper@48771: Adaptive dialogue guidance is already under
neuper@48771: construction~\cite{gdaroczy-EP-13} and the two buttons will disappear,
neuper@48771: since their presence is not wanted in many learning scenarios (in
neuper@48771: particular, {\em not} in written exams).
jan@48767: 
neuper@48771: The buttons \framebox{Theories}, \framebox{Problems} and
neuper@48771: \framebox{Methods} are the entry points for interactive lookup of the
neuper@48771: underlying knowledge.  For instance, pushing \framebox{Theories} in
neuper@48771: the configuration shown in Fig.\ref{fig-interactive}, pops up a
neuper@48771: ``Theory browser'' displaying the theorem(s) justifying the current
neuper@48772: step.  The browser allows to lookup all other theories, thus
neuper@48771: supporting indepentend investigation of underlying definitions,
neuper@48771: theorems, proofs --- where the HTML representation of the browsers is
neuper@48772: ready for arbitrary multimedia add-ons. Likewise, the browsers for
neuper@48772: \framebox{Problems} and \framebox{Methods} support context sensitive
neuper@48772: as well as interactive access to specifications and programs
neuper@48772: respectively. 
neuper@48772: 
neuper@48772: There is also a simple web-based representation of knowledge items;
neuper@48772: the items under consideration in this paper can be looked up as
jan@48778: well
jan@48778: ~\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}}}
jan@48778: ~\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}}}
jan@48778: ~\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}}}.
jan@48767: 
neuper@48771: % can be explained by having a look at 
neuper@48771: % Fig.\ref{fig-interactive} which shows the beginning of the interactive 
neuper@48771: % construction of a solution for the problem. This construction is done in the 
neuper@48771: % right window named ``Worksheet''.
neuper@48771: % \par
neuper@48771: % User-interaction on the Worksheet is {\em checked} and {\em guided} by
neuper@48771: % TP services:
neuper@48771: % \begin{enumerate}
neuper@48771: % \item Formulas input by the user are {\em checked} by TP: such a
neuper@48771: % formula establishes a proof situation --- the prover has to derive the
neuper@48771: % formula from the logical context. The context is built up from the
neuper@48771: % formal specification of the problem (here hidden from the user) by the
neuper@48771: % Lucas-Interpreter.
neuper@48771: % \item If the user gets stuck, the program developed below in this
neuper@48771: % paper ``knows the next step'' and Lucas-Interpretation provides services
neuper@48771: % featuring so-called ``next-step-guidance''; this is out of scope of this
neuper@48771: % paper and can be studied in~\cite{gdaroczy-EP-13}.
neuper@48771: % \end{enumerate} It should be noted that the programmer using the
neuper@48771: % TP-based language is not concerned with interaction at all; we will
neuper@48771: % see that the program contains neither input-statements nor
neuper@48771: % output-statements. Rather, interaction is handled by the interpreter
neuper@48771: % of the language.
neuper@48771: % 
neuper@48771: % So there is a clear separation of concerns: Dialogues are adapted by
neuper@48771: % dialogue authors (in Java-based tools), using TP services provided by
neuper@48771: % Lucas-Interpretation. The latter acts on programs developed by
neuper@48771: % mathematics-authors (in Isabelle/ML); their task is concern of this
neuper@48771: % paper.
jan@48767: 
neuper@48771: \bigskip The paper is structured as follows: The introduction
neuper@42464: \S\ref{intro} is followed by a brief re-introduction of the TP-based
neuper@42464: programming language in \S\ref{PL}, which extends the executable
neuper@42464: fragment of Isabelle's language (\S\ref{PL-isab}) by tactics which
neuper@42464: play a specific role in Lucas-Interpretation and in providing the TP
neuper@42504: services (\S\ref{PL-tacs}). The main part \S\ref{trial} describes
neuper@42464: the main steps in developing the program for the running example:
neuper@42464: prepare domain knowledge, implement the formal specification of the
neuper@42504: problem, prepare the environment for the interpreter, implement the
neuper@42504: program in \S\ref{isabisac} to \S\ref{progr} respectively. 
jan@42511: The work-flow of programming, debugging and testing is
neuper@42464: described in \S\ref{workflow}. The conclusion \S\ref{conclusion} will
neuper@42464: give directions identified for future development. 
neuper@42464: 
jan@42463: 
jan@42463: \section{\isac's Prototype for a Programming Language}\label{PL} 
jan@48768: The prototype of the language and of the Lucas-Interpreter is briefly
neuper@42504: described from the point of view of a programmer. The language extends
neuper@48771: the executable fragment of Higher-Order Logic (HOL) in the theorem prover
neuper@42504: Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}.
jan@42463: 
jan@42463: \subsection{The Executable Fragment of Isabelle's Language}\label{PL-isab}
jan@42463: The executable fragment consists of data-type and function
jan@42463: definitions.  It's usability even suggests that fragment for
neuper@48771: introductory courses \cite{nipkow-prog-prove}. HOL is a typed logic whose type system resembles that of functional programming
jan@42463: languages. Thus there are
jan@42463: \begin{description}
jan@42463: \item[base types,] in particular \textit{bool}, the type of truth
jan@42463: values, \textit{nat}, \textit{int}, \textit{complex}, and the types of
jan@42463: natural, integer and complex numbers respectively in mathematics.
jan@42463: \item[type constructors] allow to define arbitrary types, from
jan@42463: \textit{set}, \textit{list} to advanced data-structures like
jan@42463: \textit{trees}, red-black-trees etc.
jan@42463: \item[function types,] denoted by $\Rightarrow$.
jan@42463: \item[type variables,] denoted by $^\prime a, ^\prime b$ etc, provide
jan@42463: type polymorphism. Isabelle automatically computes the type of each
jan@42463: variable in a term by use of Hindley-Milner type inference
jan@42463: \cite{pl:hind97,Milner-78}.
jan@42463: \end{description}
jan@42463: 
jan@42463: \textbf{Terms} are formed as in functional programming by applying
jan@42463: functions to arguments. If $f$ is a function of type
jan@42463: $\tau_1\Rightarrow \tau_2$ and $t$ is a term of type $\tau_1$ then
jan@42463: $f\;t$ is a term of type~$\tau_2$. $t\;::\;\tau$ means that term $t$
jan@42463: has type $\tau$. There are many predefined infix symbols like $+$ and
jan@42463: $\leq$ most of which are overloaded for various types.
jan@42463: 
jan@42463: HOL also supports some basic constructs from functional programming:
jan@42512: {\footnotesize\it\label{isabelle-stmts}
jan@42463: \begin{tabbing} 123\=\kill
jan@42513: 01\>$( \; {\tt if} \; b \; {\tt then} \; t_1 \; {\tt else} \; t_2 \;)$\\
jan@42513: 02\>$( \; {\tt let} \; x=t \; {\tt in} \; u \; )$\\
jan@42513: 03\>$( \; {\tt case} \; t \; {\tt of} \; {\it pat}_1
jan@42463:   \Rightarrow t_1 \; |\dots| \; {\it pat}_n\Rightarrow t_n \; )$
jan@42512: \end{tabbing}}
neuper@42482: \noindent The running example's program uses some of these elements
neuper@42482: (marked by {\tt tt-font} on p.\pageref{s:impl}): for instance {\tt
neuper@42482: let}\dots{\tt in} in lines {\rm 02} \dots {\rm 13}. In fact, the whole program
neuper@42482: is an Isabelle term with specific function constants like {\tt
neuper@42482: program}, {\tt Take}, {\tt Rewrite}, {\tt Subproblem} and {\tt
neuper@42482: Rewrite\_Set} in lines {\rm 01, 03. 04, 07, 10} and {\rm 11, 12}
neuper@42482: respectively.
jan@42463: 
jan@42463: % Terms may also contain $\lambda$-abstractions. For example, $\lambda
jan@42463: % x. \; x$ is the identity function.
jan@42463: 
neuper@42467: %JR warum auskommentiert? WN2...
neuper@42467: %WN2 weil ein Punkt wie dieser in weiteren Zusammenh"angen innerhalb
neuper@42467: %WN2 des Papers auftauchen m"usste; nachdem ich einen solchen
neuper@42467: %WN2 Zusammenhang _noch_ nicht sehe, habe ich den Punkt _noch_ nicht
neuper@42467: %WN2 gel"oscht.
neuper@42467: %WN2 Wenn der Punkt nicht weiter gebraucht wird, nimmt er nur wertvollen
neuper@42467: %WN2 Platz f"ur Anderes weg.
jan@42466: 
neuper@42464: \textbf{Formulae} are terms of type \textit{bool}. There are the basic
jan@42463: constants \textit{True} and \textit{False} and the usual logical
jan@42463: connectives (in decreasing order of precedence): $\neg, \land, \lor,
jan@42463: \rightarrow$.
jan@42463: 
neuper@42464: \textbf{Equality} is available in the form of the infix function $=$
neuper@42464: of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for
neuper@42464: formulas, where it means ``if and only if''.
jan@42463: 
jan@42463: \textbf{Quantifiers} are written $\forall x. \; P$ and $\exists x. \;
jan@42463: P$.  Quantifiers lead to non-executable functions, so functions do not
jan@42463: always correspond to programs, for instance, if comprising \\$(
jan@42463: \;{\it if} \; \exists x.\;P \; {\it then} \; e_1 \; {\it else} \; e_2
jan@42463: \;)$.
jan@42463: 
jan@42463: \subsection{\isac's Tactics for Lucas-Interpretation}\label{PL-tacs}
jan@42463: The prototype extends Isabelle's language by specific statements
neuper@48775: called tactics~\footnote{{\sisac}'s. These tactics are different from
jan@42463: Isabelle's tactics: the former concern steps in a calculation, the
neuper@48775: latter concern proofs.}. For the programmer these
jan@42463: statements are functions with the following signatures:
jan@42463: 
jan@42463: \begin{description}
jan@42463: \item[Rewrite:] ${\it theorem}\Rightarrow{\it term}\Rightarrow{\it
jan@42463: term} * {\it term}\;{\it list}$:
jan@42511: this tactic applies {\it theorem} to a {\it term} yielding a {\it
jan@42463: term} and a {\it term list}, the list are assumptions generated by
jan@42463: conditional rewriting. For instance, the {\it theorem}
jan@42463: $b\not=0\land c\not=0\Rightarrow\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$
jan@42463: applied to the {\it term} $\frac{2\cdot x}{3\cdot x}$ yields
jan@42463: $(\frac{2}{3}, [x\not=0])$.
jan@42463: 
jan@42463: \item[Rewrite\_Set:] ${\it ruleset}\Rightarrow{\it
jan@42463: term}\Rightarrow{\it term} * {\it term}\;{\it list}$:
jan@42511: this tactic applies {\it ruleset} to a {\it term}; {\it ruleset} is
jan@42463: a confluent and terminating term rewrite system, in general. If
jan@42463: none of the rules ({\it theorem}s) is applicable on interpretation
jan@42463: of this tactic, an exception is thrown.
jan@42463: 
jan@42463: % \item[Rewrite\_Inst:] ${\it substitution}\Rightarrow{\it
jan@42463: % theorem}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
jan@42463: % list}$:
jan@42463: % 
jan@42463: % \item[Rewrite\_Set\_Inst:] ${\it substitution}\Rightarrow{\it
jan@42463: % ruleset}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
jan@42463: % list}$:
jan@42463: 
neuper@42504: %SPACEvvv
jan@42463: \item[Substitute:] ${\it substitution}\Rightarrow{\it
neuper@42482: term}\Rightarrow{\it term}$: allows to access sub-terms.
neuper@42504: %SPACE^^^
jan@42463: 
jan@42463: \item[Take:] ${\it term}\Rightarrow{\it term}$:
jan@42463: this tactic has no effect in the program; but it creates a side-effect
jan@42463: by Lucas-Interpretation (see below) and writes {\it term} to the
jan@42463: Worksheet.
jan@42463: 
jan@42463: \item[Subproblem:] ${\it theory} * {\it specification} * {\it
jan@42463: method}\Rightarrow{\it argument}\;{\it list}\Rightarrow{\it term}$:
neuper@42482: this tactic is a generalisation of a function call: it takes an
neuper@42482: \textit{argument list} as usual, and additionally a triple consisting
neuper@42482: of an Isabelle \textit{theory}, an implicit \textit{specification} of the
neuper@42482: program and a \textit{method} containing data for Lucas-Interpretation,
neuper@42482: last not least a program (as an explicit specification)~\footnote{In
neuper@42482: interactive tutoring these three items can be determined explicitly
neuper@42482: by the user.}.
jan@42463: \end{description}
jan@42463: The tactics play a specific role in
jan@42463: Lucas-Interpretation~\cite{wn:lucas-interp-12}: they are treated as
neuper@42482: break-points where, as a side-effect, a line is added to a calculation
neuper@42483: as a protocol for proceeding towards a solution in step-wise problem
neuper@42483: solving. At the same points Lucas-Interpretation serves interactive
neuper@42504: tutoring and hands over control to the user. The user is free to
neuper@42483: investigate underlying knowledge, applicable theorems, etc.  And the
neuper@42483: user can proceed constructing a solution by input of a tactic to be
neuper@42483: applied or by input of a formula; in the latter case the
jan@42463: Lucas-Interpreter has built up a logical context (initialised with the
jan@42463: precondition of the formal specification) such that Isabelle can
jan@42463: derive the formula from this context --- or give feedback, that no
jan@42463: derivation can be found.
jan@42463: 
jan@42511: \subsection{Tactics as Control Flow Statements}
jan@42463: The flow of control in a program can be determined by {\tt if then else}
jan@42463: and {\tt case of} as mentioned on p.\pageref{isabelle-stmts} and also
jan@42511: by additional tactics:
jan@42463: \begin{description}
jan@42463: \item[Repeat:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it
jan@42463: term}$: iterates over tactics which take a {\it term} as argument as
neuper@42482: long as a tactic is applicable (for instance, {\tt Rewrite\_Set} might
jan@42463: not be applicable).
jan@42463: 
jan@42463: \item[Try:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it term}$:
jan@42463: if {\it tactic} is applicable, then it is applied to {\it term},
neuper@42483: otherwise {\it term} is passed on without changes.
jan@42463: 
jan@42463: \item[Or:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
neuper@42483: term}\Rightarrow{\it term}$: If the first {\it tactic} is applicable,
neuper@42483: it is applied to the first {\it term} yielding another {\it term},
neuper@42483: otherwise the second {\it tactic} is applied; if none is applicable an
neuper@42483: exception is raised.
jan@42463: 
jan@42463: \item[@@:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
neuper@42483: term}\Rightarrow{\it term}$: applies the first {\it tactic} to the
neuper@42483: first {\it term} yielding an intermediate term (not appearing in the
neuper@42483: signature) to which the second {\it tactic} is applied.
jan@42463: 
jan@42463: \item[While:] ${\it term::bool}\Rightarrow{\it tactic}\Rightarrow{\it
neuper@42483: term}\Rightarrow{\it term}$: if the first {\it term} is true, then the
neuper@42483: {\it tactic} is applied to the first {\it term} yielding an
neuper@42483: intermediate term (not appearing in the signature); the intermediate
neuper@42483: term is added to the environment the first {\it term} is evaluated in
jan@42511: etc. as long as the first {\it term} is true.
jan@42463: \end{description}
jan@42511: The tactics are not treated as break-points by Lucas-Interpretation
neuper@42504: and thus do neither contribute to the calculation nor to interaction.
jan@42463: 
neuper@42498: \section{Concepts and Tasks in TP-based Programming}\label{trial}
neuper@42498: %\section{Development of a Program on Trial}
neuper@42498: 
neuper@42498: This section presents all the concepts involved in TP-based
neuper@42498: programming and all the tasks to be accomplished by programmers. The
neuper@42504: presentation uses the running example from
neuper@42498: Fig.\ref{fig-interactive} on p.\pageref{fig-interactive}.
jan@42466: 
jan@42466: \subsection{Mechanization of Math --- Domain Engineering}\label{isabisac}
jan@42466: 
neuper@42467: %WN was Fachleute unter obigem Titel interessiert findet sich
jan@42466: %WN unterhalb des auskommentierten Textes.
jan@42466: 
jan@42466: %WN der Text unten spricht Benutzer-Aspekte anund ist nicht speziell
jan@42466: %WN auf Computer-Mathematiker fokussiert.
neuper@42464: % \paragraph{As mentioned in the introduction,} a prototype of an
neuper@42464: % educational math assistant called
neuper@42464: % {{\sisac}}\footnote{{{\sisac}}=\textbf{Isa}belle for
neuper@42464: % \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges
neuper@42464: % the gap between (1) introducation and (2) application of mathematics:
neuper@42464: % {{\sisac}} is based on Computer Theorem Proving (TP), a technology which
neuper@42464: % requires each fact and each action justified by formal logic, so
neuper@42464: % {{{\sisac}{}}} makes justifications transparent to students in
neuper@42464: % interactive step-wise problem solving. By that way {{\sisac}} already
neuper@42464: % can serve both:
neuper@42464: % \begin{enumerate}
neuper@42464: %   \item Introduction of math stuff (in e.g. partial fraction
neuper@42464: % decomposition) by stepwise explaining and exercising respective
neuper@42464: % symbolic calculations with ``next step guidance (NSG)'' and rigorously
neuper@42464: % checking steps freely input by students --- this also in context with
neuper@42464: % advanced applications (where the stuff to be taught in higher
neuper@42464: % semesters can be skimmed through by NSG), and
neuper@42464: %   \item Application of math stuff in advanced engineering courses
neuper@42464: % (e.g. problems to be solved by inverse Z-transform in a Signal
neuper@42464: % Processing Lab) and now without much ado about basic math techniques
neuper@42464: % (like partial fraction decomposition): ``next step guidance'' supports
neuper@42464: % students in independently (re-)adopting such techniques.
neuper@42464: % \end{enumerate} 
neuper@42464: % Before the question is answers, how {{\sisac}}
neuper@42464: % accomplishes this task from a technical point of view, some remarks on
neuper@42464: % the state-of-the-art is given, therefor follow up Section~\ref{emas}.
neuper@42464: % 
neuper@42464: % \subsection{Educational Mathematics Assistants (EMAs)}\label{emas}
neuper@42464: % 
jan@42466: % \paragraph{Educational software in mathematics} is, if at all, based
jan@42466: % on Computer Algebra Systems (CAS, for instance), Dynamic Geometry
jan@42466: % Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org}
jan@42466: % \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC
jan@42466: % http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These
jan@42466: % base technologies are used to program math lessons and sometimes even
jan@42466: % exercises. The latter are cumbersome: the steps towards a solution of
jan@42466: % such an interactive exercise need to be provided with feedback, where
jan@42466: % at each step a wide variety of possible input has to be foreseen by
jan@42466: % the programmer - so such interactive exercises either require high
neuper@42464: % development efforts or the exercises constrain possible inputs.
neuper@42464: % 
jan@42466: % \subparagraph{A new generation} of educational math assistants (EMAs)
jan@42466: % is emerging presently, which is based on Theorem Proving (TP). TP, for
jan@42466: % instance Isabelle and Coq, is a technology which requires each fact
jan@42466: % and each action justified by formal logic. Pushed by demands for
jan@42466: % \textit{proven} correctness of safety-critical software TP advances
jan@42466: % into software engineering; from these advancements computer
jan@42466: % mathematics benefits in general, and math education in particular. Two
neuper@42464: % features of TP are immediately beneficial for learning:
neuper@42464: % 
jan@42466: % \paragraph{TP have knowledge in human readable format,} that is in
jan@42466: % standard predicate calculus. TP following the LCF-tradition have that
jan@42466: % knowledge down to the basic definitions of set, equality,
jan@42466: % etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html};
jan@42466: % following the typical deductive development of math, natural numbers
jan@42466: % are defined and their properties
jan@42466: % proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html},
jan@42466: % etc. Present knowledge mechanized in TP exceeds high-school
jan@42466: % mathematics by far, however by knowledge required in software
neuper@42464: % technology, and not in other engineering sciences.
neuper@42464: % 
jan@42466: % \paragraph{TP can model the whole problem solving process} in
jan@42466: % mathematical problem solving {\em within} a coherent logical
jan@42466: % framework. This is already being done by three projects, by
neuper@42464: % Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.
neuper@42464: % \par
jan@42466: % Having the whole problem solving process within a logical coherent
jan@42466: % system, such a design guarantees correctness of intermediate steps and
jan@42466: % of the result (which seems essential for math software); and the
jan@42466: % second advantage is that TP provides a wealth of theories which can be
jan@42466: % exploited for mechanizing other features essential for educational
neuper@42464: % software.
neuper@42464: % 
neuper@42464: % \subsubsection{Generation of User Guidance in EMAs}\label{user-guid}
neuper@42464: % 
jan@42466: % One essential feature for educational software is feedback to user
neuper@42464: % input and assistance in coming to a solution.
neuper@42464: % 
jan@42466: % \paragraph{Checking user input} by ATP during stepwise problem solving
jan@42466: % is being accomplished by the three projects mentioned above
jan@42466: % exclusively. They model the whole problem solving process as mentioned
jan@42466: % above, so all what happens between formalized assumptions (or formal
jan@42466: % specification) and goal (or fulfilled postcondition) can be
jan@42466: % mechanized. Such mechanization promises to greatly extend the scope of
neuper@42464: % educational software in stepwise problem solving.
neuper@42464: % 
jan@42466: % \paragraph{NSG (Next step guidance)} comprises the system's ability to
jan@42466: % propose a next step; this is a challenge for TP: either a radical
jan@42466: % restriction of the search space by restriction to very specific
jan@42466: % problem classes is required, or much care and effort is required in
jan@42466: % designing possible variants in the process of problem solving
neuper@42464: % \cite{proof-strategies-11}.
neuper@42464: % \par
jan@42466: % Another approach is restricted to problem solving in engineering
jan@42466: % domains, where a problem is specified by input, precondition, output
jan@42466: % and postcondition, and where the postcondition is proven by ATP behind
jan@42466: % the scenes: Here the possible variants in the process of problem
jan@42466: % solving are provided with feedback {\em automatically}, if the problem
jan@42466: % is described in a TP-based programing language: \cite{plmms10} the
jan@42466: % programmer only describes the math algorithm without caring about
jan@42466: % interaction (the respective program is functional and even has no
jan@42466: % input or output statements!); interaction is generated as a
jan@42466: % side-effect by the interpreter --- an efficient separation of concern
jan@42466: % between math programmers and dialog designers promising application
neuper@42464: % all over engineering disciplines.
neuper@42464: % 
neuper@42464: % 
neuper@42464: % \subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}
jan@42466: % Authoring new mathematics knowledge in {{\sisac}} can be compared with
jan@42466: % ``application programing'' of engineering problems; most of such
jan@42466: % programing uses CAS-based programing languages (CAS = Computer Algebra
neuper@42464: % Systems; e.g. Mathematica's or Maple's programing language).
neuper@42464: % 
jan@42466: % \paragraph{A novel type of TP-based language} is used by {{\sisac}{}}
jan@42466: % \cite{plmms10} for describing how to construct a solution to an
jan@42466: % engineering problem and for calling equation solvers, integration,
jan@42466: % etc~\footnote{Implementation of CAS-like functionality in TP is not
jan@42466: % primarily concerned with efficiency, but with a didactic question:
jan@42466: % What to decide for: for high-brow algorithms at the state-of-the-art
jan@42466: % or for elementary algorithms comprehensible for students?} within TP;
jan@42466: % TP can ensure ``systems that never make a mistake'' \cite{casproto} -
neuper@42464: % are impossible for CAS which have no logics underlying.
neuper@42464: % 
jan@42466: % \subparagraph{Authoring is perfect} by writing such TP based programs;
jan@42466: % the application programmer is not concerned with interaction or with
jan@42466: % user guidance: this is concern of a novel kind of program interpreter
jan@42466: % called Lucas-Interpreter. This interpreter hands over control to a
jan@42466: % dialog component at each step of calculation (like a debugger at
jan@42466: % breakpoints) and calls automated TP to check user input following
neuper@42464: % personalized strategies according to a feedback module.
neuper@42464: % \par
jan@42466: % However ``application programing with TP'' is not done with writing a
jan@42466: % program: according to the principles of TP, each step must be
jan@42466: % justified. Such justifications are given by theorems. So all steps
jan@42466: % must be related to some theorem, if there is no such theorem it must
jan@42466: % be added to the existing knowledge, which is organized in so-called
jan@42466: % \textbf{theories} in Isabelle. A theorem must be proven; fortunately
jan@42466: % Isabelle comprises a mechanism (called ``axiomatization''), which
jan@42466: % allows to omit proofs. Such a theorem is shown in
neuper@42464: % Example~\ref{eg:neuper1}.
jan@42466: 
jan@48774: The running example requires to determine the inverse ${\cal Z}$-transform
jan@48774: for a class of functions. The domain of Signal Processing
jan@42466: is accustomed to specific notation for the resulting functions, which
jan@42511: are absolutely capable of being totalled and are called step-response: $u[n]$, where $u$ is the
jan@42466: function, $n$ is the argument and the brackets indicate that the
neuper@42504: arguments are discrete. Surprisingly, Isabelle accepts the rules for
jan@42513: $z^{-1}$ in this traditional notation~\footnote{Isabelle
jan@42466: experts might be particularly surprised, that the brackets do not
jan@42466: cause errors in typing (as lists).}:
neuper@42464: %\vbox{
neuper@42464: % \begin{example}
jan@42463:   \label{eg:neuper1}
jan@42509:   {\footnotesize\begin{tabbing}
jan@42463:   123\=123\=123\=123\=\kill
jan@42509: 
jan@42513:   01\>axiomatization where \\
jan@42513:   02\>\>  rule1: ``$z^{-1}\;1 = \delta [n]$'' and\\
jan@42513:   03\>\>  rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow z^{-1}\;z / (z - 1) = u [n]$'' and\\
jan@42513:   04\>\>  rule3: ``$\vert\vert z \vert\vert < 1 \Rightarrow z / (z - 1) = -u [-n - 1]$'' and \\
jan@42513:   05\>\>  rule4: ``$\vert\vert z \vert\vert > \vert\vert$ $\alpha$ $\vert\vert \Rightarrow z / (z - \alpha) = \alpha^n \cdot u [n]$'' and\\
jan@42513:   06\>\>  rule5: ``$\vert\vert z \vert\vert < \vert\vert \alpha \vert\vert \Rightarrow z / (z - \alpha) = -(\alpha^n) \cdot u [-n - 1]$'' and\\
jan@42513:   07\>\>  rule6: ``$\vert\vert z \vert\vert > 1 \Rightarrow z/(z - 1)^2 = n \cdot u [n]$''
jan@42509:   \end{tabbing}}
neuper@42464: % \end{example}
jan@42466: %}
jan@42466: These 6 rules can be used as conditional rewrite rules, depending on
jan@48774: the respective convergence radius. Satisfaction from accordance with traditional
jan@48774: notation contrasts with the above word {\em axiomatization}: As TP-based, the
jan@42466: programming language expects these rules as {\em proved} theorems, and
jan@42466: not as axioms implemented in the above brute force manner; otherwise
jan@42466: all the verification efforts envisaged (like proof of the
jan@42466: post-condition, see below) would be meaningless.
jan@42466: 
neuper@42514: Isabelle provides a large body of knowledge, rigorously proved from
jan@42466: the basic axioms of mathematics~\footnote{This way of rigorously
jan@42466: deriving all knowledge from first principles is called the
jan@48774: LCF-paradigm in TP.}. In the case of the ${\cal Z}$-transform the most advanced
jan@42511: knowledge can be found in the theories on Multivariate
jan@42466: Analysis~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate\_Analysis}. However,
jan@42466: building up knowledge such that a proof for the above rules would be
jan@42466: reasonably short and easily comprehensible, still requires lots of
jan@42466: work (and is definitely out of scope of our case study).
jan@42466: 
neuper@42508: %REMOVED DUE TO SPACE CONSTRAINTS
neuper@42508: %At the state-of-the-art in mechanization of knowledge in engineering
neuper@42508: %sciences, the process does not stop with the mechanization of
neuper@42508: %mathematics traditionally used in these sciences. Rather, ``Formal
neuper@42508: %Methods''~\cite{ fm-03} are expected to proceed to formal and explicit
neuper@42508: %description of physical items.  Signal Processing, for instance is
neuper@42508: %concerned with physical devices for signal acquisition and
neuper@42508: %reconstruction, which involve measuring a physical signal, storing it,
neuper@42508: %and possibly later rebuilding the original signal or an approximation
neuper@42508: %thereof. For digital systems, this typically includes sampling and
neuper@42508: %quantization; devices for signal compression, including audio
neuper@42508: %compression, image compression, and video compression, etc.  ``Domain
neuper@42508: %engineering''\cite{db:dom-eng} is concerned with {\em specification}
neuper@42508: %of these devices' components and features; this part in the process of
neuper@42508: %mechanization is only at the beginning in domains like Signal
neuper@42508: %Processing.
neuper@42508: %
neuper@42508: %TP-based programming, concern of this paper, is determined to
neuper@42508: %add ``algorithmic knowledge'' to the mechanised body of knowledge.
neuper@42508: %% in Fig.\ref{fig:mathuni} on
neuper@42508: %% p.\pageref{fig:mathuni}.  As we shall see below, TP-based programming
neuper@42508: %% starts with a formal {\em specification} of the problem to be solved.
neuper@42508: %% \begin{figure}
neuper@42508: %%   \begin{center}
neuper@42508: %%     \includegraphics[width=110mm]{../../fig/jrocnik/math-universe-small}
neuper@42508: %%     \caption{The three-dimensional universe of mathematics knowledge}
neuper@42508: %%     \label{fig:mathuni}
neuper@42508: %%   \end{center}
neuper@42508: %% \end{figure}
neuper@42508: %% The language for both axes is defined in the axis at the bottom, deductive
neuper@42508: %% knowledge, in {\sisac} represented by Isabelle's theories.
jan@42466: 
jan@42466: \subsection{Preparation of Simplifiers for the Program}\label{simp}
jan@42469: 
jan@42511: All evaluation in the prototype's Lucas-Interpreter is done by term rewriting on
neuper@42507: Isabelle's terms, see \S\ref{meth} below; in this section some of respective
jan@42505: preparations are described. In order to work reliably with term rewriting, the
jan@42505: respective rule-sets must be confluent and terminating~\cite{nipk:rew-all-that},
jan@42505: then they are called (canonical) simplifiers. These properties do not go without
jan@42505: saying, their establishment is a difficult task for the programmer; this task is
neuper@42508: not yet supported in the prototype.
jan@42505: 
jan@42505: The prototype rewrites using theorems only. Axioms which are theorems as well 
jan@42505: have been already shown in \S\ref{eg:neuper1} on p.\pageref{eg:neuper1} , we
jan@42512: assemble them in a rule-set and apply them in ML as follows:
jan@42505: 
neuper@42508: {\footnotesize
neuper@42508: \begin{verbatim}
jan@42512:    01  val inverse_z = Rls 
jan@42512:    02      {id       = "inverse_z",
jan@42512:    03       rew_ord  = dummy_ord,
jan@42512:    04       erls     = Erls,
jan@42512:    05       rules    = [Thm ("rule1", @{thm rule1}), Thm ("rule2", @{thm rule1}), 
jan@42512:    06                   Thm ("rule3", @{thm rule3}), Thm ("rule4", @{thm rule4}), 
jan@42512:    07                   Thm ("rule5", @{thm rule5}), Thm ("rule6", @{thm rule6})],
jan@42512:    08       errpatts = [],
jan@42512:    09       scr      = ""}
neuper@42508: \end{verbatim}}
jan@42505: 
neuper@42508: \noindent The items, line by line, in the above record have the following purpose:
neuper@42508: \begin{description}
jan@42512: \item[01..02] the ML-value \textit{inverse\_z} stores it's identifier
neuper@42508: as a string for ``reflection'' when switching between the language
neuper@42508: layers of Isabelle/ML (like in the Lucas-Interpreter) and
neuper@42508: Isabelle/Isar (like in the example program on p.\pageref{s:impl} on
neuper@42508: line {\rm 12}).
jan@42475: 
jan@42512: \item[03..04] both, (a) the rewrite-order~\cite{nipk:rew-all-that}
neuper@42508: \textit{rew\_ord} and (b) the rule-set \textit{erls} are trivial here:
neuper@42508: (a) the \textit{rules} in {\rm 07..12} don't need ordered rewriting
neuper@42508: and (b) the assumptions of the \textit{rules} need not be evaluated
neuper@42508: (they just go into the context during rewriting).
jan@42505: 
jan@42512: \item[05..07] the \textit{rules} are the axioms from p.\pageref{eg:neuper1};
neuper@42508: also ML-functions (\S\ref{funs}) can come into this list as shown in
neuper@42508: \S\ref{flow-prep}; so they are distinguished by type-constructors \textit{Thm}
neuper@42508: and \textit{Calc} respectively; for the purpose of reflection both
neuper@42508: contain their identifiers.
jan@42502: 
jan@42512: \item[08..09] are error-patterns not discussed here and \textit{scr}
neuper@42508: is prepared to get a program, automatically generated by {\sisac} for
neuper@42508: producing intermediate rewrites when requested by the user.
jan@42502: 
neuper@42508: \end{description}
jan@42505: 
neuper@42514: %OUTCOMMENTED DUE TO SPACE RESTRICTIONS
neuper@42514: % \noindent It is advisable to immediately test rule-sets; for that
neuper@42514: % purpose an appropriate term has to be created; \textit{parse} takes a
neuper@42514: % context \textit{ctxt} and a string (with \textit{ZZ\_1} denoting ${\cal
neuper@42514: % Z}^{-1}$) and creates a term:
neuper@42514: % 
neuper@42514: % {\footnotesize
neuper@42514: % \begin{verbatim}
neuper@42514: %    01 ML {*
neuper@42514: %    02   val t = parse ctxt "ZZ_1 (z / (z - 1) + z / (z - </alpha>) + 1)";
neuper@42514: %    03 *}
neuper@42514: %    04 val t = Const ("Build_Inverse_Z_Transform.ZZ_1", 
neuper@42514: %    05   "RealDef.real => RealDef.real => RealDef.real") $
neuper@42514: %    06     (Const (...) $ (Const (...) $ Free (...) $ (Const (...) $ Free (...) 
neuper@42514: % \end{verbatim}}
neuper@42514: % 
neuper@42514: % \noindent The internal representation of the term, as required for
neuper@42514: % rewriting, consists of \textit{Const}ants, a pair of a string
neuper@42514: % \textit{"Groups.plus\_class.plus"} for $+$ and a type, variables
neuper@42514: % \textit{Free} and the respective constructor \textit{\$}. Now the
neuper@42514: % term can be rewritten by the rule-set \textit{inverse\_z}:
neuper@42514: % 
neuper@42514: % {\footnotesize
neuper@42514: % \begin{verbatim}
neuper@42514: %    01 ML {*
neuper@42514: %    02   val SOME (t', asm) = rewrite_set_ @{theory} inverse\_z t;
neuper@42514: %    03   term2str t';
neuper@42514: %    04   terms2str asm;
neuper@42514: %    05 *}
neuper@42514: %    06 val it = "u[n] + </alpha> ^ n * u[n] + </delta>[n]" : string
neuper@42514: %    07 val it = "|| z || > 1 & || z || > </alpha>" : string
neuper@42514: % \end{verbatim}}
neuper@42514: % 
neuper@42514: % \noindent The resulting term \textit{t} and the assumptions
neuper@42514: % \textit{asm} are converted to readable strings by \textit{term2str}
neuper@42514: % and \textit{terms2str}.
jan@42505: 
jan@42466: \subsection{Preparation of ML-Functions}\label{funs}
neuper@42504: Some functionality required in programming, cannot be accomplished by
neuper@42504: rewriting. So the prototype has a mechanism to call functions within
neuper@42514: the rewrite-engine: certain redexes in Isabelle terms call these
neuper@42504: functions written in SML~\cite{pl:milner97}, the implementation {\em
neuper@42504: and} meta-language of Isabelle. The programmer has to use this
neuper@42504: mechanism.
jan@42469: 
neuper@42498: In the running example's program on p.\pageref{s:impl} the lines {\rm
neuper@42498: 05} and {\rm 06} contain such functions; we go into the details with
neuper@42498: \textit{argument\_in X\_z;}. This function fetches the argument from a
neuper@42498: function application: Line {\rm 03} in the example calculation on
neuper@42498: p.\pageref{exp-calc} is created by line {\rm 06} of the example
neuper@42498: program on p.\pageref{s:impl} where the program's environment assigns
neuper@42498: the value \textit{X z} to the variable \textit{X\_z}; so the function
neuper@42498: shall extract the argument \textit{z}.
jan@42469: 
neuper@42498: \medskip In order to be recognised as a function constant in the
neuper@42499: program source the constant needs to be declared in a theory, here in
neuper@42498: \textit{Build\_Inverse\_Z\_Transform.thy}; then it can be parsed in
neuper@42498: the context \textit{ctxt} of that theory:
neuper@42504: 
neuper@42498: {\footnotesize
neuper@42498: \begin{verbatim}
jan@42513: 01   consts
jan@42513: 02     argument'_in :: "real => real" ("argument'_in _" 10)
neuper@42507: \end{verbatim}}
neuper@42498:    
neuper@42507: %^3.2^    ML {* val SOME t = parse ctxt "argument_in (X z)"; *}
neuper@42507: %^3.2^    val t = Const ("Build_Inverse_Z_Transform.argument'_in", "RealDef.real ⇒ RealDef.real") 
neuper@42507: %^3.2^              $ (Free ("X", "RealDef.real ⇒ RealDef.real") $ Free ("z", "RealDef.real")): term
neuper@42507: %^3.2^ \end{verbatim}}
neuper@42507: %^3.2^ 
neuper@42507: %^3.2^ \noindent Parsing produces a term \texttt{t} in internal
neuper@42507: %^3.2^ representation~\footnote{The attentive reader realizes the 
neuper@42507: %^3.2^ differences between interal and extermal representation even in the
neuper@42507: %^3.2^ strings, i.e \texttt{'\_}}, consisting of \texttt{Const
neuper@42507: %^3.2^ ("argument'\_in", type)} and the two variables \texttt{Free ("X",
neuper@42507: %^3.2^ type)} and \texttt{Free ("z", type)}, \texttt{\$} is the term
neuper@42507: %^3.2^ constructor. 
neuper@42507: The function body below is implemented directly in SML,
neuper@42499: i.e in an \texttt{ML \{* *\}} block; the function definition provides
neuper@42499: a unique prefix \texttt{eval\_} to the function name:
jan@42473: 
neuper@42498: {\footnotesize
jan@42470: \begin{verbatim}
jan@42513: 01   ML {*
jan@42513: 02     fun eval_argument_in _ 
jan@42513: 03       "Build_Inverse_Z_Transform.argument'_in" 
jan@42513: 04       (t as (Const ("Build_Inverse_Z_Transform.argument'_in", _) $(f $arg))) _ =
jan@42513: 05         if is_Free arg (*could be something to be simplified before*)
jan@42513: 06         then SOME (term2str t ^"="^ term2str arg, Trueprop $(mk_equality (t, arg)))
jan@42513: 07         else NONE
jan@42513: 08     | eval_argument_in _ _ _ _ = NONE;
jan@42513: 09   *}
neuper@42498: \end{verbatim}}
jan@42469: 
jan@48766: \noindent The function body creates either \texttt{NONE}
neuper@42514: telling the rewrite-engine to search for the next redex, or creates an
neuper@42498: ad-hoc theorem for rewriting, thus the programmer needs to adopt many
neuper@42498: technicalities of Isabelle, for instance, the \textit{Trueprop}
neuper@42498: constant.
jan@42469: 
neuper@42498: \bigskip This sub-task particularly sheds light on basic issues in the
jan@42511: design of a programming language, the integration of differential language
neuper@42498: layers, the layer of Isabelle/Isar and Isabelle/ML.
jan@42469: 
neuper@42498: Another point of improvement for the prototype is the rewrite-engine: The
neuper@42498: program on p.\pageref{s:impl} would not allow to contract the two lines {\rm 05}
neuper@42498: and {\rm 06} to
jan@42469: 
neuper@42498: {\small\it\label{s:impl}
neuper@42498: \begin{tabbing}
neuper@42498: 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
jan@42512: \>{\rm 05/06}\>\>\>  (z::real) = argument\_in (lhs X\_eq) ;
neuper@42498: \end{tabbing}}
jan@42469: 
neuper@42498: \noindent because nested function calls would require creating redexes
neuper@42498: inside-out; however, the prototype's rewrite-engine only works top down
neuper@42498: from the root of a term down to the leaves.
jan@42469: 
neuper@42504: How all these technicalities are to be checked in the prototype is 
neuper@42498: shown in \S\ref{flow-prep} below.
jan@42473: 
neuper@42498: % \paragraph{Explicit Problems} require explicit methods to solve them, and within
neuper@42498: % this methods we have some explicit steps to do. This steps can be unique for
neuper@42498: % a special problem or refindable in other problems. No mather what case, such
neuper@42498: % steps often require some technical functions behind. For the solving process
neuper@42498: % of the Inverse Z Transformation and the corresponding partial fraction it was
neuper@42498: % neccessary to build helping functions like \texttt{get\_denominator},
neuper@42498: % \texttt{get\_numerator} or \texttt{argument\_in}. First two functions help us
neuper@42498: % to filter the denominator or numerator out of a fraction, last one helps us to
neuper@42498: % get to know the bound variable in a equation.
neuper@42498: % \par
neuper@42498: % By taking \texttt{get\_denominator} as an example, we want to explain how to 
neuper@42498: % implement new functions into the existing system and how we can later use them
neuper@42498: % in our program.
neuper@42498: % 
neuper@42498: % \subsubsection{Find a place to Store the Function}
neuper@42498: % 
neuper@42498: % The whole system builds up on a well defined structure of Knowledge. This
neuper@42498: % Knowledge sets up at the Path:
neuper@42498: % \begin{center}\ttfamily src/Tools/isac/Knowledge\normalfont\end{center}
neuper@42498: % For implementing the Function \texttt{get\_denominator} (which let us extract
neuper@42498: % the denominator out of a fraction) we have choosen the Theory (file)
neuper@42498: % \texttt{Rational.thy}.
neuper@42498: % 
neuper@42498: % \subsubsection{Write down the new Function}
neuper@42498: % 
neuper@42498: % In upper Theory we now define the new function and its purpose:
neuper@42498: % \begin{verbatim}
neuper@42498: %   get_denominator :: "real => real"
neuper@42498: % \end{verbatim}
neuper@42498: % This command tells the machine that a function with the name
neuper@42498: % \texttt{get\_denominator} exists which gets a real expression as argument and
neuper@42498: % returns once again a real expression. Now we are able to implement the function
neuper@42498: % itself, upcoming example now shows the implementation of
neuper@42498: % \texttt{get\_denominator}.
neuper@42498: % 
neuper@42498: % %\begin{example}
neuper@42498: %   \label{eg:getdenom}
neuper@42498: %   \begin{verbatim}
neuper@42498: % 
neuper@42498: % 01  (*
neuper@42498: % 02   *("get_denominator",
neuper@42498: % 03   *  ("Rational.get_denominator", eval_get_denominator ""))
neuper@42498: % 04   *)
neuper@42498: % 05  fun eval_get_denominator (thmid:string) _ 
neuper@42498: % 06            (t as Const ("Rational.get_denominator", _) $
neuper@42498: % 07                (Const ("Rings.inverse_class.divide", _) $num 
neuper@42498: % 08                  $denom)) thy = 
neuper@42498: % 09          SOME (mk_thmid thmid "" 
neuper@42498: % 10              (Print_Mode.setmp [] 
neuper@42498: % 11                (Syntax.string_of_term (thy2ctxt thy)) denom) "", 
neuper@42498: % 12              Trueprop $ (mk_equality (t, denom)))
neuper@42498: % 13    | eval_get_denominator _ _ _ _ = NONE;\end{verbatim}
neuper@42498: % %\end{example}
neuper@42498: % 
neuper@42498: % Line \texttt{07} and \texttt{08} are describing the mode of operation the best -
neuper@42498: % there is a fraction\\ (\ttfamily Rings.inverse\_class.divide\normalfont) 
neuper@42498: % splittet
neuper@42498: % into its two parts (\texttt{\$num \$denom}). The lines before are additionals
neuper@42498: % commands for declaring the function and the lines after are modeling and 
neuper@42498: % returning a real variable out of \texttt{\$denom}.
neuper@42498: % 
neuper@42498: % \subsubsection{Add a test for the new Function}
neuper@42498: % 
neuper@42498: % \paragraph{Everytime when adding} a new function it is essential also to add
neuper@42498: % a test for it. Tests for all functions are sorted in the same structure as the
neuper@42498: % knowledge it self and can be found up from the path:
neuper@42498: % \begin{center}\ttfamily test/Tools/isac/Knowledge\normalfont\end{center}
neuper@42498: % This tests are nothing very special, as a first prototype the functionallity
neuper@42498: % of a function can be checked by evaluating the result of a simple expression
neuper@42498: % passed to the function. Example~\ref{eg:getdenomtest} shows the test for our
neuper@42498: % \textit{just} created function \texttt{get\_denominator}.
neuper@42498: % 
neuper@42498: % %\begin{example}
neuper@42498: % \label{eg:getdenomtest}
neuper@42498: % \begin{verbatim}
neuper@42498: % 
neuper@42498: % 01 val thy = @{theory Isac};
neuper@42498: % 02 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
neuper@42498: % 03 val SOME (_, t') = eval_get_denominator "" 0 t thy;
neuper@42498: % 04 if term2str t' = "get_denominator ((a + x) / b) = b" then ()
neuper@42498: % 05 else error "get_denominator ((a + x) / b) = b" \end{verbatim}
neuper@42498: % %\end{example}
neuper@42498: % 
neuper@42498: % \begin{description}
neuper@42498: % \item[01] checks if the proofer set up on our {\sisac{}} System.
neuper@42498: % \item[02] passes a simple expression (fraction) to our suddenly created
neuper@42498: %           function.
neuper@42498: % \item[04] checks if the resulting variable is the correct one (in this case
neuper@42498: %           ``b'' the denominator) and returns.
neuper@42498: % \item[05] handels the error case and reports that the function is not able to
neuper@42498: %           solve the given problem.
neuper@42498: % \end{description}
jan@42469: 
jan@42491: \subsection{Specification of the Problem}\label{spec}
jan@42491: %WN <--> \chapter 7 der Thesis
jan@42491: %WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..
jan@42491: 
neuper@42504: Mechanical treatment requires to translate a textual problem
neuper@42504: description like in Fig.\ref{fig-interactive} on
neuper@42504: p.\pageref{fig-interactive} into a {\em formal} specification. The
neuper@48777: formal specification of the running example could look like is this
neuper@48777: ~\footnote{The ``TODO'' in the postcondition indicates, that postconditions
neuper@48777: are not yet handled in the prototype; in particular, the postcondition, i.e.
neuper@48777: the correctness of the result is not yet automatically proved.}:
jan@42491: 
jan@42491: %WN Hier brauchen wir die Spezifikation des 'running example' ...
jan@42491: %JR Habe input, output und precond vom Beispiel eingefügt brauche aber Hilfe bei
jan@42491: %JR der post condition - die existiert für uns ja eigentlich nicht aka
jan@42491: %JR haben sie bis jetzt nicht beachtet WN...
jan@42491: %WN2 Mein Vorschlag ist, das TODO zu lassen und deutlich zu kommentieren.
jan@42491: %JR2 done
jan@42491: 
neuper@42504: \label{eg:neuper2}
neuper@42504: {\small\begin{tabbing}
neuper@42504:   123\=123\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill
neuper@42504:   %\hfill \\
neuper@42504:   \>Specification:\\
neuper@42507:   \>  \>input    \>: ${\it filterExpression} \;\;X\;z=\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}, \;{\it domain}\;\mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$\\
neuper@42504:   \>\>precond  \>: $\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}\;\; {\it continuous\_on}\; \mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$ \\
neuper@42504:   \>\>output   \>: stepResponse $x[n]$ \\
neuper@42504:   \>\>postcond \>: TODO
neuper@42504: \end{tabbing}}
jan@42491: 
jan@42500: %JR wie besprochen, kein remark, keine begründung, nur simples "nicht behandelt"
jan@42500: 
jan@42500: % \begin{remark}
jan@42500: %    Defining the postcondition requires a high amount mathematical 
jan@42500: %    knowledge, the difficult part in our case is not to set up this condition 
jan@42500: %    nor it is more to define it in a way the interpreter is able to handle it. 
jan@42500: %    Due the fact that implementing that mechanisms is quite the same amount as 
jan@42500: %    creating the programm itself, it is not avaible in our prototype.
jan@42500: %    \label{rm:postcond}
jan@42500: % \end{remark}
jan@42491: 
neuper@42504: The implementation of the formal specification in the present
neuper@42504: prototype, still bar-bones without support for authoring, is done
neuper@42504: like that:
jan@42491: %WN Kopie von Inverse_Z_Transform.thy, leicht versch"onert:
neuper@42504: 
jan@42491: {\footnotesize\label{exp-spec}
jan@42491: \begin{verbatim}
neuper@42504:    00 ML {*
jan@42491:    01  store_specification
jan@42491:    02    (prepare_specification
neuper@42504:    03      "pbl_SP_Ztrans_inv"
neuper@42504:    04      ["Jan Rocnik"]
jan@42491:    05      thy
jan@42491:    06      ( ["Inverse", "Z_Transform", "SignalProcessing"],
neuper@42507:    07        [ ("#Given", ["filterExpression X_eq", "domain D"]),
neuper@42507:    08          ("#Pre"  , ["(rhs X_eq) is_continuous_in D"]),
jan@42494:    09          ("#Find" , ["stepResponse n_eq"]),
neuper@42507:    10          ("#Post" , [" TODO "])])
neuper@42507:    11        prls
neuper@42507:    12        NONE
neuper@42507:    13        [["SignalProcessing","Z_Transform","Inverse"]]);
neuper@42504:    14 *}
jan@42491: \end{verbatim}}
neuper@42504: 
jan@42491: Although the above details are partly very technical, we explain them
jan@42491: in order to document some intricacies of TP-based programming in the
jan@42491: present state of the {\sisac} prototype:
jan@42491: \begin{description}
jan@42491: \item[01..02]\textit{store\_specification:} stores the result of the
jan@42491: function \textit{prep\_specification} in a global reference
jan@42491: \textit{Unsynchronized.ref}, which causes principal conflicts with
jan@42511: Isabelle's asynchronous document model~\cite{Wenzel-11:doc-orient} and
jan@42491: parallel execution~\cite{Makarius-09:parall-proof} and is under
jan@42491: reconstruction already.
jan@42491: 
neuper@42504: \textit{prep\_specification:} translates the specification to an internal format
jan@42491: which allows efficient processing; see for instance line {\rm 07}
jan@42491: below.
neuper@42504: \item[03..04] are a unique identifier for the specification within {\sisac}
neuper@42504: and the ``mathematics author'' holding the copy-rights.
jan@42491: \item[05] is the Isabelle \textit{theory} required to parse the
jan@42491: specification in lines {\rm 07..10}.
jan@42491: \item[06] is a key into the tree of all specifications as presented to
jan@42511: the user (where some branches might be hidden by the dialogue
jan@42491: component).
jan@42491: \item[07..10] are the specification with input, pre-condition, output
neuper@42507: and post-condition respectively; note that the specification contains
neuper@42507: variables to be instantiated with concrete values for a concrete problem ---
neuper@42507: thus the specification actually captures a class of problems. The post-condition is not handled in
neuper@42504: the prototype presently.
neuper@42507: \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
neuper@42507: rewriting determined by rule-sets.
jan@42491: \item[12]\textit{NONE:} could be \textit{SOME ``solve ...''} for a
jan@42491: problem associated to a function from Computer Algebra (like an
jan@42491: equation solver) which is not the case here.
neuper@42504: \item[13] is a list of methods solving the specified problem (here
neuper@42504: only one list item) represented analogously to {\rm 06}.
jan@42491: \end{description}
jan@42491: 
jan@42491: 
jan@42491: %WN die folgenden Erkl"arungen finden sich durch "grep -r 'datatype pbt' *"
jan@42491: %WN ...
jan@42491: %  type pbt = 
jan@42491: %     {guh  : guh,         (*unique within this isac-knowledge*)
jan@42491: %      mathauthors: string list, (*copyright*)
jan@42491: %      init  : pblID,      (*to start refinement with*)
jan@42491: %      thy   : theory,     (* which allows to compile that pbt
jan@42491: %			  TODO: search generalized for subthy (ref.p.69*)
jan@42491: %      (*^^^ WN050912 NOT used during application of the problem,
jan@42491: %       because applied terms may be from 'subthy' as well as from super;
jan@42491: %       thus we take 'maxthy'; see match_ags !*)
jan@42491: %      cas   : term option,(*'CAS-command'*)
jan@42491: %      prls  : rls,        (* for preds in where_*)
jan@42491: %      where_: term list,  (* where - predicates*)
jan@42491: %      ppc   : pat list,
jan@42491: %      (*this is the model-pattern; 
jan@42491: %       it contains "#Given","#Where","#Find","#Relate"-patterns
Walther@60469: %       for constraints on identifiers see "fun copy_name"*)
jan@42491: %      met   : metID list}; (* methods solving the pbt*)
jan@42491: %
jan@42491: %WN weil dieser Code sehr unaufger"aumt ist, habe ich die Erkl"arungen
jan@42491: %WN oben selbst geschrieben.
jan@42491: 
jan@42491: 
jan@42491: 
jan@42491: 
jan@42491: %WN das w"urde ich in \sec\label{progr} verschieben und
jan@42491: %WN das SubProblem partial fractions zum Erkl"aren verwenden.
jan@42491: % Such a specification is checked before the execution of a program is
jan@42491: % started, the same applies for sub-programs. In the following example
jan@42491: % (Example~\ref{eg:subprob}) shows the call of such a subproblem:
jan@42491: % 
jan@42491: % \vbox{
jan@42491: %   \begin{example}
jan@42491: %   \label{eg:subprob}
jan@42491: %   \hfill \\
jan@42491: %   {\ttfamily \begin{tabbing}
jan@42491: %   ``(L\_L::bool list) = (\=SubProblem (\=Test','' \\
jan@42491: %   ``\>\>[linear,univariate,equation,test],'' \\
jan@42491: %   ``\>\>[Test,solve\_linear])'' \\
jan@42491: %   ``\>[BOOL equ, REAL z])'' \\
jan@42491: %   \end{tabbing}
jan@42491: %   }
jan@42491: %   {\small\textit{
jan@42491: %     \noindent If a program requires a result which has to be
jan@42491: % calculated first we can use a subproblem to do so. In our specific
jan@42491: % case we wanted to calculate the zeros of a fraction and used a
jan@42491: % subproblem to calculate the zeros of the denominator polynom.
jan@42491: %     }}
jan@42491: %   \end{example}
jan@42491: % }
jan@42491: 
jan@42491: \subsection{Implementation of the Method}\label{meth}
neuper@42504: A method collects all data required to interpret a certain program by
neuper@42504: Lucas-Interpretation. The \texttt{program} from p.\pageref{s:impl} of
neuper@42507: the running example is embedded on the last line in the following method:
neuper@42504: %The methods represent the different ways a problem can be solved. This can
neuper@42504: %include mathematical tactics as well as tactics taught in different courses.
neuper@42504: %Declaring the Method itself gives us the possibilities to describe the way of 
neuper@42504: %calculation in deep, as well we get the oppertunities to build in different
neuper@42504: %rulesets.
jan@42491: 
jan@42502: {\footnotesize
jan@42491: \begin{verbatim}
neuper@42504:    00 ML {*
neuper@42504:    01  store_method
neuper@42504:    02    (prep_method
neuper@42504:    03      "SP_InverseZTransformation_classic" 
neuper@42504:    04      ["Jan Rocnik"]
neuper@42504:    05      thy 
neuper@42507:    06      ( ["SignalProcessing", "Z_Transform", "Inverse"], 
neuper@42507:    07        [ ("#Given", ["filterExpression X_eq", "domain D"]),
neuper@42507:    08          ("#Pre"  , ["(rhs X_eq) is_continuous_in D"]),
neuper@42507:    09          ("#Find" , ["stepResponse n_eq"]),
neuper@42507:    10        rew_ord  erls
neuper@42507:    11        srls  prls  nrls
neuper@42507:    12        errpats 
neuper@42507:    13        program);
neuper@42507:    14 *}
neuper@42504: \end{verbatim}}
jan@42494: 
neuper@42504: \noindent The above code stores the whole structure analogously to a
neuper@42507: specification as described above:
neuper@42504: \begin{description}
neuper@42504: \item[01..06] are identical to those for the example specification on
neuper@42504: p.\pageref{exp-spec}.
jan@42494: 
neuper@42504: \item[07..09] show something looking like the specification; this is a
neuper@42507: {\em guard}: as long as not all \textit{Given} items are present and
neuper@42507: the \textit{Pre}-conditions is not true, interpretation of the program
neuper@42504: is not started.
neuper@42504: 
neuper@42507: \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
neuper@42507: \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 
neuper@42507: \textit{srls, prls, nrls} feature evaluating (a) the ML-functions in the program (e.g.
jan@42511: \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}
neuper@42507: and (c) is required for the derivation-machinery checking user-input formulas.
neuper@42504: 
neuper@42507: \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}.
jan@42494: \end{description}
neuper@42507: The many rule-sets above cause considerable efforts for the
neuper@42507: programmers, in particular, because there are no tools for checking
neuper@42507: essential features of rule-sets.
neuper@42504: 
neuper@42504: % is again very technical and goes hard in detail. Unfortunataly
neuper@42504: % most declerations are not essential for a basic programm but leads us to a huge
neuper@42504: % range of powerful possibilities.
neuper@42504: % 
neuper@42504: % \begin{description}
neuper@42504: % \item[01..02] stores the method with the given name into the system under a global
neuper@42504: % reference.
neuper@42504: % \item[03] specifies the topic within which context the method can be found.
neuper@42504: % \item[04..05] as the requirements for different methods can be deviant we 
neuper@42504: % declare what is \emph{given} and and what to \emph{find} for this specific method.
neuper@42504: % The code again helds on the topic of the case studie, where the inverse 
neuper@42504: % z-transformation does a switch between a term describing a electrical filter into
neuper@42504: % its step response. Also the datatype has to be declared (bool - due the fact that 
neuper@42504: % we handle equations).
neuper@42504: % \item[06] \emph{rewrite order} is the order of this rls (ruleset), where one 
neuper@42504: % theorem of it is used for rewriting one single step.
neuper@42504: % \item[07] \texttt{rls} is the currently used ruleset for this method. This set
neuper@42504: % has already been defined before.
neuper@42504: % \item[08] we would have the possiblitiy to add this method to a predefined tree of
neuper@42504: % calculations, i.eg. if it would be a sub of a bigger problem, here we leave it
neuper@42504: % independend.
neuper@42504: % \item[09] The \emph{source ruleset}, can be used to evaluate list expressions in 
neuper@42504: % the source.
neuper@42504: % \item[10] \emph{predicates ruleset} can be used to indicates predicates within 
neuper@42504: % model patterns.
neuper@42504: % \item[11] The \emph{check ruleset} summarizes rules for checking formulas 
neuper@42504: % elementwise.
neuper@42504: % \item[12] \emph{error patterns} which are expected in this kind of method can be
neuper@42504: % pre-specified to recognize them during the method.
neuper@42504: % \item[13] finally the \emph{canonical ruleset}, declares the canonical simplifier 
neuper@42504: % of the specific method.
neuper@42504: % \item[14] for this code snipset we don't specify the programm itself and keep it 
neuper@42504: % empty. Follow up \S\ref{progr} for informations on how to implement this
neuper@42504: % \textit{main} part.
neuper@42504: % \end{description}
neuper@42504: 
neuper@42478: \subsection{Implementation of the TP-based Program}\label{progr} 
neuper@42507: So finally all the prerequisites are described and the final task can
neuper@42480: be addressed. The program below comes back to the running example: it
neuper@42480: computes a solution for the problem from Fig.\ref{fig-interactive} on
neuper@42480: p.\pageref{fig-interactive}. The reader is reminded of
neuper@42480: \S\ref{PL-isab}, the introduction of the programming language:
jan@42502: 
jan@42502: {\footnotesize\it\label{s:impl}
neuper@42482: \begin{tabbing}
neuper@42478: 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
neuper@42507: \>{\rm 00}\>ML \{*\\
neuper@42480: \>{\rm 00}\>val program =\\
neuper@42480: \>{\rm 01}\>  "{\tt Program} InverseZTransform (X\_eq::bool) =   \\
neuper@42482: \>{\rm 02}\>\>  {\tt let}                                       \\
neuper@42468: \>{\rm 03}\>\>\>  X\_eq = {\tt Take} X\_eq ;   \\
neuper@42507: \>{\rm 04}\>\>\>  X\_eq = {\tt Rewrite} prep\_for\_part\_frac X\_eq ; \\
neuper@42468: \>{\rm 05}\>\>\>  (X\_z::real) = lhs X\_eq ;       \\ %no inside-out evaluation
neuper@42468: \>{\rm 06}\>\>\>  (z::real) = argument\_in X\_z; \\
neuper@42468: \>{\rm 07}\>\>\>  (part\_frac::real) = {\tt SubProblem} \\
neuper@42478: \>{\rm 08}\>\>\>\>\>\>\>\>  ( Isac, [partial\_fraction, rational, simplification], [] )\\
neuper@42478: %\>{\rm 10}\>\>\>\>\>\>\>\>\>  [simplification, of\_rationals, to\_partial\_fraction] ) \\
neuper@42478: \>{\rm 09}\>\>\>\>\>\>\>\>  [ (rhs X\_eq)::real, z::real ]; \\
neuper@42478: \>{\rm 10}\>\>\>  (X'\_eq::bool) = {\tt Take} ((X'::real =$>$ bool) z = ZZ\_1 part\_frac) ; \\
neuper@42507: \>{\rm 11}\>\>\>  X'\_eq = (({\tt Rewrite\_Set} prep\_for\_inverse\_z) @@   \\
neuper@42478: \>{\rm 12}\>\>\>\>\>  $\;\;$ ({\tt Rewrite\_Set} inverse\_z)) X'\_eq \\
neuper@42482: \>{\rm 13}\>\>  {\tt in } \\
neuper@42504: \>{\rm 14}\>\>\>  X'\_eq"\\
neuper@42507: \>{\rm 15}\>*\}
neuper@42478: \end{tabbing}}
neuper@42468: % ORIGINAL FROM Inverse_Z_Transform.thy
neuper@42468: % "Script InverseZTransform (X_eq::bool) =            "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
neuper@42468: % "(let X = Take X_eq;                                "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
neuper@42468: % "  X' = Rewrite ruleZY False X;                     "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
neuper@42468: % "  (X'_z::real) = lhs X';                           "^(*            ?X' z*)
neuper@42468: % "  (zzz::real) = argument_in X'_z;                  "^(*            z *)
neuper@42468: % "  (funterm::real) = rhs X';                        "^(*            3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
neuper@42468: %
neuper@42468: % "  (pbz::real) = (SubProblem (Isac',                "^(**)
neuper@42468: % "    [partial_fraction,rational,simplification],    "^
neuper@42468: % "    [simplification,of_rationals,to_partial_fraction]) "^
neuper@42468: % "    [REAL funterm, REAL zzz]);                     "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
neuper@42468: %
neuper@42468: % "  (pbz_eq::bool) = Take (X'_z = pbz);              "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
neuper@42468: % "  pbz_eq = Rewrite ruleYZ False pbz_eq;            "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
neuper@42468: % "  pbz_eq = drop_questionmarks pbz_eq;              "^(*               4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
neuper@42468: % "  (X_zeq::bool) = Take (X_z = rhs pbz_eq);         "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
neuper@42468: % "  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]*)
neuper@42468: % "  n_eq = drop_questionmarks n_eq                   "^(*            X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
neuper@42468: % "in n_eq)"                                            (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
neuper@42480: The program is represented as a string and part of the method in
neuper@42480: \S\ref{meth}.  As mentioned in \S\ref{PL} the program is purely
neuper@42480: functional and lacks any input statements and output statements. So
neuper@42480: the steps of calculation towards a solution (and interactive tutoring
neuper@42480: in step-wise problem solving) are created as a side-effect by
neuper@42480: Lucas-Interpretation.  The side-effects are triggered by the tactics
neuper@42482: \texttt{Take}, \texttt{Rewrite}, \texttt{SubProblem} and
neuper@42482: \texttt{Rewrite\_Set} in the above lines {\rm 03, 04, 07, 10, 11} and
neuper@42507: {\rm 12} respectively. These tactics produce the respective lines in the
neuper@42480: calculation on p.\pageref{flow-impl}.
neuper@42478: 
neuper@42480: The above lines {\rm 05, 06} do not contain a tactics, so they do not
neuper@42480: immediately contribute to the calculation on p.\pageref{flow-impl};
neuper@42482: rather, they compute actual arguments for the \texttt{SubProblem} in
neuper@42480: line {\rm 09}~\footnote{The tactics also are break-points for the
neuper@42480: interpreter, where control is handed over to the user in interactive
neuper@42482: tutoring.}. Line {\rm 11} contains tactical \textit{@@}.
neuper@42480: 
neuper@42480: \medskip The above program also indicates the dominant role of interactive
neuper@42478: selection of knowledge in the three-dimensional universe of
jan@48766: mathematics. The \texttt{SubProblem} in the above lines
neuper@42478: {\rm 07..09} is more than a function call with the actual arguments
neuper@42478: \textit{[ (rhs X\_eq)::real, z::real ]}. The programmer has to determine
neuper@42478: three items:
neuper@42480: 
neuper@42478: \begin{enumerate}
neuper@42478: \item the theory, in the example \textit{Isac} because different
neuper@42478: methods can be selected in Pt.3 below, which are defined in different
neuper@42478: theories with \textit{Isac} collecting them.
neuper@42480: \item the specification identified by \textit{[partial\_fraction,
neuper@42480: rational, simplification]} in the tree of specifications; this
neuper@42480: specification is analogous to the specification of the main program
neuper@42480: described in \S\ref{spec}; the problem is to find a ``partial fraction
neuper@42480: decomposition'' for a univariate rational polynomial.
neuper@42480: \item the method in the above example is \textit{[ ]}, i.e. empty,
neuper@42480: which supposes the interpreter to select one of the methods predefined
neuper@42480: in the specification, for instance in line {\rm 13} in the running
neuper@42480: example's specification on p.\pageref{exp-spec}~\footnote{The freedom
neuper@42480: (or obligation) for selection carries over to the student in
neuper@42480: interactive tutoring.}.
neuper@42478: \end{enumerate}
neuper@42478: 
neuper@42480: The program code, above presented as a string, is parsed by Isabelle's
neuper@42480: parser --- the program is an Isabelle term. This fact is expected to
neuper@42480: simplify verification tasks in the future; on the other hand, this
jan@42511: fact causes troubles in error detection which are discussed as part
neuper@42514: of the work-flow in the subsequent section.
neuper@42467: 
neuper@42514: \section{Work-flow of Programming in the Prototype}\label{workflow}
neuper@42498: The new prover IDE Isabelle/jEdit~\cite{makar-jedit-12} is a great
neuper@42498: step forward for interactive theory and proof development. The
neuper@42498: {\sisac}-prototype re-uses this IDE as a programming environment.  The
neuper@42498: experiences from this re-use show, that the essential components are
neuper@42498: available from Isabelle/jEdit. However, additional tools and features
jan@42511: are required to achieve acceptable usability.
neuper@42498: 
neuper@42498: So notable experiences are reported here, also as a requirement
neuper@42498: capture for further development of TP-based languages and respective
neuper@42498: IDEs.
neuper@42468: 
jan@42466: \subsection{Preparations and Trials}\label{flow-prep}
neuper@42499: The many sub-tasks to be accomplished {\em before} the first line of
neuper@42499: program code can be written and tested suggest an approach which
neuper@42499: step-wise establishes the prerequisites. The case study underlying
neuper@42499: this paper~\cite{jrocnik-bakk} documents the approach in a separate
neuper@42499: Isabelle theory,
neuper@42499: \textit{Build\_Inverse\_Z\_Transform.thy}~\footnote{http://www.ist.tugraz.at/projects/isac/publ/Build\_Inverse\_Z\_Transform.thy}. Part
neuper@42499: II in the study comprises this theory, \LaTeX ed from the theory by
neuper@42499: use of Isabelle's document preparation system. This paper resembles
neuper@42499: the approach in \S\ref{isabisac} to \S\ref{meth}, which in actual
neuper@42499: implementation work involves several iterations.
neuper@42498: 
neuper@42499: \bigskip For instance, only the last step, implementing the program
neuper@42499: described in \S\ref{meth}, reveals details required. Let us assume,
neuper@42499: this is the ML-function \textit{argument\_in} required in line {\rm 06}
neuper@42499: of the example program on p.\pageref{s:impl}; how this function needs
neuper@42499: to be implemented in the prototype has been discussed in \S\ref{funs}
neuper@42499: already.
neuper@42498: 
neuper@42499: Now let us assume, that calling this function from the program code
neuper@42499: does not work; so testing this function is required in order to find out
neuper@42499: the reason: type errors, a missing entry of the function somewhere or
neuper@42499: even more nasty technicalities \dots
neuper@42498: 
neuper@42499: {\footnotesize
neuper@42482: \begin{verbatim}
jan@42513: 01   ML {*
jan@42513: 02     val SOME t = parseNEW ctxt "argument_in (X (z::real))";
jan@42513: 03     val SOME (str, t') = eval_argument_in "" 
jan@42513: 04       "Build_Inverse_Z_Transform.argument'_in" t 0;
jan@42513: 05     term2str t';
jan@42513: 06   *}
jan@42513: 07   val it = "(argument_in X z) = z": string\end{verbatim}}
neuper@42499: 
neuper@42499: \noindent So, this works: we get an ad-hoc theorem, which used in
neuper@42499: rewriting would reduce \texttt{argument\_in X z} to \texttt{z}. Now we check this
neuper@42499: reduction and create a rule-set \texttt{rls} for that purpose:
neuper@42499: 
neuper@42499: {\footnotesize
neuper@42482: \begin{verbatim}
jan@42513: 01   ML {*
jan@42513: 02     val rls = append_rls "test" e_rls 
jan@42513: 03       [Calc ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")]
jan@42513: 04     val SOME (t', asm) = rewrite_set_ @{theory} rls t;
jan@42513: 05   *}
jan@42513: 06   val t' = Free ("z", "RealDef.real"): term
jan@42513: 07   val asm = []: term list\end{verbatim}}
neuper@42499: 
neuper@42499: \noindent The resulting term \texttt{t'} is \texttt{Free ("z",
neuper@42499: "RealDef.real")}, i.e the variable \texttt{z}, so all is
neuper@42499: perfect. Probably we have forgotten to store this function correctly~?
neuper@42499: We review the respective \texttt{calclist} (again an
neuper@42499: \textit{Unsynchronized.ref} to be removed in order to adjust to
neuper@42514: Isabelle/Isar's asynchronous document model):
neuper@42499: 
neuper@42499: {\footnotesize
neuper@42499: \begin{verbatim}
jan@42513: 01   calclist:= overwritel (! calclist, 
jan@42513: 02    [("argument_in",
jan@42513: 03     ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")),
jan@42513: 04       ...
jan@42513: 05    ]);\end{verbatim}}
neuper@42499: 
neuper@42499: \noindent The entry is perfect. So what is the reason~? Ah, probably there
neuper@42499: is something messed up with the many rule-sets in the method, see \S\ref{meth} ---
neuper@42499: right, the function \texttt{argument\_in} is not contained in the respective
neuper@42499: rule-set \textit{srls} \dots this just as an example of the intricacies in
neuper@42499: debugging a program in the present state of the prototype.
neuper@42499: 
neuper@42499: \subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}
neuper@42499: Given all the prerequisites from \S\ref{isabisac} to \S\ref{meth},
neuper@42499: usually developed within several iterations, the program can be
neuper@42499: assembled; on p.\pageref{s:impl} there is the complete program of the
neuper@42499: running example.
neuper@42499: 
neuper@42499: The completion of this program required efforts for several weeks
neuper@42499: (after some months of familiarisation with {\sisac}), caused by the
neuper@42499: abundance of intricacies indicated above. Also writing the program is
neuper@42499: not pleasant, given Isabelle/Isar/ without add-ons for
neuper@42499: programming. Already writing and parsing a few lines of program code
neuper@42499: is a challenge: the program is an Isabelle term; Isabelle's parser,
neuper@42499: however, is not meant for huge terms like the program of the running
neuper@42499: example. So reading out the specific error (usually type errors) from
neuper@42499: Isabelle's message is difficult.
neuper@42499: 
neuper@42499: \medskip Testing the evaluation of the program has to rely on very
neuper@42514: simple tools. Step-wise execution is modeled by a function
neuper@42499: \texttt{me}, short for mathematics-engine~\footnote{The interface used
neuper@42514: by the front-end which created the calculation on
neuper@42499: p.\pageref{fig-interactive} is different from this function}:
neuper@42499: %the following is a simplification of the actual function 
neuper@42499: 
neuper@42499: {\footnotesize
neuper@42499: \begin{verbatim}
jan@42513: 01   ML {* me; *}
jan@42513: 02   val it = tac -> ctree * pos -> mout * tac * ctree * pos\end{verbatim}} 
neuper@42499: 
neuper@42499: \noindent This function takes as arguments a tactic \texttt{tac} which
neuper@42499: determines the next step, the step applied to the interpreter-state
neuper@42499: \texttt{ctree * pos} as last argument taken. The interpreter-state is
neuper@42499: a pair of a tree \texttt{ctree} representing the calculation created
neuper@42499: (see the example below) and a position \texttt{pos} in the
jan@42511: calculation. The function delivers a quadruple, beginning with the new
neuper@42499: formula \texttt{mout} and the next tactic followed by the new
neuper@42499: interpreter-state.
neuper@42499: 
neuper@42499: This function allows to stepwise check the program:
neuper@42499: 
neuper@48771: {\footnotesize\label{ml-check-program}
neuper@42482: \begin{verbatim}
jan@42513: 01   ML {*
jan@42513: 02     val fmz =
jan@42513: 03       ["filterExpression (X z = 3 / ((z::real) + 1/10 - 1/50*(1/z)))",
jan@42513: 04        "stepResponse (x[n::real]::bool)"];     
jan@42513: 05     val (dI,pI,mI) =
jan@42513: 06       ("Isac", 
jan@42513: 07        ["Inverse", "Z_Transform", "SignalProcessing"], 
jan@42513: 08        ["SignalProcessing","Z_Transform","Inverse"]);
jan@42513: 09     val (mout, tac, ctree, pos)  = CalcTreeTEST [(fmz, (dI, pI, mI))];
jan@42513: 10     val (mout, tac, ctree, pos)  = me tac (ctree, pos);
jan@42513: 11     val (mout, tac, ctree, pos)  = me tac (ctree, pos);
jan@42513: 12     val (mout, tac, ctree, pos)  = me tac (ctree, pos);
neuper@48771: 13     ...
neuper@48771: \end{verbatim}} 
neuper@42481: 
jan@42511: \noindent Several dozens of calls for \texttt{me} are required to
neuper@42499: create the lines in the calculation below (including the sub-problems
neuper@42499: not shown). When an error occurs, the reason might be located
neuper@42499: many steps before: if evaluation by rewriting, as done by the prototype,
neuper@42499: fails, then first nothing happens --- the effects come later and
neuper@42499: cause unpleasant checks.
neuper@42481: 
neuper@42499: The checks comprise watching the rewrite-engine for many different
neuper@42499: kinds of rule-sets (see \S\ref{meth}), the interpreter-state, in
neuper@42499: particular the environment and the context at the states position ---
neuper@42499: all checks have to rely on simple functions accessing the
neuper@42499: \texttt{ctree}. So getting the calculation below (which resembles the
neuper@42499: calculation in Fig.\ref{fig-interactive} on p.\pageref{fig-interactive})
neuper@42507: is the result of several weeks of development:
jan@42469: 
neuper@42498: {\small\it\label{exp-calc}
neuper@42468: \begin{tabbing}
neuper@42468: 123l\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=\kill
neuper@42468: \>{\rm 01}\> $\bullet$  \> {\tt Problem } (Inverse\_Z\_Transform, [Inverse, Z\_Transform, SignalProcessing])       \`\\
neuper@42468: \>{\rm 02}\>\> $\vdash\;\;X z = \frac{3}{z - \frac{1}{4} - \frac{1}{8} \cdot z^{-1}}$       \`{\footnotesize {\tt Take} X\_eq}\\
neuper@42507: \>{\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}\\
neuper@42468: \>{\rm 04}\>\> $\bullet$\> {\tt Problem } [partial\_fraction,rational,simplification]        \`{\footnotesize {\tt SubProblem} \dots}\\
neuper@42468: \>{\rm 05}\>\>\>  $\vdash\;\;\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=$    \`- - -\\
neuper@42468: \>{\rm 06}\>\>\>  $\frac{24}{-1 + -2 \cdot z + 8 \cdot z^2}$                                   \`- - -\\
neuper@42468: \>{\rm 07}\>\>\>  $\bullet$\> solve ($-1 + -2 \cdot z + 8 \cdot z^2,\;z$ )                      \`- - -\\
neuper@42468: \>{\rm 08}\>\>\>\>   $\vdash$ \> $\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=0$ \`- - -\\
neuper@42468: \>{\rm 09}\>\>\>\>   $z = \frac{2+\sqrt{-4+8}}{16}\;\lor\;z = \frac{2-\sqrt{-4+8}}{16}$           \`- - -\\
neuper@42468: \>{\rm 10}\>\>\>\>   $z = \frac{1}{2}\;\lor\;z =$ \_\_\_                                           \`- - -\\
neuper@42468: \>        \>\>\>\>   \_\_\_                                                                        \`- - -\\
neuper@42468: \>{\rm 11}\>\> \dots\> $\frac{4}{z - \frac{1}{2}} + \frac{-4}{z - \frac{-1}{4}}$                   \`\\
jan@42512: \>{\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)}\\
jan@42512: \>{\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 }\\
neuper@42468: \>{\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}\\
neuper@42468: \>{\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}}
neuper@42468: \end{tabbing}}
neuper@42507: The tactics on the right margin of the above calculation are those in
neuper@42507: the program on p.\pageref{s:impl} which create the respective formulas
neuper@42507: on the left.
neuper@42468: % ORIGINAL FROM Inverse_Z_Transform.thy
neuper@42468: %    "Script InverseZTransform (X_eq::bool) =            "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
neuper@42468: %    "(let X = Take X_eq;                                "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
neuper@42468: %    "  X' = Rewrite ruleZY False X;                     "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
neuper@42468: %    "  (X'_z::real) = lhs X';                           "^(*            ?X' z*)
neuper@42468: %    "  (zzz::real) = argument_in X'_z;                  "^(*            z *)
neuper@42468: %    "  (funterm::real) = rhs X';                        "^(*            3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
neuper@42468: % 
neuper@42468: %    "  (pbz::real) = (SubProblem (Isac',                "^(**)
neuper@42468: %    "    [partial_fraction,rational,simplification],    "^
neuper@42468: %    "    [simplification,of_rationals,to_partial_fraction]) "^
neuper@42468: %    "    [REAL funterm, REAL zzz]);                     "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
neuper@42468: % 
neuper@42468: %    "  (pbz_eq::bool) = Take (X'_z = pbz);              "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
neuper@42468: %    "  pbz_eq = Rewrite ruleYZ False pbz_eq;            "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
neuper@42468: %    "  pbz_eq = drop_questionmarks pbz_eq;              "^(*               4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
neuper@42468: %    "  (X_zeq::bool) = Take (X_z = rhs pbz_eq);         "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
neuper@42468: %    "  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]*)
neuper@42468: %    "  n_eq = drop_questionmarks n_eq                   "^(*            X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
neuper@42468: %    "in n_eq)"                                            (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
neuper@42468: 
neuper@42499: \subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}
neuper@42499: Finally \textit{Build\_Inverse\_Z\_Transform.thy} has got the job done
neuper@42499: and the knowledge accumulated in it can be distributed to appropriate
neuper@42499: theories: the program to \textit{Inverse\_Z\_Transform.thy}, the
neuper@42499: sub-problem accomplishing the partial fraction decomposition to
neuper@42499: \textit{Partial\_Fractions.thy}. Since there are hacks into Isabelle's
neuper@42499: internals, this kind of distribution is not trivial. For instance, the
neuper@42499: function \texttt{argument\_in} in \S\ref{funs} explicitly contains a
neuper@42499: string with the theory it has been defined in, so this string needs to
neuper@42499: be updated from \texttt{Build\_Inverse\_Z\_Transform} to
neuper@42499: \texttt{Atools} if that function is transferred to theory
neuper@42499: \textit{Atools.thy}.
neuper@42468: 
neuper@42499: 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.
neuper@42499: This process is also rather bare-bones without authoring tools and is
neuper@42499: described in detail in the {\sisac} wiki~\footnote{http://www.ist.tugraz.at/isac/index.php/Generate\_representations\_for\_ISAC\_Knowledge}.
neuper@42468: 
neuper@42478: % \newpage
neuper@42478: % -------------------------------------------------------------------
neuper@42478: % 
neuper@42478: % Material, falls noch Platz bleibt ...
neuper@42478: % 
neuper@42478: % -------------------------------------------------------------------
neuper@42478: % 
neuper@42478: % 
neuper@42478: % \subsubsection{Trials on Notation and Termination}
neuper@42478: % 
neuper@42478: % \paragraph{Technical notations} are a big problem for our piece of software,
neuper@42478: % but the reason for that isn't a fault of the software itself, one of the
neuper@42478: % troubles comes out of the fact that different technical subtopics use different
neuper@42478: % symbols and notations for a different purpose. The most famous example for such
neuper@42478: % a symbol is the complex number $i$ (in cassique math) or $j$ (in technical
neuper@42478: % math). In the specific part of signal processing one of this notation issues is
neuper@42478: % the use of brackets --- we use round brackets for analoge signals and squared
neuper@42478: % brackets for digital samples. Also if there is no problem for us to handle this
neuper@42478: % fact, we have to tell the machine what notation leads to wich meaning and that
neuper@42478: % this purpose seperation is only valid for this special topic - signal
neuper@42478: % processing.
neuper@42478: % \subparagraph{In the programming language} itself it is not possible to declare
neuper@42478: % fractions, exponents, absolutes and other operators or remarks in a way to make
neuper@42478: % them pretty to read; our only posssiblilty were ASCII characters and a handfull
neuper@42478: % greek symbols like: $\alpha, \beta, \gamma, \phi,\ldots$.
neuper@42478: % \par
neuper@42478: % With the upper collected knowledge it is possible to check if we were able to
neuper@42478: % donate all required terms and expressions.
neuper@42478: % 
neuper@42478: % \subsubsection{Definition and Usage of Rules}
neuper@42478: % 
neuper@42478: % \paragraph{The core} of our implemented problem is the Z-Transformation, due
neuper@42478: % the fact that the transformation itself would require higher math which isn't
neuper@42478: % yet avaible in our system we decided to choose the way like it is applied in
neuper@42478: % labratory and problem classes at our university - by applying transformation
neuper@42478: % rules (collected in transformation tables).
neuper@42478: % \paragraph{Rules,} in {\sisac{}}'s programming language can be designed by the
neuper@42478: % use of axiomatizations like shown in Example~\ref{eg:ruledef}
neuper@42478: % 
neuper@42478: % \begin{example}
neuper@42478: %   \label{eg:ruledef}
neuper@42478: %   \hfill\\
neuper@42478: %   \begin{verbatim}
neuper@42478: %   axiomatization where
neuper@42478: %     rule1: ``1 = $\delta$[n]'' and
neuper@42478: %     rule2: ``|| z || > 1 ==> z / (z - 1) = u [n]'' and
neuper@42478: %     rule3: ``|| z || < 1 ==> z / (z - 1) = -u [-n - 1]''
neuper@42478: %   \end{verbatim}
neuper@42478: % \end{example}
neuper@42478: % 
neuper@42478: % This rules can be collected in a ruleset and applied to a given expression as
neuper@42478: % follows in Example~\ref{eg:ruleapp}.
neuper@42478: % 
neuper@42478: % \begin{example}
neuper@42478: %   \hfill\\
neuper@42478: %   \label{eg:ruleapp}
neuper@42478: %   \begin{enumerate}
neuper@42478: %   \item Store rules in ruleset:
neuper@42478: %   \begin{verbatim}
neuper@42478: %   val inverse_Z = append_rls "inverse_Z" e_rls
neuper@42478: %     [ Thm ("rule1",num_str @{thm rule1}),
neuper@42478: %       Thm ("rule2",num_str @{thm rule2}),
neuper@42478: %       Thm ("rule3",num_str @{thm rule3})
neuper@42478: %     ];\end{verbatim}
neuper@42478: %   \item Define exression:
neuper@42478: %   \begin{verbatim}
Walther@60566: %   val sample_term = TermC.parse_test @{context} "z/(z-1)+z/(z-</delta>)+1";\end{verbatim}
neuper@42478: %   \item Apply ruleset:
neuper@42478: %   \begin{verbatim}
neuper@42478: %   val SOME (sample_term', asm) = 
neuper@42478: %     rewrite_set_ thy true inverse_Z sample_term;\end{verbatim}
neuper@42478: %   \end{enumerate}
neuper@42478: % \end{example}
neuper@42478: % 
neuper@42478: % The use of rulesets makes it much easier to develop our designated applications,
neuper@42478: % but the programmer has to be careful and patient. When applying rulesets
neuper@42478: % two important issues have to be mentionend:
neuper@42478: % \subparagraph{How often} the rules have to be applied? In case of
neuper@42478: % transformations it is quite clear that we use them once but other fields
neuper@42478: % reuqire to apply rules until a special condition is reached (e.g.
neuper@42478: % a simplification is finished when there is nothing to be done left).
neuper@42478: % \subparagraph{The order} in which rules are applied often takes a big effect
neuper@42478: % and has to be evaluated for each purpose once again.
neuper@42478: % \par
neuper@42478: % In our special case of Signal Processing and the rules defined in
neuper@42478: % Example~\ref{eg:ruledef} we have to apply rule~1 first of all to transform all
neuper@42478: % constants. After this step has been done it no mather which rule fit's next.
neuper@42478: % 
neuper@42478: % \subsubsection{Helping Functions}
neuper@42478: % 
neuper@42478: % \paragraph{New Programms require,} often new ways to get through. This new ways
neuper@42478: % means that we handle functions that have not been in use yet, they can be 
neuper@42478: % something special and unique for a programm or something famous but unneeded in
neuper@42478: % the system yet. In our dedicated example it was for example neccessary to split
neuper@42478: % a fraction into numerator and denominator; the creation of such function and
neuper@42478: % even others is described in upper Sections~\ref{simp} and \ref{funs}.
neuper@42478: % 
neuper@42478: % \subsubsection{Trials on equation solving}
neuper@42478: % %simple eq and problem with double fractions/negative exponents
neuper@42478: % \paragraph{The Inverse Z-Transformation} makes it neccessary to solve
neuper@42478: % equations degree one and two. Solving equations in the first degree is no 
neuper@42478: % problem, wether for a student nor for our machine; but even second degree
neuper@42478: % equations can lead to big troubles. The origin of this troubles leads from
neuper@42478: % the build up process of our equation solving functions; they have been
neuper@42478: % implemented some time ago and of course they are not as good as we want them to
neuper@42478: % be. Wether or not following we only want to show how cruel it is to build up new
neuper@42478: % work on not well fundamentials.
neuper@42478: % \subparagraph{A simple equation solving,} can be set up as shown in the next
neuper@42478: % example:
neuper@42478: % 
neuper@42478: % \begin{example}
neuper@42478: % \begin{verbatim}
neuper@42478: %   
neuper@42478: %   val fmz =
neuper@42478: %     ["equality (-1 + -2 * z + 8 * z ^^^ 2 = (0::real))",
neuper@42478: %      "solveFor z",
neuper@42478: %      "solutions L"];                                    
neuper@42478: % 
neuper@42478: %   val (dI',pI',mI') =
neuper@42478: %     ("Isac", 
neuper@42478: %       ["abcFormula","degree_2","polynomial","univariate","equation"],
neuper@42478: %       ["no_met"]);\end{verbatim}
neuper@42478: % \end{example}
neuper@42478: % 
neuper@42478: % Here we want to solve the equation: $-1+-2\cdot z+8\cdot z^{2}=0$. (To give
neuper@42478: % a short overview on the commands; at first we set up the equation and tell the
neuper@42478: % machine what's the bound variable and where to store the solution. Second step 
neuper@42478: % is to define the equation type and determine if we want to use a special method
neuper@42478: % to solve this type.) Simple checks tell us that the we will get two results for
neuper@42478: % this equation and this results will be real.
neuper@42478: % So far it is easy for us and for our machine to solve, but
neuper@42478: % mentioned that a unvariate equation second order can have three different types
neuper@42478: % of solutions it is getting worth.
neuper@42478: % \subparagraph{The solving of} all this types of solutions is not yet supported.
neuper@42478: % Luckily it was needed for us; but something which has been needed in this 
neuper@42478: % context, would have been the solving of an euation looking like:
neuper@42478: % $-z^{-2}+-2\cdot z^{-1}+8=0$ which is basically the same equation as mentioned
neuper@42478: % before (remember that befor it was no problem to handle for the machine) but
neuper@42478: % now, after a simple equivalent transformation, we are not able to solve
neuper@42478: % it anymore.
neuper@42478: % \subparagraph{Error messages} we get when we try to solve something like upside
neuper@42478: % were very confusing and also leads us to no special hint about a problem.
neuper@42478: % \par The fault behind is, that we have no well error handling on one side and
neuper@42478: % no sufficient formed equation solving on the other side. This two facts are
neuper@42478: % making the implemention of new material very difficult.
neuper@42478: % 
neuper@42478: % \subsection{Formalization of missing knowledge in Isabelle}
neuper@42478: % 
neuper@42478: % \paragraph{A problem} behind is the mechanization of mathematic
neuper@42478: % theories in TP-bases languages. There is still a huge gap between
neuper@42478: % these algorithms and this what we want as a solution - in Example
neuper@42478: % Signal Processing. 
neuper@42478: % 
neuper@42478: % \vbox{
neuper@42478: %   \begin{example}
neuper@42478: %     \label{eg:gap}
neuper@42478: %     \[
neuper@42478: %       X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY
neuper@42478: %     \]
neuper@42478: %     {\small\textit{
neuper@42478: %       \noindent A very simple example on this what we call gap is the
neuper@42478: % simplification above. It is needles to say that it is correct and also
neuper@42478: % Isabelle for fills it correct - \emph{always}. But sometimes we don't
neuper@42478: % want expand such terms, sometimes we want another structure of
neuper@42478: % them. Think of a problem were we now would need only the coefficients
neuper@42478: % of $X$ and $Y$. This is what we call the gap between mechanical
neuper@42478: % simplification and the solution.
neuper@42478: %     }}
neuper@42478: %   \end{example}
neuper@42478: % }
neuper@42478: % 
neuper@42478: % \paragraph{We are not able to fill this gap,} until we have to live
neuper@42478: % with it but first have a look on the meaning of this statement:
neuper@42478: % Mechanized math starts from mathematical models and \emph{hopefully}
neuper@42478: % proceeds to match physics. Academic engineering starts from physics
neuper@42478: % (experimentation, measurement) and then proceeds to mathematical
neuper@42478: % modeling and formalization. The process from a physical observance to
neuper@42478: % a mathematical theory is unavoidable bound of setting up a big
neuper@42478: % collection of standards, rules, definition but also exceptions. These
neuper@42478: % are the things making mechanization that difficult.
neuper@42478: % 
neuper@42478: % \vbox{
neuper@42478: %   \begin{example}
neuper@42478: %     \label{eg:units}
neuper@42478: %     \[
neuper@42478: %       m,\ kg,\ s,\ldots
neuper@42478: %     \]
neuper@42478: %     {\small\textit{
neuper@42478: %       \noindent Think about some units like that one's above. Behind
neuper@42478: % each unit there is a discerning and very accurate definition: One
neuper@42478: % Meter is the distance the light travels, in a vacuum, through the time
neuper@42478: % of 1 / 299.792.458 second; one kilogram is the weight of a
neuper@42478: % platinum-iridium cylinder in paris; and so on. But are these
neuper@42478: % definitions usable in a computer mechanized world?!
neuper@42478: %     }}
neuper@42478: %   \end{example}
neuper@42478: % }
neuper@42478: % 
neuper@42478: % \paragraph{A computer} or a TP-System builds on programs with
neuper@42478: % predefined logical rules and does not know any mathematical trick
neuper@42478: % (follow up example \ref{eg:trick}) or recipe to walk around difficult
neuper@42478: % expressions. 
neuper@42478: % 
neuper@42478: % \vbox{
neuper@42478: %   \begin{example}
neuper@42478: %     \label{eg:trick}
neuper@42478: %   \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]
neuper@42478: %   \[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)=
neuper@42478: %      \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]
neuper@42478: %   \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]
neuper@42478: %     {\small\textit{
neuper@42478: %       \noindent Sometimes it is also useful to be able to apply some
neuper@42478: % \emph{tricks} to get a beautiful and particularly meaningful result,
neuper@42478: % which we are able to interpret. But as seen in this example it can be
neuper@42478: % hard to find out what operations have to be done to transform a result
neuper@42478: % into a meaningful one.
neuper@42478: %     }}
neuper@42478: %   \end{example}
neuper@42478: % }
neuper@42478: % 
neuper@42478: % \paragraph{The only possibility,} for such a system, is to work
neuper@42478: % through its known definitions and stops if none of these
neuper@42478: % fits. Specified on Signal Processing or any other application it is
neuper@42478: % often possible to walk through by doing simple creases. This creases
neuper@42478: % are in general based on simple math operational but the challenge is
neuper@42478: % to teach the machine \emph{all}\footnote{Its pride to call it
neuper@42478: % \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to
neuper@42478: % reach a high level of \emph{all} but it in real it will still be a
neuper@42478: % survey of knowledge which links to other knowledge and {{\sisac}{}} a
neuper@42478: % trainer and helper but no human compensating calculator. 
neuper@42478: % \par
neuper@42478: % {{{\sisac}{}}} itself aims to adds \emph{Algorithmic Knowledge} (formal
neuper@42478: % specifications of problems out of topics from Signal Processing, etc.)
neuper@42478: % and \emph{Application-oriented Knowledge} to the \emph{deductive} axis of
neuper@42478: % physical knowledge. The result is a three-dimensional universe of
neuper@42478: % mathematics seen in Figure~\ref{fig:mathuni}.
neuper@42478: % 
neuper@42478: % \begin{figure}
neuper@42478: %   \begin{center}
neuper@42478: %     \includegraphics{fig/universe}
neuper@42478: %     \caption{Didactic ``Math-Universe'': Algorithmic Knowledge (Programs) is
neuper@42478: %              combined with Application-oriented Knowledge (Specifications) and Deductive Knowledge (Axioms, Definitions, Theorems). The Result
neuper@42478: %              leads to a three dimensional math universe.\label{fig:mathuni}}
neuper@42478: %   \end{center}
neuper@42478: % \end{figure}
neuper@42478: % 
neuper@42478: % %WN Deine aktuelle Benennung oben wird Dir kein Fachmann abnehmen;
neuper@42478: % %WN bitte folgende Bezeichnungen nehmen:
neuper@42478: % %WN 
neuper@42478: % %WN axis 1: Algorithmic Knowledge (Programs)
neuper@42478: % %WN axis 2: Application-oriented Knowledge (Specifications)
neuper@42478: % %WN axis 3: Deductive Knowledge (Axioms, Definitions, Theorems)
neuper@42478: % %WN 
neuper@42478: % %WN und bitte die R"ander von der Grafik wegschneiden (was ich f"ur *.pdf
neuper@42478: % %WN nicht hinkriege --- weshalb ich auch die eJMT-Forderung nicht ganz
neuper@42478: % %WN verstehe, separierte PDFs zu schicken; ich w"urde *.png schicken)
neuper@42478: % 
neuper@42478: % %JR Ränder und beschriftung geändert. Keine Ahnung warum eJMT sich pdf's
neuper@42478: % %JR wünschen, würde ebenfalls png oder ähnliches verwenden, aber wenn pdf's
neuper@42478: % %JR gefordert werden WN2...
neuper@42478: % %WN2 meiner Meinung nach hat sich eJMT unklar ausgedr"uckt (z.B. kann
neuper@42478: % %WN2 man meines Wissens pdf-figures nicht auf eine bestimmte Gr"osse
neuper@42478: % %WN2 zusammenschneiden um die R"ander weg zu bekommen)
neuper@42478: % %WN2 Mein Vorschlag ist, in umserem tex-file bei *.png zu bleiben und
neuper@42478: % %WN2 png + pdf figures mitzuschicken.
neuper@42478: % 
neuper@42478: % \subsection{Notes on Problems with Traditional Notation}
neuper@42478: % 
neuper@42478: % \paragraph{During research} on these topic severely problems on
neuper@42478: % traditional notations have been discovered. Some of them have been
neuper@42478: % known in computer science for many years now and are still unsolved,
neuper@42478: % one of them aggregates with the so called \emph{Lambda Calculus},
neuper@42478: % Example~\ref{eg:lamda} provides a look on the problem that embarrassed
neuper@42478: % us.
neuper@42478: % 
neuper@42478: % \vbox{
neuper@42478: %   \begin{example}
neuper@42478: %     \label{eg:lamda}
neuper@42478: % 
neuper@42478: %   \[ f(x)=\ldots\;  \quad R \rightarrow \quad R \]
neuper@42478: % 
neuper@42478: % 
neuper@42478: %   \[ f(p)=\ldots\;  p \in \quad R \]
neuper@42478: % 
neuper@42478: %     {\small\textit{
neuper@42478: %       \noindent Above we see two equations. The first equation aims to
neuper@42478: % be a mapping of an function from the reel range to the reel one, but
neuper@42478: % when we change only one letter we get the second equation which
neuper@42478: % usually aims to insert a reel point $p$ into the reel function. In
neuper@42478: % computer science now we have the problem to tell the machine (TP) the
neuper@42478: % difference between this two notations. This Problem is called
neuper@42478: % \emph{Lambda Calculus}.
neuper@42478: %     }}
neuper@42478: %   \end{example}
neuper@42478: % }
neuper@42478: % 
neuper@42478: % \paragraph{An other problem} is that terms are not full simplified in
neuper@42478: % traditional notations, in {{\sisac}} we have to simplify them complete
neuper@42478: % to check weather results are compatible or not. in e.g. the solutions
neuper@42478: % of an second order linear equation is an rational in {{\sisac}} but in
neuper@42478: % tradition we keep fractions as long as possible and as long as they
neuper@42478: % aim to be \textit{beautiful} (1/8, 5/16,...).
neuper@42478: % \subparagraph{The math} which should be mechanized in Computer Theorem
neuper@42478: % Provers (\emph{TP}) has (almost) a problem with traditional notations
neuper@42478: % (predicate calculus) for axioms, definitions, lemmas, theorems as a
neuper@42478: % computer program or script is not able to interpret every Greek or
neuper@42478: % Latin letter and every Greek, Latin or whatever calculations
neuper@42478: % symbol. Also if we would be able to handle these symbols we still have
neuper@42478: % a problem to interpret them at all. (Follow up \hbox{Example
neuper@42478: % \ref{eg:symbint1}})
neuper@42478: % 
neuper@42478: % \vbox{
neuper@42478: %   \begin{example}
neuper@42478: %     \label{eg:symbint1}
neuper@42478: %     \[
neuper@42478: %       u\left[n\right] \ \ldots \ unitstep
neuper@42478: %     \]
neuper@42478: %     {\small\textit{
neuper@42478: %       \noindent The unitstep is something we need to solve Signal
neuper@42478: % Processing problem classes. But in {{{\sisac}{}}} the rectangular
neuper@42478: % brackets have a different meaning. So we abuse them for our
neuper@42478: % requirements. We get something which is not defined, but usable. The
neuper@42478: % Result is syntax only without semantic.
neuper@42478: %     }}
neuper@42478: %   \end{example}
neuper@42478: % }
neuper@42478: % 
neuper@42478: % In different problems, symbols and letters have different meanings and
neuper@42478: % ask for different ways to get through. (Follow up \hbox{Example
neuper@42478: % \ref{eg:symbint2}}) 
neuper@42478: % 
neuper@42478: % \vbox{
neuper@42478: %   \begin{example}
neuper@42478: %     \label{eg:symbint2}
neuper@42478: %     \[
neuper@42478: %       \widehat{\ }\ \widehat{\ }\ \widehat{\ } \  \ldots \  exponent
neuper@42478: %     \]
neuper@42478: %     {\small\textit{
neuper@42478: %     \noindent For using exponents the three \texttt{widehat} symbols
neuper@42478: % are required. The reason for that is due the development of
neuper@42478: % {{{\sisac}{}}} the single \texttt{widehat} and also the double were
neuper@42478: % already in use for different operations.
neuper@42478: %     }}
neuper@42478: %   \end{example}
neuper@42478: % }
neuper@42478: % 
neuper@42478: % \paragraph{Also the output} can be a problem. We are familiar with a
neuper@42478: % specified notations and style taught in university but a computer
neuper@42478: % program has no knowledge of the form proved by a professor and the
neuper@42478: % machines themselves also have not yet the possibilities to print every
neuper@42478: % symbol (correct) Recent developments provide proofs in a human
neuper@42478: % readable format but according to the fact that there is no money for
neuper@42478: % good working formal editors yet, the style is one thing we have to
neuper@42478: % live with.
neuper@42478: % 
neuper@42478: % \section{Problems rising out of the Development Environment}
neuper@42478: % 
neuper@42478: % fehlermeldungen! TODO
jan@42463: 
neuper@42492: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{verbatim}
neuper@42492: 
neuper@48771: \section{Summary and Conclusions}\label{conclusion}
jan@42463: 
jan@42512: %JR obvious
jan@42512: 
jan@42512: %This paper gives a first experience report about programming with a
jan@42512: %TP-based programming language.
jan@42512: 
jan@42512: A brief re-introduction of the novel kind of programming
neuper@42492: language by example of the {\sisac}-prototype makes the paper
neuper@42492: self-contained. The main section describes all the main concepts
neuper@42492: involved in TP-based programming and all the sub-tasks concerning
neuper@48771: respective implementation in the {\sisac} prototype: mechanisation of mathematics and domain
neuper@42514: modeling, implementation of term rewriting systems for the
neuper@42492: rewriting-engine, formal (implicit) specification of the problem to be
neuper@42507: (explicitly) described by the program, implementation of the many components
neuper@42492: required for Lucas-Interpretation and finally implementation of the
neuper@42492: program itself.
neuper@42492: 
neuper@42492: The many concepts and sub-tasks involved in programming require a
neuper@42514: comprehensive work-flow; first experiences with the work-flow as
neuper@42492: supported by the present prototype are described as well: Isabelle +
neuper@42492: Isar + jEdit provide appropriate components for establishing an
neuper@42492: efficient development environment integrating computation and
neuper@42492: deduction. However, the present state of the prototype is far off a
neuper@42492: state appropriate for wide-spread use: the prototype of the program
neuper@42492: language lacks expressiveness and elegance, the prototype of the
neuper@42492: development environment is hardly usable: error messages still address
neuper@42492: the developer of the prototype's interpreter rather than the
neuper@42492: application programmer, implementation of the many settings for the
neuper@48771: Lucas-Interpreter is cumbersome. 
neuper@42492: 
neuper@48773: \subsection{Conclusions for Future Development}
neuper@48771: From the above mentioned experiences a successful proof of concept can be concluded:
neuper@42492: programming arbitrary problems from engineering sciences is possible,
neuper@42492: in principle even in the prototype. Furthermore the experiences allow
neuper@42492: to conclude detailed requirements for further development:
neuper@48771: \begin{enumerate}
neuper@42492: \item Clarify underlying logics such that programming is smoothly
neuper@42492: integrated with verification of the program; the post-condition should
neuper@42492: be proved more or less automatically, otherwise working engineers
neuper@42492: would not encounter such programming.
neuper@42492: \item Combine the prototype's programming language with Isabelle's
neuper@42492: powerful function package and probably with more of SML's
neuper@42492: pattern-matching features; include parallel execution on multi-core
jan@42511: machines into the language design.
neuper@42492: \item Extend the prototype's Lucas-Interpreter such that it also
neuper@42492: handles functions defined by use of Isabelle's functions package; and
neuper@42492: generalize Isabelle's code generator such that efficient code for the
neuper@42507: whole definition of the programming language can be generated (for
neuper@42492: multi-core machines).
neuper@42492: \item Develop an efficient development environment with
neuper@42492: integration of programming and proving, with management not only of
neuper@42492: Isabelle theories, but also of large collections of specifications and
neuper@42492: of programs.
neuper@48771: \item\label{CAS} Extend Isabelle's computational features in direction of
neuper@48771: \textit{verfied} Computer Algebra: simplification extended by
neuper@48771: algorithms beyond rewriting (cancellation of multivariate rationals,
neuper@48771: factorisation, partial fraction decomposition, etc), equation solving
neuper@48771: , integration, etc.
neuper@48771: \end{enumerate} 
neuper@42492: Provided successful accomplishment, these points provide distinguished
neuper@48771: components for virtual workbenches appealing to practitioners of
neuper@42492: engineering in the near future.
neuper@42492: 
neuper@48771: \subsection{Preview to Development of Course Material}
neuper@48771: Interactive course material, as addressed by the title,
neuper@42507: can comprise step-wise problem solving created as a side-effect of a
neuper@48771: TP-based program: The introduction \S\ref{intro} briefly shows that Lucas-Interpretation not only provides an
neuper@42507: interactive programming environment, Lucas-Interpretation also can
jan@42511: provide TP-based services for a flexible dialogue component with
neuper@42507: adaptive user guidance for independent and inquiry-based learning.
neuper@42492: 
neuper@48771: However, the {\sisac} prototype is not ready for use in field-tests,
neuper@48771: not only due to the above five requirements not sufficiently
neuper@48771: accomplished, but also due to usability of the fron-end, in particular
neuper@48771: the lack of an editor for formulas in 2-dimension representation.
neuper@48771: 
neuper@48771: Nevertheless, the experiences from the case study described in this
neuper@48771: paper, allow to give a preview to the development of course material,
neuper@48771: if based on Lucas-Interpretation:
neuper@48771: 
neuper@48771: \paragraph{Development of material from scratch} is too much effort
neuper@48771: just for e-learning; this has become clear with the case study.  For
neuper@48771: getting support for stepwise problem solving just in {\em one} example
neuper@48771: class, the one presented in this paper, involved the following tasks:
neuper@48771: \begin{itemize}
neuper@48771: \item Adapt the equation solver; since that was too laborous, the
neuper@48771: program has been adapted in an unelegant way.
neuper@48771: \item Implement an algorithms for partial fraction decomposition,
neuper@48771: which is considered a standard normal form in Computer Algebra.
neuper@48771: \item Implement a specification for partial fraction decomposition and
neuper@48771: locate it appropriately in the hierarchy of specification.
jan@48774: \item Declare definitions and theorems within the theory of
jan@48774: ${\cal Z}$-transform, and prove the theorems (which was not done in the
neuper@48771: case study).
neuper@48771: \end{itemize}
neuper@48771: On the other hand, for the one the class of problems implemented,
neuper@48771: adding an arbitrary number of examples within this class requires a
neuper@48771: few minutes~\footnote{As shown in Fig.\ref{fig-interactive}, an
neuper@48771: example is called from an HTML-file by an URL, which addresses an
neuper@48771: XML-structure holding the respective data as shown on
neuper@48771: p.\pageref{ml-check-program}.} and the support for individual stepwise
neuper@48771: problem solving comes for free.
neuper@48771: 
neuper@48771: \paragraph{E-learning benefits from Formal Domain Engineering} which can be
neuper@48771: expected for various domains in the near future. In order to cope with
neuper@48771: increasing complexity in domain of technology, specific domain
neuper@48771: knowledge is beeing mechanised, not only for software technology
neuper@48771: \footnote{For instance, the Archive of Formal Proofs
neuper@48771: http://afp.sourceforge.net/} but also for other engineering domains
neuper@48771: \cite{Dehbonei&94,Hansen94b,db:dom-eng}.  This fairly new part of
neuper@48771: engineering sciences is called ``domain engineering'' in
neuper@48771: \cite{db:SW-engIII}.
neuper@48771: 
neuper@48771: Given this kind of mechanised knowledge including mathematical
neuper@48771: theories, domain specific definitions, specifications and algorithms,
neuper@48771: theorems and proofs, then e-learning with support for individual
neuper@48771: stepwise problem solving will not be much ado anymore; then e-learning
neuper@48771: media in technology education can be derived from this knowledge with
neuper@48771: reasonable effort.
neuper@48771: 
neuper@48771: \paragraph{Development differentiates into tasks} more separated than
neuper@48771: without Lucas-Interpretation and more challenginging in specific
neuper@48771: expertise. These are the kinds of experts expected to cooperate in
neuper@48771: development of
neuper@48771: \begin{itemize}
neuper@48773: \item ``Domain engineers'', who accomplish fairly novel tasks
neuper@48773: described in this paper.
neuper@48771: \item Course designers, who provide the instructional design according
neuper@48771: to curricula, together with usability experts and media designers, are
neuper@48771: indispensable in production of e-learning media at the state-of-the
neuper@48771: art.
neuper@48771: \item ``Dialog designers'', whose part of development is clearly
neuper@48773: separated from the part of domain engineers as a consequence of
neuper@48773: Lucas-Interpretation: TP-based programs are functional, as mentioned,
neuper@48773: and are only concerned with describing mathematics --- and not at all
neuper@48773: concerned with interaction, psychology, learning theory and the like,
neuper@48773: because there are no in/output statements. Dialog designers can expect
neuper@48773: a high-level rule-based language~\cite{gdaroczy-EP-13} for describing
neuper@48773: their part.
neuper@48771: \end{itemize}
neuper@48771: 
neuper@48771: % response-to-referees:
neuper@48771: % (2.1) details of novel technology in order to estimate the impact
neuper@48771: % (2.2) which kinds of expertise are required for production of e-learning media (instructional design, math authoring, dialog authoring, media design)
neuper@48771: % (2.3) what in particular is required for programming new exercises supported by next-step-guidance (expertise / efforts)
neuper@48771: % (2.4) estimation of break-even points for development of next-step-guidance
neuper@48771: % (2.5) usability of ISAC prototype at the present state
neuper@48771: % 
neuper@48771: % 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."".
neuper@48771: 
neuper@48773: \bigskip\noindent For this decade there seems to be a window of opportunity opening from
neuper@48771: one side inreasing demand for formal domain engineering and from the
neuper@48771: other side from TP more and more gaining industrial relevance. Within
neuper@48771: this window, development of TP-based educational software can take
neuper@48775: benefit from the fact, that the TPs leading in Europe, Coq~\cite{coq-team-10} and
neuper@48771: Isabelle are still open source together with the major part of
neuper@48771: mechanised knowledge.%~\footnote{NICTA}.
jan@42463: 
jan@42463: \bibliographystyle{alpha}
neuper@42507: {\small\bibliography{references}}
jan@42463: 
neuper@42514: \end{document}
neuper@42514: % LocalWords:  TP IST SPSC Telematics Dialogues dialogue HOL bool nat Hindley
neuper@42514: % LocalWords:  Milner tt Subproblem Formulae ruleset generalisation initialised
neuper@42514: % LocalWords:  axiomatization LCF Simplifiers simplifiers Isar rew Thm Calc SML
neuper@42514: % LocalWords:  recognised hoc Trueprop redexes Unsynchronized pre rhs ord erls
neuper@42514: % LocalWords:  srls prls nrls lhs errpats InverseZTransform SubProblem IDE IDEs
neuper@42514: % LocalWords:  univariate jEdit rls RealDef calclist familiarisation ons pos eq
neuper@42514: % LocalWords:  mout ctree SignalProcessing frac ZZ Postcond Atools wiki SML's
neuper@42514: % LocalWords:  mechanisation multi




