jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % Electronic Journal of Mathematics and Technology (eJMT) % jan@42463: % style sheet for LaTeX. Please do not modify sections % jan@42463: % or commands marked 'eJMT'. % jan@42463: % % jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % % jan@42463: % eJMT commands % jan@42463: % % jan@42463: \documentclass[12pt,a4paper]{article}% % jan@42463: \usepackage{times} % jan@42463: \usepackage{amsfonts,amsmath,amssymb} % jan@42463: \usepackage[a4paper]{geometry} % jan@42463: \usepackage{fancyhdr} % jan@42463: \usepackage{color} % jan@42463: \usepackage[pdftex]{hyperref} % see note below % jan@42463: \usepackage{graphicx}% % jan@42463: \hypersetup{ % jan@42463: a4paper, % jan@42463: breaklinks % jan@42463: } % jan@42463: % % jan@42463: \newtheorem{theorem}{Theorem} % jan@42463: \newtheorem{acknowledgement}[theorem]{Acknowledgement} % jan@42463: \newtheorem{algorithm}[theorem]{Algorithm} % jan@42463: \newtheorem{axiom}[theorem]{Axiom} % jan@42463: \newtheorem{case}[theorem]{Case} % jan@42463: \newtheorem{claim}[theorem]{Claim} % jan@42463: \newtheorem{conclusion}[theorem]{Conclusion} % jan@42463: \newtheorem{condition}[theorem]{Condition} % jan@42463: \newtheorem{conjecture}[theorem]{Conjecture} % jan@42463: \newtheorem{corollary}[theorem]{Corollary} % jan@42463: \newtheorem{criterion}[theorem]{Criterion} % jan@42463: \newtheorem{definition}[theorem]{Definition} % jan@42463: \newtheorem{example}[theorem]{Example} % jan@42463: \newtheorem{exercise}[theorem]{Exercise} % jan@42463: \newtheorem{lemma}[theorem]{Lemma} % jan@42463: \newtheorem{notation}[theorem]{Notation} % jan@42463: \newtheorem{problem}[theorem]{Problem} % jan@42463: \newtheorem{proposition}[theorem]{Proposition} % jan@42463: \newtheorem{remark}[theorem]{Remark} % jan@42463: \newtheorem{solution}[theorem]{Solution} % jan@42463: \newtheorem{summary}[theorem]{Summary} % jan@42463: \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} } % jan@42463: {\ \rule{0.5em}{0.5em}} % jan@42463: % % jan@42463: % eJMT page dimensions % jan@42463: % % jan@42463: \geometry{left=2cm,right=2cm,top=3.2cm,bottom=4cm} % jan@42463: % % jan@42463: % eJMT header & footer % jan@42463: % % jan@42463: \newcounter{ejmtFirstpage} % jan@42463: \setcounter{ejmtFirstpage}{1} % jan@42463: \pagestyle{empty} % jan@42463: \setlength{\headheight}{14pt} % jan@42463: \geometry{left=2cm,right=2cm,top=3.2cm,bottom=4cm} % jan@42463: \pagestyle{fancyplain} % jan@42463: \fancyhf{} % jan@42463: \fancyhead[c]{\small The Electronic Journal of Mathematics% jan@42463: \ and Technology, Volume 1, Number 1, ISSN 1933-2823} % jan@42463: \cfoot{% % jan@42463: \ifnum\value{ejmtFirstpage}=0% % jan@42463: {\vtop to\hsize{\hrule\vskip .2cm\thepage}}% % jan@42463: \else\setcounter{ejmtFirstpage}{0}\fi% % jan@42463: } % jan@42463: % % jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % jan@42463: % Please place your own definitions here jan@42463: % jan@42463: \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$} jan@42463: \def\sisac{\footnotesize${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$} jan@42463: jan@42463: \usepackage{color} jan@42463: \definecolor{lgray}{RGB}{238,238,238} jan@42463: jan@48778: \usepackage{hyperref} jan@48778: jan@42463: % jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % % jan@42463: % How to use hyperref % jan@42463: % ------------------- % jan@42463: % % jan@42463: % Probably the only way you will need to use the hyperref % jan@42463: % package is as follows. To make some text, say % jan@42463: % "My Text Link", into a link to the URL % jan@42463: % http://something.somewhere.com/mystuff, use % jan@42463: % % jan@42463: % \href{http://something.somewhere.com/mystuff}{My Text Link} jan@42463: % % jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % jan@42463: \begin{document} jan@42463: % jan@42463: % document title jan@42463: % neuper@42464: \title{Trials with TP-based Programming neuper@42464: \\ neuper@42464: for Interactive Course Material}% jan@42463: % jan@42463: % Single author. Please supply at least your name, jan@42463: % email address, and affiliation here. jan@42463: % 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} jan@42463: }% jan@42463: % jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % % jan@42463: % eJMT commands - do not change these % jan@42463: % % jan@42463: \date{} % jan@42463: \maketitle % jan@42463: % % jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % jan@42463: % abstract jan@42463: % 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. jan@42463: % jan@42463: \end{abstract}% jan@42463: % jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % % jan@42463: % eJMT command % jan@42463: % % jan@42463: \thispagestyle{fancy} % jan@42463: % % jan@42463: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jan@42463: % jan@42463: % Please use the following to indicate sections, subsections, jan@42463: % etc. Please also use \subsubsection{...}, \paragraph{...} jan@42463: % and \subparagraph{...} as necessary. jan@42463: % 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 - ) + 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] + ^ n * u[n] + [n]" : string neuper@42514: % 07 val it = "|| z || > 1 & || z || > " : 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-)+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