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61 \ and Technology, Volume 1, Number 1, ISSN 1933-2823} %
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97 \title{Trials with TP-based Programming
99 for Interactive Course Material}%
101 % Single author. Please supply at least your name,
102 % email address, and affiliation here.
104 \author{\begin{tabular}{c}
105 \textit{Jan Ro\v{c}nik} \\
106 jan.rocnik@student.tugraz.at \\
108 Graz University of Technology\\
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125 Traditional course material in engineering disciplines lacks an
126 important component, interactive support for step-wise problem
127 solving. Theorem-Proving (TP) technology is appropriate for one part
128 of such support, in checking user-input. For the other part of such
129 support, guiding the learner towards a solution, another kind of
130 technology is required.
132 Both kinds of support can be achieved by so-called
133 Lucas-Interpretation which combines deduction and computation and, for
134 the latter, uses a novel kind of programming language. This language
135 is based on (Computer) Theorem Proving (TP), thus called a ``TP-based
136 programming language''.
138 This paper is the experience report of the first ``application
139 programmer'' using this language for creating exercises in step-wise
140 problem solving for an advanced lab in Signal Processing. The tasks
141 involved in TP-based programming are described together with the
142 experience gained from a prototype of the programming language and of
145 The report concludes with a positive proof of concept, states
146 insufficiency usability of the prototype and captures the requirements
147 for further development of both, the programming language and the
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160 % Please use the following to indicate sections, subsections,
161 % etc. Please also use \subsubsection{...}, \paragraph{...}
162 % and \subparagraph{...} as necessary.
165 \section{Introduction}\label{intro}
167 % \paragraph{Didactics of mathematics}
168 %WN: wenn man in einem high-quality paper von 'didactics' spricht,
169 %WN muss man am state-of-the-art ankn"upfen -- siehe
170 %WN W.Neuper, On the Emergence of TP-based Educational Math Assistants
171 % faces a specific issue, a gap
172 % between (1) introduction of math concepts and skills and (2)
173 % application of these concepts and skills, which usually are separated
174 % into different units in curricula (for good reasons). For instance,
175 % (1) teaching partial fraction decomposition is separated from (2)
176 % application for inverse Z-transform in signal processing.
178 % \par This gap is an obstacle for applying math as an fundamental
179 % thinking technology in engineering: In (1) motivation is lacking
180 % because the question ``What is this stuff good for?'' cannot be
181 % treated sufficiently, and in (2) the ``stuff'' is not available to
182 % students in higher semesters as widespread experience shows.
184 % \paragraph{Motivation} taken by this didactic issue on the one hand,
185 % and ongoing research and development on a novel kind of educational
186 % mathematics assistant at Graz University of
187 % Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to
188 % scope with this issue on the other hand, several institutes are
189 % planning to join their expertise: the Institute for Information
190 % Systems and Computer Media (IICM), the Institute for Software
191 % Technology (IST), the Institutes for Mathematics, the Institute for
192 % Signal Processing and Speech Communication (SPSC), the Institute for
193 % Structural Analysis and the Institute of Electrical Measurement and
194 % Measurement Signal Processing.
195 %WN diese Information ist f"ur das Paper zu spezielle, zu aktuell
196 %WN und damit zu verg"anglich.
197 % \par This thesis is the first attempt to tackle the above mentioned
198 % issue, it focuses on Telematics, because these specific studies focus
199 % on mathematics in \emph{STEOP}, the introductory orientation phase in
200 % Austria. \emph{STEOP} is considered an opportunity to investigate the
201 % impact of {\sisac}'s prototype on the issue and others.
204 Traditional course material in engineering disciplines lacks an
205 important component, interactive support for step-wise problem
206 solving. Theorem-Proving (TP) technology can provide such support by
207 specific services. An important part of such services is called
208 ``next-step-guidance'', generated by a specific kind of ``TP-based
209 programming language''. In the
210 {\sisac}-project~\footnote{http://www.ist.tugraz.at/projects/isac/} such
211 a language is prototyped in line with~\cite{plmms10} and built upon
212 the theorem prover Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}
213 \footnote{http://isabelle.in.tum.de/}.
214 The TP services are coordinated by a specific interpreter for the
215 programming language, called
216 Lucas-Interpreter~\cite{wn:lucas-interp-12}. The language and the
217 interpreter will be briefly re-introduced in order to make the paper
220 The main part of the paper is an account of first experiences
221 with programming in this TP-based language. The experience was gained
222 in a case study by the author. The author was considered an ideal
223 candidate for this study for the following reasons: as a student in
224 Telematics (computer science with focus on Signal Processing) he had
225 general knowledge in programming as well as specific domain knowledge
226 in Signal Processing; and he was {\em not} involved in the development of
227 {\sisac}'s programming language and interpreter, thus being a novice to the
230 The goals of the case study were: (1) to identify some TP-based programs for
231 interactive course material for a specific ``Advanced Signal
232 Processing Lab'' in a higher semester, (2) respective program
233 development with as little advice as possible from the {\sisac}-team and (3)
234 to document records and comments for the main steps of development in an
235 Isabelle theory; this theory should provide guidelines for future programmers.
236 An excerpt from this theory is the main part of this paper.
238 The paper will use the problem in Fig.\ref{fig-interactive} as a
242 \includegraphics[width=140mm]{fig/isac-Ztrans-math-3}
243 %\includegraphics[width=140mm]{fig/isac-Ztrans-math}
244 \caption{Step-wise problem solving guided by the TP-based program
245 \label{fig-interactive}}
249 \paragraph{The Engineering Background of the Problem} comes out of the domain
250 Signal Processing, which takes a major part n the authors field of education.
251 The given Problem requests to determine the inverse $z$-transform for a
254 ``The $z$-Transform for discrete-time signals is the counterpart of the
255 Laplace transform for continuous-time signals, and they each have a similar
256 relationship to the corresponding Fourier transform. One motivation for
257 introducing this generalization is that the Fourier transform does not
258 converge for all sequences, and it is useful to have a generalization of the
259 Fourier transform that encompasses a broader class of signals. A second
260 advantage is that in analytic problems, the $z$-transform notation is often
261 more convenient than the Fourier transform notation.''
262 ~\cite[p. 128]{oppenheim2010discrete}
264 The $z$-transform can be defined as:
266 X(z)=\sum_{n=-\infty }^{\infty }x[n]z^{-n}
268 Upper equation transforms a discrete time sequence $x[n]$ into the function
269 $X(z)$ where $z$ is a continuous complex variable. The inverse function (as it
270 is used in the given problem) is defined as:
272 x[n]=\frac{1}{2\pi j} \oint_{C} X(z)\cdot z^{n-1} dz
274 The letter $C$ represents a contour within the range of converge of the $z$-
275 transform. The unit circle can be a special case of this contour. Remember
276 that $j$ is the complex number in the field of engineering.
277 As this transformation requires high effort to be solved, tables of
278 common transform pairs are used in education as well as in (TODO: real); such
279 tables can be found at~\cite{wiki:1} or~\cite[Table~3.1]{oppenheim2010discrete} as well.
280 A completely solved and more detailed example can be found at
281 ~\cite[p. 149f]{oppenheim2010discrete}. The upcoming implementation tries to
282 fit this example in the way it is toughed at the authors university.
286 \paragraph{The educational aspect} can be explained by having a look at
287 Fig.\ref{fig-interactive} which shows the beginning of the interactive
288 construction of a solution for the problem. This construction is done in the
289 right window named ``Worksheet''.
291 User-interaction on the Worksheet is {\em checked} and {\em guided} by
294 \item Formulas input by the user are {\em checked} by TP: such a
295 formula establishes a proof situation --- the prover has to derive the
296 formula from the logical context. The context is built up from the
297 formal specification of the problem (here hidden from the user) by the
299 \item If the user gets stuck, the program developed below in this
300 paper ``knows the next step'' and Lucas-Interpretation provides services
301 featuring so-called ``next-step-guidance''; this is out of scope of this
302 paper and can be studied in~\cite{gdaroczy-EP-13}.
303 \end{enumerate} It should be noted that the programmer using the
304 TP-based language is not concerned with interaction at all; we will
305 see that the program contains neither input-statements nor
306 output-statements. Rather, interaction is handled by the interpreter
309 So there is a clear separation of concerns: Dialogues are adapted by
310 dialogue authors (in Java-based tools), using TP services provided by
311 Lucas-Interpretation. The latter acts on programs developed by
312 mathematics-authors (in Isabelle/ML); their task is concern of this
315 The paper is structured as follows: The introduction
316 \S\ref{intro} is followed by a brief re-introduction of the TP-based
317 programming language in \S\ref{PL}, which extends the executable
318 fragment of Isabelle's language (\S\ref{PL-isab}) by tactics which
319 play a specific role in Lucas-Interpretation and in providing the TP
320 services (\S\ref{PL-tacs}). The main part \S\ref{trial} describes
321 the main steps in developing the program for the running example:
322 prepare domain knowledge, implement the formal specification of the
323 problem, prepare the environment for the interpreter, implement the
324 program in \S\ref{isabisac} to \S\ref{progr} respectively.
325 The work-flow of programming, debugging and testing is
326 described in \S\ref{workflow}. The conclusion \S\ref{conclusion} will
327 give directions identified for future development.
330 \section{\isac's Prototype for a Programming Language}\label{PL}
331 The prototype of the language and of the Lucas-Interpreter are briefly
332 described from the point of view of a programmer. The language extends
333 the executable fragment in the language of the theorem prover
334 Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}.
336 \subsection{The Executable Fragment of Isabelle's Language}\label{PL-isab}
337 The executable fragment consists of data-type and function
338 definitions. It's usability even suggests that fragment for
339 introductory courses \cite{nipkow-prog-prove}. HOL (Higher-Order Logic)
340 is a typed logic whose type system resembles that of functional programming
341 languages. Thus there are
343 \item[base types,] in particular \textit{bool}, the type of truth
344 values, \textit{nat}, \textit{int}, \textit{complex}, and the types of
345 natural, integer and complex numbers respectively in mathematics.
346 \item[type constructors] allow to define arbitrary types, from
347 \textit{set}, \textit{list} to advanced data-structures like
348 \textit{trees}, red-black-trees etc.
349 \item[function types,] denoted by $\Rightarrow$.
350 \item[type variables,] denoted by $^\prime a, ^\prime b$ etc, provide
351 type polymorphism. Isabelle automatically computes the type of each
352 variable in a term by use of Hindley-Milner type inference
353 \cite{pl:hind97,Milner-78}.
356 \textbf{Terms} are formed as in functional programming by applying
357 functions to arguments. If $f$ is a function of type
358 $\tau_1\Rightarrow \tau_2$ and $t$ is a term of type $\tau_1$ then
359 $f\;t$ is a term of type~$\tau_2$. $t\;::\;\tau$ means that term $t$
360 has type $\tau$. There are many predefined infix symbols like $+$ and
361 $\leq$ most of which are overloaded for various types.
363 HOL also supports some basic constructs from functional programming:
364 {\footnotesize\it\label{isabelle-stmts}
365 \begin{tabbing} 123\=\kill
366 01\>$( \; {\tt if} \; b \; {\tt then} \; t_1 \; {\tt else} \; t_2 \;)$\\
367 02\>$( \; {\tt let} \; x=t \; {\tt in} \; u \; )$\\
368 03\>$( \; {\tt case} \; t \; {\tt of} \; {\it pat}_1
369 \Rightarrow t_1 \; |\dots| \; {\it pat}_n\Rightarrow t_n \; )$
371 \noindent The running example's program uses some of these elements
372 (marked by {\tt tt-font} on p.\pageref{s:impl}): for instance {\tt
373 let}\dots{\tt in} in lines {\rm 02} \dots {\rm 13}. In fact, the whole program
374 is an Isabelle term with specific function constants like {\tt
375 program}, {\tt Take}, {\tt Rewrite}, {\tt Subproblem} and {\tt
376 Rewrite\_Set} in lines {\rm 01, 03. 04, 07, 10} and {\rm 11, 12}
379 % Terms may also contain $\lambda$-abstractions. For example, $\lambda
380 % x. \; x$ is the identity function.
382 %JR warum auskommentiert? WN2...
383 %WN2 weil ein Punkt wie dieser in weiteren Zusammenh"angen innerhalb
384 %WN2 des Papers auftauchen m"usste; nachdem ich einen solchen
385 %WN2 Zusammenhang _noch_ nicht sehe, habe ich den Punkt _noch_ nicht
387 %WN2 Wenn der Punkt nicht weiter gebraucht wird, nimmt er nur wertvollen
388 %WN2 Platz f"ur Anderes weg.
390 \textbf{Formulae} are terms of type \textit{bool}. There are the basic
391 constants \textit{True} and \textit{False} and the usual logical
392 connectives (in decreasing order of precedence): $\neg, \land, \lor,
395 \textbf{Equality} is available in the form of the infix function $=$
396 of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for
397 formulas, where it means ``if and only if''.
399 \textbf{Quantifiers} are written $\forall x. \; P$ and $\exists x. \;
400 P$. Quantifiers lead to non-executable functions, so functions do not
401 always correspond to programs, for instance, if comprising \\$(
402 \;{\it if} \; \exists x.\;P \; {\it then} \; e_1 \; {\it else} \; e_2
405 \subsection{\isac's Tactics for Lucas-Interpretation}\label{PL-tacs}
406 The prototype extends Isabelle's language by specific statements
407 called tactics~\footnote{{\sisac}'s. This tactics are different from
408 Isabelle's tactics: the former concern steps in a calculation, the
409 latter concern proofs.} and tactics. For the programmer these
410 statements are functions with the following signatures:
413 \item[Rewrite:] ${\it theorem}\Rightarrow{\it term}\Rightarrow{\it
414 term} * {\it term}\;{\it list}$:
415 this tactic applies {\it theorem} to a {\it term} yielding a {\it
416 term} and a {\it term list}, the list are assumptions generated by
417 conditional rewriting. For instance, the {\it theorem}
418 $b\not=0\land c\not=0\Rightarrow\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$
419 applied to the {\it term} $\frac{2\cdot x}{3\cdot x}$ yields
420 $(\frac{2}{3}, [x\not=0])$.
422 \item[Rewrite\_Set:] ${\it ruleset}\Rightarrow{\it
423 term}\Rightarrow{\it term} * {\it term}\;{\it list}$:
424 this tactic applies {\it ruleset} to a {\it term}; {\it ruleset} is
425 a confluent and terminating term rewrite system, in general. If
426 none of the rules ({\it theorem}s) is applicable on interpretation
427 of this tactic, an exception is thrown.
429 % \item[Rewrite\_Inst:] ${\it substitution}\Rightarrow{\it
430 % theorem}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
433 % \item[Rewrite\_Set\_Inst:] ${\it substitution}\Rightarrow{\it
434 % ruleset}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
438 \item[Substitute:] ${\it substitution}\Rightarrow{\it
439 term}\Rightarrow{\it term}$: allows to access sub-terms.
442 \item[Take:] ${\it term}\Rightarrow{\it term}$:
443 this tactic has no effect in the program; but it creates a side-effect
444 by Lucas-Interpretation (see below) and writes {\it term} to the
447 \item[Subproblem:] ${\it theory} * {\it specification} * {\it
448 method}\Rightarrow{\it argument}\;{\it list}\Rightarrow{\it term}$:
449 this tactic is a generalisation of a function call: it takes an
450 \textit{argument list} as usual, and additionally a triple consisting
451 of an Isabelle \textit{theory}, an implicit \textit{specification} of the
452 program and a \textit{method} containing data for Lucas-Interpretation,
453 last not least a program (as an explicit specification)~\footnote{In
454 interactive tutoring these three items can be determined explicitly
457 The tactics play a specific role in
458 Lucas-Interpretation~\cite{wn:lucas-interp-12}: they are treated as
459 break-points where, as a side-effect, a line is added to a calculation
460 as a protocol for proceeding towards a solution in step-wise problem
461 solving. At the same points Lucas-Interpretation serves interactive
462 tutoring and hands over control to the user. The user is free to
463 investigate underlying knowledge, applicable theorems, etc. And the
464 user can proceed constructing a solution by input of a tactic to be
465 applied or by input of a formula; in the latter case the
466 Lucas-Interpreter has built up a logical context (initialised with the
467 precondition of the formal specification) such that Isabelle can
468 derive the formula from this context --- or give feedback, that no
469 derivation can be found.
471 \subsection{Tactics as Control Flow Statements}
472 The flow of control in a program can be determined by {\tt if then else}
473 and {\tt case of} as mentioned on p.\pageref{isabelle-stmts} and also
474 by additional tactics:
476 \item[Repeat:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it
477 term}$: iterates over tactics which take a {\it term} as argument as
478 long as a tactic is applicable (for instance, {\tt Rewrite\_Set} might
481 \item[Try:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it term}$:
482 if {\it tactic} is applicable, then it is applied to {\it term},
483 otherwise {\it term} is passed on without changes.
485 \item[Or:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
486 term}\Rightarrow{\it term}$: If the first {\it tactic} is applicable,
487 it is applied to the first {\it term} yielding another {\it term},
488 otherwise the second {\it tactic} is applied; if none is applicable an
491 \item[@@:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
492 term}\Rightarrow{\it term}$: applies the first {\it tactic} to the
493 first {\it term} yielding an intermediate term (not appearing in the
494 signature) to which the second {\it tactic} is applied.
496 \item[While:] ${\it term::bool}\Rightarrow{\it tactic}\Rightarrow{\it
497 term}\Rightarrow{\it term}$: if the first {\it term} is true, then the
498 {\it tactic} is applied to the first {\it term} yielding an
499 intermediate term (not appearing in the signature); the intermediate
500 term is added to the environment the first {\it term} is evaluated in
501 etc. as long as the first {\it term} is true.
503 The tactics are not treated as break-points by Lucas-Interpretation
504 and thus do neither contribute to the calculation nor to interaction.
506 \section{Concepts and Tasks in TP-based Programming}\label{trial}
507 %\section{Development of a Program on Trial}
509 This section presents all the concepts involved in TP-based
510 programming and all the tasks to be accomplished by programmers. The
511 presentation uses the running example from
512 Fig.\ref{fig-interactive} on p.\pageref{fig-interactive}.
514 \subsection{Mechanization of Math --- Domain Engineering}\label{isabisac}
516 %WN was Fachleute unter obigem Titel interessiert findet sich
517 %WN unterhalb des auskommentierten Textes.
519 %WN der Text unten spricht Benutzer-Aspekte anund ist nicht speziell
520 %WN auf Computer-Mathematiker fokussiert.
521 % \paragraph{As mentioned in the introduction,} a prototype of an
522 % educational math assistant called
523 % {{\sisac}}\footnote{{{\sisac}}=\textbf{Isa}belle for
524 % \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges
525 % the gap between (1) introducation and (2) application of mathematics:
526 % {{\sisac}} is based on Computer Theorem Proving (TP), a technology which
527 % requires each fact and each action justified by formal logic, so
528 % {{{\sisac}{}}} makes justifications transparent to students in
529 % interactive step-wise problem solving. By that way {{\sisac}} already
532 % \item Introduction of math stuff (in e.g. partial fraction
533 % decomposition) by stepwise explaining and exercising respective
534 % symbolic calculations with ``next step guidance (NSG)'' and rigorously
535 % checking steps freely input by students --- this also in context with
536 % advanced applications (where the stuff to be taught in higher
537 % semesters can be skimmed through by NSG), and
538 % \item Application of math stuff in advanced engineering courses
539 % (e.g. problems to be solved by inverse Z-transform in a Signal
540 % Processing Lab) and now without much ado about basic math techniques
541 % (like partial fraction decomposition): ``next step guidance'' supports
542 % students in independently (re-)adopting such techniques.
544 % Before the question is answers, how {{\sisac}}
545 % accomplishes this task from a technical point of view, some remarks on
546 % the state-of-the-art is given, therefor follow up Section~\ref{emas}.
548 % \subsection{Educational Mathematics Assistants (EMAs)}\label{emas}
550 % \paragraph{Educational software in mathematics} is, if at all, based
551 % on Computer Algebra Systems (CAS, for instance), Dynamic Geometry
552 % Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org}
553 % \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC
554 % http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These
555 % base technologies are used to program math lessons and sometimes even
556 % exercises. The latter are cumbersome: the steps towards a solution of
557 % such an interactive exercise need to be provided with feedback, where
558 % at each step a wide variety of possible input has to be foreseen by
559 % the programmer - so such interactive exercises either require high
560 % development efforts or the exercises constrain possible inputs.
562 % \subparagraph{A new generation} of educational math assistants (EMAs)
563 % is emerging presently, which is based on Theorem Proving (TP). TP, for
564 % instance Isabelle and Coq, is a technology which requires each fact
565 % and each action justified by formal logic. Pushed by demands for
566 % \textit{proven} correctness of safety-critical software TP advances
567 % into software engineering; from these advancements computer
568 % mathematics benefits in general, and math education in particular. Two
569 % features of TP are immediately beneficial for learning:
571 % \paragraph{TP have knowledge in human readable format,} that is in
572 % standard predicate calculus. TP following the LCF-tradition have that
573 % knowledge down to the basic definitions of set, equality,
574 % etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html};
575 % following the typical deductive development of math, natural numbers
576 % are defined and their properties
577 % proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html},
578 % etc. Present knowledge mechanized in TP exceeds high-school
579 % mathematics by far, however by knowledge required in software
580 % technology, and not in other engineering sciences.
582 % \paragraph{TP can model the whole problem solving process} in
583 % mathematical problem solving {\em within} a coherent logical
584 % framework. This is already being done by three projects, by
585 % Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.
587 % Having the whole problem solving process within a logical coherent
588 % system, such a design guarantees correctness of intermediate steps and
589 % of the result (which seems essential for math software); and the
590 % second advantage is that TP provides a wealth of theories which can be
591 % exploited for mechanizing other features essential for educational
594 % \subsubsection{Generation of User Guidance in EMAs}\label{user-guid}
596 % One essential feature for educational software is feedback to user
597 % input and assistance in coming to a solution.
599 % \paragraph{Checking user input} by ATP during stepwise problem solving
600 % is being accomplished by the three projects mentioned above
601 % exclusively. They model the whole problem solving process as mentioned
602 % above, so all what happens between formalized assumptions (or formal
603 % specification) and goal (or fulfilled postcondition) can be
604 % mechanized. Such mechanization promises to greatly extend the scope of
605 % educational software in stepwise problem solving.
607 % \paragraph{NSG (Next step guidance)} comprises the system's ability to
608 % propose a next step; this is a challenge for TP: either a radical
609 % restriction of the search space by restriction to very specific
610 % problem classes is required, or much care and effort is required in
611 % designing possible variants in the process of problem solving
612 % \cite{proof-strategies-11}.
614 % Another approach is restricted to problem solving in engineering
615 % domains, where a problem is specified by input, precondition, output
616 % and postcondition, and where the postcondition is proven by ATP behind
617 % the scenes: Here the possible variants in the process of problem
618 % solving are provided with feedback {\em automatically}, if the problem
619 % is described in a TP-based programing language: \cite{plmms10} the
620 % programmer only describes the math algorithm without caring about
621 % interaction (the respective program is functional and even has no
622 % input or output statements!); interaction is generated as a
623 % side-effect by the interpreter --- an efficient separation of concern
624 % between math programmers and dialog designers promising application
625 % all over engineering disciplines.
628 % \subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}
629 % Authoring new mathematics knowledge in {{\sisac}} can be compared with
630 % ``application programing'' of engineering problems; most of such
631 % programing uses CAS-based programing languages (CAS = Computer Algebra
632 % Systems; e.g. Mathematica's or Maple's programing language).
634 % \paragraph{A novel type of TP-based language} is used by {{\sisac}{}}
635 % \cite{plmms10} for describing how to construct a solution to an
636 % engineering problem and for calling equation solvers, integration,
637 % etc~\footnote{Implementation of CAS-like functionality in TP is not
638 % primarily concerned with efficiency, but with a didactic question:
639 % What to decide for: for high-brow algorithms at the state-of-the-art
640 % or for elementary algorithms comprehensible for students?} within TP;
641 % TP can ensure ``systems that never make a mistake'' \cite{casproto} -
642 % are impossible for CAS which have no logics underlying.
644 % \subparagraph{Authoring is perfect} by writing such TP based programs;
645 % the application programmer is not concerned with interaction or with
646 % user guidance: this is concern of a novel kind of program interpreter
647 % called Lucas-Interpreter. This interpreter hands over control to a
648 % dialog component at each step of calculation (like a debugger at
649 % breakpoints) and calls automated TP to check user input following
650 % personalized strategies according to a feedback module.
652 % However ``application programing with TP'' is not done with writing a
653 % program: according to the principles of TP, each step must be
654 % justified. Such justifications are given by theorems. So all steps
655 % must be related to some theorem, if there is no such theorem it must
656 % be added to the existing knowledge, which is organized in so-called
657 % \textbf{theories} in Isabelle. A theorem must be proven; fortunately
658 % Isabelle comprises a mechanism (called ``axiomatization''), which
659 % allows to omit proofs. Such a theorem is shown in
660 % Example~\ref{eg:neuper1}.
662 The running example requires to determine the inverse $\cal
663 Z$-transform for a class of functions. The domain of Signal Processing
664 is accustomed to specific notation for the resulting functions, which
665 are absolutely capable of being totalled and are called step-response: $u[n]$, where $u$ is the
666 function, $n$ is the argument and the brackets indicate that the
667 arguments are discrete. Surprisingly, Isabelle accepts the rules for
668 $z^{-1}$ in this traditional notation~\footnote{Isabelle
669 experts might be particularly surprised, that the brackets do not
670 cause errors in typing (as lists).}:
674 {\footnotesize\begin{tabbing}
675 123\=123\=123\=123\=\kill
677 01\>axiomatization where \\
678 02\>\> rule1: ``$z^{-1}\;1 = \delta [n]$'' and\\
679 03\>\> rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow z^{-1}\;z / (z - 1) = u [n]$'' and\\
680 04\>\> rule3: ``$\vert\vert z \vert\vert < 1 \Rightarrow z / (z - 1) = -u [-n - 1]$'' and \\
681 05\>\> rule4: ``$\vert\vert z \vert\vert > \vert\vert$ $\alpha$ $\vert\vert \Rightarrow z / (z - \alpha) = \alpha^n \cdot u [n]$'' and\\
682 06\>\> rule5: ``$\vert\vert z \vert\vert < \vert\vert \alpha \vert\vert \Rightarrow z / (z - \alpha) = -(\alpha^n) \cdot u [-n - 1]$'' and\\
683 07\>\> rule6: ``$\vert\vert z \vert\vert > 1 \Rightarrow z/(z - 1)^2 = n \cdot u [n]$''
687 These 6 rules can be used as conditional rewrite rules, depending on
688 the respective convergence radius. Satisfaction from accordance with traditional notation
689 contrasts with the above word {\em axiomatization}: As TP-based, the
690 programming language expects these rules as {\em proved} theorems, and
691 not as axioms implemented in the above brute force manner; otherwise
692 all the verification efforts envisaged (like proof of the
693 post-condition, see below) would be meaningless.
695 Isabelle provides a large body of knowledge, rigorously proved from
696 the basic axioms of mathematics~\footnote{This way of rigorously
697 deriving all knowledge from first principles is called the
698 LCF-paradigm in TP.}. In the case of the ${\cal z}$-Transform the most advanced
699 knowledge can be found in the theories on Multivariate
700 Analysis~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate\_Analysis}. However,
701 building up knowledge such that a proof for the above rules would be
702 reasonably short and easily comprehensible, still requires lots of
703 work (and is definitely out of scope of our case study).
705 %REMOVED DUE TO SPACE CONSTRAINTS
706 %At the state-of-the-art in mechanization of knowledge in engineering
707 %sciences, the process does not stop with the mechanization of
708 %mathematics traditionally used in these sciences. Rather, ``Formal
709 %Methods''~\cite{ fm-03} are expected to proceed to formal and explicit
710 %description of physical items. Signal Processing, for instance is
711 %concerned with physical devices for signal acquisition and
712 %reconstruction, which involve measuring a physical signal, storing it,
713 %and possibly later rebuilding the original signal or an approximation
714 %thereof. For digital systems, this typically includes sampling and
715 %quantization; devices for signal compression, including audio
716 %compression, image compression, and video compression, etc. ``Domain
717 %engineering''\cite{db:dom-eng} is concerned with {\em specification}
718 %of these devices' components and features; this part in the process of
719 %mechanization is only at the beginning in domains like Signal
722 %TP-based programming, concern of this paper, is determined to
723 %add ``algorithmic knowledge'' to the mechanised body of knowledge.
724 %% in Fig.\ref{fig:mathuni} on
725 %% p.\pageref{fig:mathuni}. As we shall see below, TP-based programming
726 %% starts with a formal {\em specification} of the problem to be solved.
729 %% \includegraphics[width=110mm]{../../fig/jrocnik/math-universe-small}
730 %% \caption{The three-dimensional universe of mathematics knowledge}
731 %% \label{fig:mathuni}
734 %% The language for both axes is defined in the axis at the bottom, deductive
735 %% knowledge, in {\sisac} represented by Isabelle's theories.
737 \subsection{Preparation of Simplifiers for the Program}\label{simp}
739 All evaluation in the prototype's Lucas-Interpreter is done by term rewriting on
740 Isabelle's terms, see \S\ref{meth} below; in this section some of respective
741 preparations are described. In order to work reliably with term rewriting, the
742 respective rule-sets must be confluent and terminating~\cite{nipk:rew-all-that},
743 then they are called (canonical) simplifiers. These properties do not go without
744 saying, their establishment is a difficult task for the programmer; this task is
745 not yet supported in the prototype.
747 The prototype rewrites using theorems only. Axioms which are theorems as well
748 have been already shown in \S\ref{eg:neuper1} on p.\pageref{eg:neuper1} , we
749 assemble them in a rule-set and apply them in ML as follows:
753 01 val inverse_z = Rls
754 02 {id = "inverse_z",
755 03 rew_ord = dummy_ord,
757 05 rules = [Thm ("rule1", @{thm rule1}), Thm ("rule2", @{thm rule1}),
758 06 Thm ("rule3", @{thm rule3}), Thm ("rule4", @{thm rule4}),
759 07 Thm ("rule5", @{thm rule5}), Thm ("rule6", @{thm rule6})],
764 \noindent The items, line by line, in the above record have the following purpose:
766 \item[01..02] the ML-value \textit{inverse\_z} stores it's identifier
767 as a string for ``reflection'' when switching between the language
768 layers of Isabelle/ML (like in the Lucas-Interpreter) and
769 Isabelle/Isar (like in the example program on p.\pageref{s:impl} on
772 \item[03..04] both, (a) the rewrite-order~\cite{nipk:rew-all-that}
773 \textit{rew\_ord} and (b) the rule-set \textit{erls} are trivial here:
774 (a) the \textit{rules} in {\rm 07..12} don't need ordered rewriting
775 and (b) the assumptions of the \textit{rules} need not be evaluated
776 (they just go into the context during rewriting).
778 \item[05..07] the \textit{rules} are the axioms from p.\pageref{eg:neuper1};
779 also ML-functions (\S\ref{funs}) can come into this list as shown in
780 \S\ref{flow-prep}; so they are distinguished by type-constructors \textit{Thm}
781 and \textit{Calc} respectively; for the purpose of reflection both
782 contain their identifiers.
784 \item[08..09] are error-patterns not discussed here and \textit{scr}
785 is prepared to get a program, automatically generated by {\sisac} for
786 producing intermediate rewrites when requested by the user.
790 %OUTCOMMENTED DUE TO SPACE RESTRICTIONS
791 % \noindent It is advisable to immediately test rule-sets; for that
792 % purpose an appropriate term has to be created; \textit{parse} takes a
793 % context \textit{ctxt} and a string (with \textit{ZZ\_1} denoting ${\cal
794 % Z}^{-1}$) and creates a term:
799 % 02 val t = parse ctxt "ZZ_1 (z / (z - 1) + z / (z - </alpha>) + 1)";
801 % 04 val t = Const ("Build_Inverse_Z_Transform.ZZ_1",
802 % 05 "RealDef.real => RealDef.real => RealDef.real") $
803 % 06 (Const (...) $ (Const (...) $ Free (...) $ (Const (...) $ Free (...)
806 % \noindent The internal representation of the term, as required for
807 % rewriting, consists of \textit{Const}ants, a pair of a string
808 % \textit{"Groups.plus\_class.plus"} for $+$ and a type, variables
809 % \textit{Free} and the respective constructor \textit{\$}. Now the
810 % term can be rewritten by the rule-set \textit{inverse\_z}:
815 % 02 val SOME (t', asm) = rewrite_set_ @{theory} inverse\_z t;
819 % 06 val it = "u[n] + </alpha> ^ n * u[n] + </delta>[n]" : string
820 % 07 val it = "|| z || > 1 & || z || > </alpha>" : string
823 % \noindent The resulting term \textit{t} and the assumptions
824 % \textit{asm} are converted to readable strings by \textit{term2str}
825 % and \textit{terms2str}.
827 \subsection{Preparation of ML-Functions}\label{funs}
828 Some functionality required in programming, cannot be accomplished by
829 rewriting. So the prototype has a mechanism to call functions within
830 the rewrite-engine: certain redexes in Isabelle terms call these
831 functions written in SML~\cite{pl:milner97}, the implementation {\em
832 and} meta-language of Isabelle. The programmer has to use this
835 In the running example's program on p.\pageref{s:impl} the lines {\rm
836 05} and {\rm 06} contain such functions; we go into the details with
837 \textit{argument\_in X\_z;}. This function fetches the argument from a
838 function application: Line {\rm 03} in the example calculation on
839 p.\pageref{exp-calc} is created by line {\rm 06} of the example
840 program on p.\pageref{s:impl} where the program's environment assigns
841 the value \textit{X z} to the variable \textit{X\_z}; so the function
842 shall extract the argument \textit{z}.
844 \medskip In order to be recognised as a function constant in the
845 program source the constant needs to be declared in a theory, here in
846 \textit{Build\_Inverse\_Z\_Transform.thy}; then it can be parsed in
847 the context \textit{ctxt} of that theory:
852 02 argument'_in :: "real => real" ("argument'_in _" 10)
855 %^3.2^ ML {* val SOME t = parse ctxt "argument_in (X z)"; *}
856 %^3.2^ val t = Const ("Build_Inverse_Z_Transform.argument'_in", "RealDef.real ⇒ RealDef.real")
857 %^3.2^ $ (Free ("X", "RealDef.real ⇒ RealDef.real") $ Free ("z", "RealDef.real")): term
858 %^3.2^ \end{verbatim}}
860 %^3.2^ \noindent Parsing produces a term \texttt{t} in internal
861 %^3.2^ representation~\footnote{The attentive reader realizes the
862 %^3.2^ differences between interal and extermal representation even in the
863 %^3.2^ strings, i.e \texttt{'\_}}, consisting of \texttt{Const
864 %^3.2^ ("argument'\_in", type)} and the two variables \texttt{Free ("X",
865 %^3.2^ type)} and \texttt{Free ("z", type)}, \texttt{\$} is the term
867 The function body below is implemented directly in SML,
868 i.e in an \texttt{ML \{* *\}} block; the function definition provides
869 a unique prefix \texttt{eval\_} to the function name:
874 02 fun eval_argument_in _
875 03 "Build_Inverse_Z_Transform.argument'_in"
876 04 (t as (Const ("Build_Inverse_Z_Transform.argument'_in", _) $(f $arg))) _ =
877 05 if is_Free arg (*could be something to be simplified before*)
878 06 then SOME (term2str t ^"="^ term2str arg, Trueprop $(mk_equality (t, arg)))
880 08 | eval_argument_in _ _ _ _ = NONE;
884 \noindent The function body creates either \texttt{NONE}
885 telling the rewrite-engine to search for the next redex, or creates an
886 ad-hoc theorem for rewriting, thus the programmer needs to adopt many
887 technicalities of Isabelle, for instance, the \textit{Trueprop}
890 \bigskip This sub-task particularly sheds light on basic issues in the
891 design of a programming language, the integration of differential language
892 layers, the layer of Isabelle/Isar and Isabelle/ML.
894 Another point of improvement for the prototype is the rewrite-engine: The
895 program on p.\pageref{s:impl} would not allow to contract the two lines {\rm 05}
898 {\small\it\label{s:impl}
900 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
901 \>{\rm 05/06}\>\>\> (z::real) = argument\_in (lhs X\_eq) ;
904 \noindent because nested function calls would require creating redexes
905 inside-out; however, the prototype's rewrite-engine only works top down
906 from the root of a term down to the leaves.
908 How all these technicalities are to be checked in the prototype is
909 shown in \S\ref{flow-prep} below.
911 % \paragraph{Explicit Problems} require explicit methods to solve them, and within
912 % this methods we have some explicit steps to do. This steps can be unique for
913 % a special problem or refindable in other problems. No mather what case, such
914 % steps often require some technical functions behind. For the solving process
915 % of the Inverse Z Transformation and the corresponding partial fraction it was
916 % neccessary to build helping functions like \texttt{get\_denominator},
917 % \texttt{get\_numerator} or \texttt{argument\_in}. First two functions help us
918 % to filter the denominator or numerator out of a fraction, last one helps us to
919 % get to know the bound variable in a equation.
921 % By taking \texttt{get\_denominator} as an example, we want to explain how to
922 % implement new functions into the existing system and how we can later use them
925 % \subsubsection{Find a place to Store the Function}
927 % The whole system builds up on a well defined structure of Knowledge. This
928 % Knowledge sets up at the Path:
929 % \begin{center}\ttfamily src/Tools/isac/Knowledge\normalfont\end{center}
930 % For implementing the Function \texttt{get\_denominator} (which let us extract
931 % the denominator out of a fraction) we have choosen the Theory (file)
932 % \texttt{Rational.thy}.
934 % \subsubsection{Write down the new Function}
936 % In upper Theory we now define the new function and its purpose:
938 % get_denominator :: "real => real"
940 % This command tells the machine that a function with the name
941 % \texttt{get\_denominator} exists which gets a real expression as argument and
942 % returns once again a real expression. Now we are able to implement the function
943 % itself, upcoming example now shows the implementation of
944 % \texttt{get\_denominator}.
947 % \label{eg:getdenom}
951 % 02 *("get_denominator",
952 % 03 * ("Rational.get_denominator", eval_get_denominator ""))
954 % 05 fun eval_get_denominator (thmid:string) _
955 % 06 (t as Const ("Rational.get_denominator", _) $
956 % 07 (Const ("Rings.inverse_class.divide", _) $num
958 % 09 SOME (mk_thmid thmid ""
959 % 10 (Print_Mode.setmp []
960 % 11 (Syntax.string_of_term (thy2ctxt thy)) denom) "",
961 % 12 Trueprop $ (mk_equality (t, denom)))
962 % 13 | eval_get_denominator _ _ _ _ = NONE;\end{verbatim}
965 % Line \texttt{07} and \texttt{08} are describing the mode of operation the best -
966 % there is a fraction\\ (\ttfamily Rings.inverse\_class.divide\normalfont)
968 % into its two parts (\texttt{\$num \$denom}). The lines before are additionals
969 % commands for declaring the function and the lines after are modeling and
970 % returning a real variable out of \texttt{\$denom}.
972 % \subsubsection{Add a test for the new Function}
974 % \paragraph{Everytime when adding} a new function it is essential also to add
975 % a test for it. Tests for all functions are sorted in the same structure as the
976 % knowledge it self and can be found up from the path:
977 % \begin{center}\ttfamily test/Tools/isac/Knowledge\normalfont\end{center}
978 % This tests are nothing very special, as a first prototype the functionallity
979 % of a function can be checked by evaluating the result of a simple expression
980 % passed to the function. Example~\ref{eg:getdenomtest} shows the test for our
981 % \textit{just} created function \texttt{get\_denominator}.
984 % \label{eg:getdenomtest}
987 % 01 val thy = @{theory Isac};
988 % 02 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
989 % 03 val SOME (_, t') = eval_get_denominator "" 0 t thy;
990 % 04 if term2str t' = "get_denominator ((a + x) / b) = b" then ()
991 % 05 else error "get_denominator ((a + x) / b) = b" \end{verbatim}
994 % \begin{description}
995 % \item[01] checks if the proofer set up on our {\sisac{}} System.
996 % \item[02] passes a simple expression (fraction) to our suddenly created
998 % \item[04] checks if the resulting variable is the correct one (in this case
999 % ``b'' the denominator) and returns.
1000 % \item[05] handels the error case and reports that the function is not able to
1001 % solve the given problem.
1004 \subsection{Specification of the Problem}\label{spec}
1005 %WN <--> \chapter 7 der Thesis
1006 %WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..
1008 Mechanical treatment requires to translate a textual problem
1009 description like in Fig.\ref{fig-interactive} on
1010 p.\pageref{fig-interactive} into a {\em formal} specification. The
1011 formal specification of the running example could look like is this:
1013 %WN Hier brauchen wir die Spezifikation des 'running example' ...
1014 %JR Habe input, output und precond vom Beispiel eingefügt brauche aber Hilfe bei
1015 %JR der post condition - die existiert für uns ja eigentlich nicht aka
1016 %JR haben sie bis jetzt nicht beachtet WN...
1017 %WN2 Mein Vorschlag ist, das TODO zu lassen und deutlich zu kommentieren.
1021 {\small\begin{tabbing}
1022 123\=123\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill
1025 \> \>input \>: ${\it filterExpression} \;\;X\;z=\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}, \;{\it domain}\;\mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$\\
1026 \>\>precond \>: $\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}\;\; {\it continuous\_on}\; \mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$ \\
1027 \>\>output \>: stepResponse $x[n]$ \\
1028 \>\>postcond \>: TODO
1031 %JR wie besprochen, kein remark, keine begründung, nur simples "nicht behandelt"
1034 % Defining the postcondition requires a high amount mathematical
1035 % knowledge, the difficult part in our case is not to set up this condition
1036 % nor it is more to define it in a way the interpreter is able to handle it.
1037 % Due the fact that implementing that mechanisms is quite the same amount as
1038 % creating the programm itself, it is not avaible in our prototype.
1039 % \label{rm:postcond}
1042 The implementation of the formal specification in the present
1043 prototype, still bar-bones without support for authoring, is done
1045 %WN Kopie von Inverse_Z_Transform.thy, leicht versch"onert:
1047 {\footnotesize\label{exp-spec}
1050 01 store_specification
1051 02 (prepare_specification
1052 03 "pbl_SP_Ztrans_inv"
1055 06 ( ["Inverse", "Z_Transform", "SignalProcessing"],
1056 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1057 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1058 09 ("#Find" , ["stepResponse n_eq"]),
1059 10 ("#Post" , [" TODO "])])
1062 13 [["SignalProcessing","Z_Transform","Inverse"]]);
1066 Although the above details are partly very technical, we explain them
1067 in order to document some intricacies of TP-based programming in the
1068 present state of the {\sisac} prototype:
1070 \item[01..02]\textit{store\_specification:} stores the result of the
1071 function \textit{prep\_specification} in a global reference
1072 \textit{Unsynchronized.ref}, which causes principal conflicts with
1073 Isabelle's asynchronous document model~\cite{Wenzel-11:doc-orient} and
1074 parallel execution~\cite{Makarius-09:parall-proof} and is under
1075 reconstruction already.
1077 \textit{prep\_specification:} translates the specification to an internal format
1078 which allows efficient processing; see for instance line {\rm 07}
1080 \item[03..04] are a unique identifier for the specification within {\sisac}
1081 and the ``mathematics author'' holding the copy-rights.
1082 \item[05] is the Isabelle \textit{theory} required to parse the
1083 specification in lines {\rm 07..10}.
1084 \item[06] is a key into the tree of all specifications as presented to
1085 the user (where some branches might be hidden by the dialogue
1087 \item[07..10] are the specification with input, pre-condition, output
1088 and post-condition respectively; note that the specification contains
1089 variables to be instantiated with concrete values for a concrete problem ---
1090 thus the specification actually captures a class of problems. The post-condition is not handled in
1091 the prototype presently.
1092 \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
1093 rewriting determined by rule-sets.
1094 \item[12]\textit{NONE:} could be \textit{SOME ``solve ...''} for a
1095 problem associated to a function from Computer Algebra (like an
1096 equation solver) which is not the case here.
1097 \item[13] is a list of methods solving the specified problem (here
1098 only one list item) represented analogously to {\rm 06}.
1102 %WN die folgenden Erkl"arungen finden sich durch "grep -r 'datatype pbt' *"
1105 % {guh : guh, (*unique within this isac-knowledge*)
1106 % mathauthors: string list, (*copyright*)
1107 % init : pblID, (*to start refinement with*)
1108 % thy : theory, (* which allows to compile that pbt
1109 % TODO: search generalized for subthy (ref.p.69*)
1110 % (*^^^ WN050912 NOT used during application of the problem,
1111 % because applied terms may be from 'subthy' as well as from super;
1112 % thus we take 'maxthy'; see match_ags !*)
1113 % cas : term option,(*'CAS-command'*)
1114 % prls : rls, (* for preds in where_*)
1115 % where_: term list, (* where - predicates*)
1117 % (*this is the model-pattern;
1118 % it contains "#Given","#Where","#Find","#Relate"-patterns
1119 % for constraints on identifiers see "fun cpy_nam"*)
1120 % met : metID list}; (* methods solving the pbt*)
1122 %WN weil dieser Code sehr unaufger"aumt ist, habe ich die Erkl"arungen
1123 %WN oben selbst geschrieben.
1128 %WN das w"urde ich in \sec\label{progr} verschieben und
1129 %WN das SubProblem partial fractions zum Erkl"aren verwenden.
1130 % Such a specification is checked before the execution of a program is
1131 % started, the same applies for sub-programs. In the following example
1132 % (Example~\ref{eg:subprob}) shows the call of such a subproblem:
1136 % \label{eg:subprob}
1138 % {\ttfamily \begin{tabbing}
1139 % ``(L\_L::bool list) = (\=SubProblem (\=Test','' \\
1140 % ``\>\>[linear,univariate,equation,test],'' \\
1141 % ``\>\>[Test,solve\_linear])'' \\
1142 % ``\>[BOOL equ, REAL z])'' \\
1146 % \noindent If a program requires a result which has to be
1147 % calculated first we can use a subproblem to do so. In our specific
1148 % case we wanted to calculate the zeros of a fraction and used a
1149 % subproblem to calculate the zeros of the denominator polynom.
1154 \subsection{Implementation of the Method}\label{meth}
1155 A method collects all data required to interpret a certain program by
1156 Lucas-Interpretation. The \texttt{program} from p.\pageref{s:impl} of
1157 the running example is embedded on the last line in the following method:
1158 %The methods represent the different ways a problem can be solved. This can
1159 %include mathematical tactics as well as tactics taught in different courses.
1160 %Declaring the Method itself gives us the possibilities to describe the way of
1161 %calculation in deep, as well we get the oppertunities to build in different
1169 03 "SP_InverseZTransformation_classic"
1172 06 ( ["SignalProcessing", "Z_Transform", "Inverse"],
1173 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1174 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1175 09 ("#Find" , ["stepResponse n_eq"]),
1183 \noindent The above code stores the whole structure analogously to a
1184 specification as described above:
1186 \item[01..06] are identical to those for the example specification on
1187 p.\pageref{exp-spec}.
1189 \item[07..09] show something looking like the specification; this is a
1190 {\em guard}: as long as not all \textit{Given} items are present and
1191 the \textit{Pre}-conditions is not true, interpretation of the program
1194 \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
1195 \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
1196 \textit{srls, prls, nrls} feature evaluating (a) the ML-functions in the program (e.g.
1197 \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}
1198 and (c) is required for the derivation-machinery checking user-input formulas.
1200 \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}.
1202 The many rule-sets above cause considerable efforts for the
1203 programmers, in particular, because there are no tools for checking
1204 essential features of rule-sets.
1206 % is again very technical and goes hard in detail. Unfortunataly
1207 % most declerations are not essential for a basic programm but leads us to a huge
1208 % range of powerful possibilities.
1210 % \begin{description}
1211 % \item[01..02] stores the method with the given name into the system under a global
1213 % \item[03] specifies the topic within which context the method can be found.
1214 % \item[04..05] as the requirements for different methods can be deviant we
1215 % declare what is \emph{given} and and what to \emph{find} for this specific method.
1216 % The code again helds on the topic of the case studie, where the inverse
1217 % z-transformation does a switch between a term describing a electrical filter into
1218 % its step response. Also the datatype has to be declared (bool - due the fact that
1219 % we handle equations).
1220 % \item[06] \emph{rewrite order} is the order of this rls (ruleset), where one
1221 % theorem of it is used for rewriting one single step.
1222 % \item[07] \texttt{rls} is the currently used ruleset for this method. This set
1223 % has already been defined before.
1224 % \item[08] we would have the possiblitiy to add this method to a predefined tree of
1225 % calculations, i.eg. if it would be a sub of a bigger problem, here we leave it
1227 % \item[09] The \emph{source ruleset}, can be used to evaluate list expressions in
1229 % \item[10] \emph{predicates ruleset} can be used to indicates predicates within
1231 % \item[11] The \emph{check ruleset} summarizes rules for checking formulas
1233 % \item[12] \emph{error patterns} which are expected in this kind of method can be
1234 % pre-specified to recognize them during the method.
1235 % \item[13] finally the \emph{canonical ruleset}, declares the canonical simplifier
1236 % of the specific method.
1237 % \item[14] for this code snipset we don't specify the programm itself and keep it
1238 % empty. Follow up \S\ref{progr} for informations on how to implement this
1239 % \textit{main} part.
1242 \subsection{Implementation of the TP-based Program}\label{progr}
1243 So finally all the prerequisites are described and the final task can
1244 be addressed. The program below comes back to the running example: it
1245 computes a solution for the problem from Fig.\ref{fig-interactive} on
1246 p.\pageref{fig-interactive}. The reader is reminded of
1247 \S\ref{PL-isab}, the introduction of the programming language:
1249 {\footnotesize\it\label{s:impl}
1251 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
1252 \>{\rm 00}\>ML \{*\\
1253 \>{\rm 00}\>val program =\\
1254 \>{\rm 01}\> "{\tt Program} InverseZTransform (X\_eq::bool) = \\
1255 \>{\rm 02}\>\> {\tt let} \\
1256 \>{\rm 03}\>\>\> X\_eq = {\tt Take} X\_eq ; \\
1257 \>{\rm 04}\>\>\> X\_eq = {\tt Rewrite} prep\_for\_part\_frac X\_eq ; \\
1258 \>{\rm 05}\>\>\> (X\_z::real) = lhs X\_eq ; \\ %no inside-out evaluation
1259 \>{\rm 06}\>\>\> (z::real) = argument\_in X\_z; \\
1260 \>{\rm 07}\>\>\> (part\_frac::real) = {\tt SubProblem} \\
1261 \>{\rm 08}\>\>\>\>\>\>\>\> ( Isac, [partial\_fraction, rational, simplification], [] )\\
1262 %\>{\rm 10}\>\>\>\>\>\>\>\>\> [simplification, of\_rationals, to\_partial\_fraction] ) \\
1263 \>{\rm 09}\>\>\>\>\>\>\>\> [ (rhs X\_eq)::real, z::real ]; \\
1264 \>{\rm 10}\>\>\> (X'\_eq::bool) = {\tt Take} ((X'::real =$>$ bool) z = ZZ\_1 part\_frac) ; \\
1265 \>{\rm 11}\>\>\> X'\_eq = (({\tt Rewrite\_Set} prep\_for\_inverse\_z) @@ \\
1266 \>{\rm 12}\>\>\>\>\> $\;\;$ ({\tt Rewrite\_Set} inverse\_z)) X'\_eq \\
1267 \>{\rm 13}\>\> {\tt in } \\
1268 \>{\rm 14}\>\>\> X'\_eq"\\
1271 % ORIGINAL FROM Inverse_Z_Transform.thy
1272 % "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1273 % "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1274 % " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1275 % " (X'_z::real) = lhs X'; "^(* ?X' z*)
1276 % " (zzz::real) = argument_in X'_z; "^(* z *)
1277 % " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1279 % " (pbz::real) = (SubProblem (Isac', "^(**)
1280 % " [partial_fraction,rational,simplification], "^
1281 % " [simplification,of_rationals,to_partial_fraction]) "^
1282 % " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1284 % " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1285 % " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1286 % " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1287 % " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1288 % " 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]*)
1289 % " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1290 % "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1291 The program is represented as a string and part of the method in
1292 \S\ref{meth}. As mentioned in \S\ref{PL} the program is purely
1293 functional and lacks any input statements and output statements. So
1294 the steps of calculation towards a solution (and interactive tutoring
1295 in step-wise problem solving) are created as a side-effect by
1296 Lucas-Interpretation. The side-effects are triggered by the tactics
1297 \texttt{Take}, \texttt{Rewrite}, \texttt{SubProblem} and
1298 \texttt{Rewrite\_Set} in the above lines {\rm 03, 04, 07, 10, 11} and
1299 {\rm 12} respectively. These tactics produce the respective lines in the
1300 calculation on p.\pageref{flow-impl}.
1302 The above lines {\rm 05, 06} do not contain a tactics, so they do not
1303 immediately contribute to the calculation on p.\pageref{flow-impl};
1304 rather, they compute actual arguments for the \texttt{SubProblem} in
1305 line {\rm 09}~\footnote{The tactics also are break-points for the
1306 interpreter, where control is handed over to the user in interactive
1307 tutoring.}. Line {\rm 11} contains tactical \textit{@@}.
1309 \medskip The above program also indicates the dominant role of interactive
1310 selection of knowledge in the three-dimensional universe of
1311 mathematics. The \texttt{SubProblem} in the above lines
1312 {\rm 07..09} is more than a function call with the actual arguments
1313 \textit{[ (rhs X\_eq)::real, z::real ]}. The programmer has to determine
1317 \item the theory, in the example \textit{Isac} because different
1318 methods can be selected in Pt.3 below, which are defined in different
1319 theories with \textit{Isac} collecting them.
1320 \item the specification identified by \textit{[partial\_fraction,
1321 rational, simplification]} in the tree of specifications; this
1322 specification is analogous to the specification of the main program
1323 described in \S\ref{spec}; the problem is to find a ``partial fraction
1324 decomposition'' for a univariate rational polynomial.
1325 \item the method in the above example is \textit{[ ]}, i.e. empty,
1326 which supposes the interpreter to select one of the methods predefined
1327 in the specification, for instance in line {\rm 13} in the running
1328 example's specification on p.\pageref{exp-spec}~\footnote{The freedom
1329 (or obligation) for selection carries over to the student in
1330 interactive tutoring.}.
1333 The program code, above presented as a string, is parsed by Isabelle's
1334 parser --- the program is an Isabelle term. This fact is expected to
1335 simplify verification tasks in the future; on the other hand, this
1336 fact causes troubles in error detection which are discussed as part
1337 of the work-flow in the subsequent section.
1339 \section{Work-flow of Programming in the Prototype}\label{workflow}
1340 The new prover IDE Isabelle/jEdit~\cite{makar-jedit-12} is a great
1341 step forward for interactive theory and proof development. The
1342 {\sisac}-prototype re-uses this IDE as a programming environment. The
1343 experiences from this re-use show, that the essential components are
1344 available from Isabelle/jEdit. However, additional tools and features
1345 are required to achieve acceptable usability.
1347 So notable experiences are reported here, also as a requirement
1348 capture for further development of TP-based languages and respective
1351 \subsection{Preparations and Trials}\label{flow-prep}
1352 The many sub-tasks to be accomplished {\em before} the first line of
1353 program code can be written and tested suggest an approach which
1354 step-wise establishes the prerequisites. The case study underlying
1355 this paper~\cite{jrocnik-bakk} documents the approach in a separate
1357 \textit{Build\_Inverse\_Z\_Transform.thy}~\footnote{http://www.ist.tugraz.at/projects/isac/publ/Build\_Inverse\_Z\_Transform.thy}. Part
1358 II in the study comprises this theory, \LaTeX ed from the theory by
1359 use of Isabelle's document preparation system. This paper resembles
1360 the approach in \S\ref{isabisac} to \S\ref{meth}, which in actual
1361 implementation work involves several iterations.
1363 \bigskip For instance, only the last step, implementing the program
1364 described in \S\ref{meth}, reveals details required. Let us assume,
1365 this is the ML-function \textit{argument\_in} required in line {\rm 06}
1366 of the example program on p.\pageref{s:impl}; how this function needs
1367 to be implemented in the prototype has been discussed in \S\ref{funs}
1370 Now let us assume, that calling this function from the program code
1371 does not work; so testing this function is required in order to find out
1372 the reason: type errors, a missing entry of the function somewhere or
1373 even more nasty technicalities \dots
1378 02 val SOME t = parseNEW ctxt "argument_in (X (z::real))";
1379 03 val SOME (str, t') = eval_argument_in ""
1380 04 "Build_Inverse_Z_Transform.argument'_in" t 0;
1383 07 val it = "(argument_in X z) = z": string\end{verbatim}}
1385 \noindent So, this works: we get an ad-hoc theorem, which used in
1386 rewriting would reduce \texttt{argument\_in X z} to \texttt{z}. Now we check this
1387 reduction and create a rule-set \texttt{rls} for that purpose:
1392 02 val rls = append_rls "test" e_rls
1393 03 [Calc ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")]
1394 04 val SOME (t', asm) = rewrite_set_ @{theory} rls t;
1396 06 val t' = Free ("z", "RealDef.real"): term
1397 07 val asm = []: term list\end{verbatim}}
1399 \noindent The resulting term \texttt{t'} is \texttt{Free ("z",
1400 "RealDef.real")}, i.e the variable \texttt{z}, so all is
1401 perfect. Probably we have forgotten to store this function correctly~?
1402 We review the respective \texttt{calclist} (again an
1403 \textit{Unsynchronized.ref} to be removed in order to adjust to
1404 Isabelle/Isar's asynchronous document model):
1408 01 calclist:= overwritel (! calclist,
1410 03 ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")),
1412 05 ]);\end{verbatim}}
1414 \noindent The entry is perfect. So what is the reason~? Ah, probably there
1415 is something messed up with the many rule-sets in the method, see \S\ref{meth} ---
1416 right, the function \texttt{argument\_in} is not contained in the respective
1417 rule-set \textit{srls} \dots this just as an example of the intricacies in
1418 debugging a program in the present state of the prototype.
1420 \subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}
1421 Given all the prerequisites from \S\ref{isabisac} to \S\ref{meth},
1422 usually developed within several iterations, the program can be
1423 assembled; on p.\pageref{s:impl} there is the complete program of the
1426 The completion of this program required efforts for several weeks
1427 (after some months of familiarisation with {\sisac}), caused by the
1428 abundance of intricacies indicated above. Also writing the program is
1429 not pleasant, given Isabelle/Isar/ without add-ons for
1430 programming. Already writing and parsing a few lines of program code
1431 is a challenge: the program is an Isabelle term; Isabelle's parser,
1432 however, is not meant for huge terms like the program of the running
1433 example. So reading out the specific error (usually type errors) from
1434 Isabelle's message is difficult.
1436 \medskip Testing the evaluation of the program has to rely on very
1437 simple tools. Step-wise execution is modeled by a function
1438 \texttt{me}, short for mathematics-engine~\footnote{The interface used
1439 by the front-end which created the calculation on
1440 p.\pageref{fig-interactive} is different from this function}:
1441 %the following is a simplification of the actual function
1446 02 val it = tac -> ctree * pos -> mout * tac * ctree * pos\end{verbatim}}
1448 \noindent This function takes as arguments a tactic \texttt{tac} which
1449 determines the next step, the step applied to the interpreter-state
1450 \texttt{ctree * pos} as last argument taken. The interpreter-state is
1451 a pair of a tree \texttt{ctree} representing the calculation created
1452 (see the example below) and a position \texttt{pos} in the
1453 calculation. The function delivers a quadruple, beginning with the new
1454 formula \texttt{mout} and the next tactic followed by the new
1457 This function allows to stepwise check the program:
1463 03 ["filterExpression (X z = 3 / ((z::real) + 1/10 - 1/50*(1/z)))",
1464 04 "stepResponse (x[n::real]::bool)"];
1467 07 ["Inverse", "Z_Transform", "SignalProcessing"],
1468 08 ["SignalProcessing","Z_Transform","Inverse"]);
1469 09 val (mout, tac, ctree, pos) = CalcTreeTEST [(fmz, (dI, pI, mI))];
1470 10 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1471 11 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1472 12 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1473 13 ...\end{verbatim}}
1475 \noindent Several dozens of calls for \texttt{me} are required to
1476 create the lines in the calculation below (including the sub-problems
1477 not shown). When an error occurs, the reason might be located
1478 many steps before: if evaluation by rewriting, as done by the prototype,
1479 fails, then first nothing happens --- the effects come later and
1480 cause unpleasant checks.
1482 The checks comprise watching the rewrite-engine for many different
1483 kinds of rule-sets (see \S\ref{meth}), the interpreter-state, in
1484 particular the environment and the context at the states position ---
1485 all checks have to rely on simple functions accessing the
1486 \texttt{ctree}. So getting the calculation below (which resembles the
1487 calculation in Fig.\ref{fig-interactive} on p.\pageref{fig-interactive})
1488 is the result of several weeks of development:
1490 {\small\it\label{exp-calc}
1492 123l\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=\kill
1493 \>{\rm 01}\> $\bullet$ \> {\tt Problem } (Inverse\_Z\_Transform, [Inverse, Z\_Transform, SignalProcessing]) \`\\
1494 \>{\rm 02}\>\> $\vdash\;\;X z = \frac{3}{z - \frac{1}{4} - \frac{1}{8} \cdot z^{-1}}$ \`{\footnotesize {\tt Take} X\_eq}\\
1495 \>{\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}\\
1496 \>{\rm 04}\>\> $\bullet$\> {\tt Problem } [partial\_fraction,rational,simplification] \`{\footnotesize {\tt SubProblem} \dots}\\
1497 \>{\rm 05}\>\>\> $\vdash\;\;\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=$ \`- - -\\
1498 \>{\rm 06}\>\>\> $\frac{24}{-1 + -2 \cdot z + 8 \cdot z^2}$ \`- - -\\
1499 \>{\rm 07}\>\>\> $\bullet$\> solve ($-1 + -2 \cdot z + 8 \cdot z^2,\;z$ ) \`- - -\\
1500 \>{\rm 08}\>\>\>\> $\vdash$ \> $\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=0$ \`- - -\\
1501 \>{\rm 09}\>\>\>\> $z = \frac{2+\sqrt{-4+8}}{16}\;\lor\;z = \frac{2-\sqrt{-4+8}}{16}$ \`- - -\\
1502 \>{\rm 10}\>\>\>\> $z = \frac{1}{2}\;\lor\;z =$ \_\_\_ \`- - -\\
1503 \> \>\>\>\> \_\_\_ \`- - -\\
1504 \>{\rm 11}\>\> \dots\> $\frac{4}{z - \frac{1}{2}} + \frac{-4}{z - \frac{-1}{4}}$ \`\\
1505 \>{\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)}\\
1506 \>{\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 }\\
1507 \>{\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}\\
1508 \>{\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}}
1510 The tactics on the right margin of the above calculation are those in
1511 the program on p.\pageref{s:impl} which create the respective formulas
1513 % ORIGINAL FROM Inverse_Z_Transform.thy
1514 % "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1515 % "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1516 % " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1517 % " (X'_z::real) = lhs X'; "^(* ?X' z*)
1518 % " (zzz::real) = argument_in X'_z; "^(* z *)
1519 % " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1521 % " (pbz::real) = (SubProblem (Isac', "^(**)
1522 % " [partial_fraction,rational,simplification], "^
1523 % " [simplification,of_rationals,to_partial_fraction]) "^
1524 % " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1526 % " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1527 % " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1528 % " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1529 % " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1530 % " 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]*)
1531 % " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1532 % "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1534 \subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}
1535 Finally \textit{Build\_Inverse\_Z\_Transform.thy} has got the job done
1536 and the knowledge accumulated in it can be distributed to appropriate
1537 theories: the program to \textit{Inverse\_Z\_Transform.thy}, the
1538 sub-problem accomplishing the partial fraction decomposition to
1539 \textit{Partial\_Fractions.thy}. Since there are hacks into Isabelle's
1540 internals, this kind of distribution is not trivial. For instance, the
1541 function \texttt{argument\_in} in \S\ref{funs} explicitly contains a
1542 string with the theory it has been defined in, so this string needs to
1543 be updated from \texttt{Build\_Inverse\_Z\_Transform} to
1544 \texttt{Atools} if that function is transferred to theory
1545 \textit{Atools.thy}.
1547 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.
1548 This process is also rather bare-bones without authoring tools and is
1549 described in detail in the {\sisac} wiki~\footnote{http://www.ist.tugraz.at/isac/index.php/Generate\_representations\_for\_ISAC\_Knowledge}.
1552 % -------------------------------------------------------------------
1554 % Material, falls noch Platz bleibt ...
1556 % -------------------------------------------------------------------
1559 % \subsubsection{Trials on Notation and Termination}
1561 % \paragraph{Technical notations} are a big problem for our piece of software,
1562 % but the reason for that isn't a fault of the software itself, one of the
1563 % troubles comes out of the fact that different technical subtopics use different
1564 % symbols and notations for a different purpose. The most famous example for such
1565 % a symbol is the complex number $i$ (in cassique math) or $j$ (in technical
1566 % math). In the specific part of signal processing one of this notation issues is
1567 % the use of brackets --- we use round brackets for analoge signals and squared
1568 % brackets for digital samples. Also if there is no problem for us to handle this
1569 % fact, we have to tell the machine what notation leads to wich meaning and that
1570 % this purpose seperation is only valid for this special topic - signal
1572 % \subparagraph{In the programming language} itself it is not possible to declare
1573 % fractions, exponents, absolutes and other operators or remarks in a way to make
1574 % them pretty to read; our only posssiblilty were ASCII characters and a handfull
1575 % greek symbols like: $\alpha, \beta, \gamma, \phi,\ldots$.
1577 % With the upper collected knowledge it is possible to check if we were able to
1578 % donate all required terms and expressions.
1580 % \subsubsection{Definition and Usage of Rules}
1582 % \paragraph{The core} of our implemented problem is the Z-Transformation, due
1583 % the fact that the transformation itself would require higher math which isn't
1584 % yet avaible in our system we decided to choose the way like it is applied in
1585 % labratory and problem classes at our university - by applying transformation
1586 % rules (collected in transformation tables).
1587 % \paragraph{Rules,} in {\sisac{}}'s programming language can be designed by the
1588 % use of axiomatizations like shown in Example~\ref{eg:ruledef}
1591 % \label{eg:ruledef}
1594 % axiomatization where
1595 % rule1: ``1 = $\delta$[n]'' and
1596 % rule2: ``|| z || > 1 ==> z / (z - 1) = u [n]'' and
1597 % rule3: ``|| z || < 1 ==> z / (z - 1) = -u [-n - 1]''
1601 % This rules can be collected in a ruleset and applied to a given expression as
1602 % follows in Example~\ref{eg:ruleapp}.
1606 % \label{eg:ruleapp}
1608 % \item Store rules in ruleset:
1610 % val inverse_Z = append_rls "inverse_Z" e_rls
1611 % [ Thm ("rule1",num_str @{thm rule1}),
1612 % Thm ("rule2",num_str @{thm rule2}),
1613 % Thm ("rule3",num_str @{thm rule3})
1615 % \item Define exression:
1617 % val sample_term = str2term "z/(z-1)+z/(z-</delta>)+1";\end{verbatim}
1618 % \item Apply ruleset:
1620 % val SOME (sample_term', asm) =
1621 % rewrite_set_ thy true inverse_Z sample_term;\end{verbatim}
1625 % The use of rulesets makes it much easier to develop our designated applications,
1626 % but the programmer has to be careful and patient. When applying rulesets
1627 % two important issues have to be mentionend:
1628 % \subparagraph{How often} the rules have to be applied? In case of
1629 % transformations it is quite clear that we use them once but other fields
1630 % reuqire to apply rules until a special condition is reached (e.g.
1631 % a simplification is finished when there is nothing to be done left).
1632 % \subparagraph{The order} in which rules are applied often takes a big effect
1633 % and has to be evaluated for each purpose once again.
1635 % In our special case of Signal Processing and the rules defined in
1636 % Example~\ref{eg:ruledef} we have to apply rule~1 first of all to transform all
1637 % constants. After this step has been done it no mather which rule fit's next.
1639 % \subsubsection{Helping Functions}
1641 % \paragraph{New Programms require,} often new ways to get through. This new ways
1642 % means that we handle functions that have not been in use yet, they can be
1643 % something special and unique for a programm or something famous but unneeded in
1644 % the system yet. In our dedicated example it was for example neccessary to split
1645 % a fraction into numerator and denominator; the creation of such function and
1646 % even others is described in upper Sections~\ref{simp} and \ref{funs}.
1648 % \subsubsection{Trials on equation solving}
1649 % %simple eq and problem with double fractions/negative exponents
1650 % \paragraph{The Inverse Z-Transformation} makes it neccessary to solve
1651 % equations degree one and two. Solving equations in the first degree is no
1652 % problem, wether for a student nor for our machine; but even second degree
1653 % equations can lead to big troubles. The origin of this troubles leads from
1654 % the build up process of our equation solving functions; they have been
1655 % implemented some time ago and of course they are not as good as we want them to
1656 % be. Wether or not following we only want to show how cruel it is to build up new
1657 % work on not well fundamentials.
1658 % \subparagraph{A simple equation solving,} can be set up as shown in the next
1665 % ["equality (-1 + -2 * z + 8 * z ^^^ 2 = (0::real))",
1669 % val (dI',pI',mI') =
1671 % ["abcFormula","degree_2","polynomial","univariate","equation"],
1672 % ["no_met"]);\end{verbatim}
1675 % Here we want to solve the equation: $-1+-2\cdot z+8\cdot z^{2}=0$. (To give
1676 % a short overview on the commands; at first we set up the equation and tell the
1677 % machine what's the bound variable and where to store the solution. Second step
1678 % is to define the equation type and determine if we want to use a special method
1679 % to solve this type.) Simple checks tell us that the we will get two results for
1680 % this equation and this results will be real.
1681 % So far it is easy for us and for our machine to solve, but
1682 % mentioned that a unvariate equation second order can have three different types
1683 % of solutions it is getting worth.
1684 % \subparagraph{The solving of} all this types of solutions is not yet supported.
1685 % Luckily it was needed for us; but something which has been needed in this
1686 % context, would have been the solving of an euation looking like:
1687 % $-z^{-2}+-2\cdot z^{-1}+8=0$ which is basically the same equation as mentioned
1688 % before (remember that befor it was no problem to handle for the machine) but
1689 % now, after a simple equivalent transformation, we are not able to solve
1691 % \subparagraph{Error messages} we get when we try to solve something like upside
1692 % were very confusing and also leads us to no special hint about a problem.
1693 % \par The fault behind is, that we have no well error handling on one side and
1694 % no sufficient formed equation solving on the other side. This two facts are
1695 % making the implemention of new material very difficult.
1697 % \subsection{Formalization of missing knowledge in Isabelle}
1699 % \paragraph{A problem} behind is the mechanization of mathematic
1700 % theories in TP-bases languages. There is still a huge gap between
1701 % these algorithms and this what we want as a solution - in Example
1702 % Signal Processing.
1708 % X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY
1711 % \noindent A very simple example on this what we call gap is the
1712 % simplification above. It is needles to say that it is correct and also
1713 % Isabelle for fills it correct - \emph{always}. But sometimes we don't
1714 % want expand such terms, sometimes we want another structure of
1715 % them. Think of a problem were we now would need only the coefficients
1716 % of $X$ and $Y$. This is what we call the gap between mechanical
1717 % simplification and the solution.
1722 % \paragraph{We are not able to fill this gap,} until we have to live
1723 % with it but first have a look on the meaning of this statement:
1724 % Mechanized math starts from mathematical models and \emph{hopefully}
1725 % proceeds to match physics. Academic engineering starts from physics
1726 % (experimentation, measurement) and then proceeds to mathematical
1727 % modeling and formalization. The process from a physical observance to
1728 % a mathematical theory is unavoidable bound of setting up a big
1729 % collection of standards, rules, definition but also exceptions. These
1730 % are the things making mechanization that difficult.
1739 % \noindent Think about some units like that one's above. Behind
1740 % each unit there is a discerning and very accurate definition: One
1741 % Meter is the distance the light travels, in a vacuum, through the time
1742 % of 1 / 299.792.458 second; one kilogram is the weight of a
1743 % platinum-iridium cylinder in paris; and so on. But are these
1744 % definitions usable in a computer mechanized world?!
1749 % \paragraph{A computer} or a TP-System builds on programs with
1750 % predefined logical rules and does not know any mathematical trick
1751 % (follow up example \ref{eg:trick}) or recipe to walk around difficult
1757 % \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]
1758 % \[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)=
1759 % \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]
1760 % \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]
1762 % \noindent Sometimes it is also useful to be able to apply some
1763 % \emph{tricks} to get a beautiful and particularly meaningful result,
1764 % which we are able to interpret. But as seen in this example it can be
1765 % hard to find out what operations have to be done to transform a result
1766 % into a meaningful one.
1771 % \paragraph{The only possibility,} for such a system, is to work
1772 % through its known definitions and stops if none of these
1773 % fits. Specified on Signal Processing or any other application it is
1774 % often possible to walk through by doing simple creases. This creases
1775 % are in general based on simple math operational but the challenge is
1776 % to teach the machine \emph{all}\footnote{Its pride to call it
1777 % \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to
1778 % reach a high level of \emph{all} but it in real it will still be a
1779 % survey of knowledge which links to other knowledge and {{\sisac}{}} a
1780 % trainer and helper but no human compensating calculator.
1782 % {{{\sisac}{}}} itself aims to adds \emph{Algorithmic Knowledge} (formal
1783 % specifications of problems out of topics from Signal Processing, etc.)
1784 % and \emph{Application-oriented Knowledge} to the \emph{deductive} axis of
1785 % physical knowledge. The result is a three-dimensional universe of
1786 % mathematics seen in Figure~\ref{fig:mathuni}.
1790 % \includegraphics{fig/universe}
1791 % \caption{Didactic ``Math-Universe'': Algorithmic Knowledge (Programs) is
1792 % combined with Application-oriented Knowledge (Specifications) and Deductive Knowledge (Axioms, Definitions, Theorems). The Result
1793 % leads to a three dimensional math universe.\label{fig:mathuni}}
1797 % %WN Deine aktuelle Benennung oben wird Dir kein Fachmann abnehmen;
1798 % %WN bitte folgende Bezeichnungen nehmen:
1800 % %WN axis 1: Algorithmic Knowledge (Programs)
1801 % %WN axis 2: Application-oriented Knowledge (Specifications)
1802 % %WN axis 3: Deductive Knowledge (Axioms, Definitions, Theorems)
1804 % %WN und bitte die R"ander von der Grafik wegschneiden (was ich f"ur *.pdf
1805 % %WN nicht hinkriege --- weshalb ich auch die eJMT-Forderung nicht ganz
1806 % %WN verstehe, separierte PDFs zu schicken; ich w"urde *.png schicken)
1808 % %JR Ränder und beschriftung geändert. Keine Ahnung warum eJMT sich pdf's
1809 % %JR wünschen, würde ebenfalls png oder ähnliches verwenden, aber wenn pdf's
1810 % %JR gefordert werden WN2...
1811 % %WN2 meiner Meinung nach hat sich eJMT unklar ausgedr"uckt (z.B. kann
1812 % %WN2 man meines Wissens pdf-figures nicht auf eine bestimmte Gr"osse
1813 % %WN2 zusammenschneiden um die R"ander weg zu bekommen)
1814 % %WN2 Mein Vorschlag ist, in umserem tex-file bei *.png zu bleiben und
1815 % %WN2 png + pdf figures mitzuschicken.
1817 % \subsection{Notes on Problems with Traditional Notation}
1819 % \paragraph{During research} on these topic severely problems on
1820 % traditional notations have been discovered. Some of them have been
1821 % known in computer science for many years now and are still unsolved,
1822 % one of them aggregates with the so called \emph{Lambda Calculus},
1823 % Example~\ref{eg:lamda} provides a look on the problem that embarrassed
1830 % \[ f(x)=\ldots\; \quad R \rightarrow \quad R \]
1833 % \[ f(p)=\ldots\; p \in \quad R \]
1836 % \noindent Above we see two equations. The first equation aims to
1837 % be a mapping of an function from the reel range to the reel one, but
1838 % when we change only one letter we get the second equation which
1839 % usually aims to insert a reel point $p$ into the reel function. In
1840 % computer science now we have the problem to tell the machine (TP) the
1841 % difference between this two notations. This Problem is called
1842 % \emph{Lambda Calculus}.
1847 % \paragraph{An other problem} is that terms are not full simplified in
1848 % traditional notations, in {{\sisac}} we have to simplify them complete
1849 % to check weather results are compatible or not. in e.g. the solutions
1850 % of an second order linear equation is an rational in {{\sisac}} but in
1851 % tradition we keep fractions as long as possible and as long as they
1852 % aim to be \textit{beautiful} (1/8, 5/16,...).
1853 % \subparagraph{The math} which should be mechanized in Computer Theorem
1854 % Provers (\emph{TP}) has (almost) a problem with traditional notations
1855 % (predicate calculus) for axioms, definitions, lemmas, theorems as a
1856 % computer program or script is not able to interpret every Greek or
1857 % Latin letter and every Greek, Latin or whatever calculations
1858 % symbol. Also if we would be able to handle these symbols we still have
1859 % a problem to interpret them at all. (Follow up \hbox{Example
1860 % \ref{eg:symbint1}})
1864 % \label{eg:symbint1}
1866 % u\left[n\right] \ \ldots \ unitstep
1869 % \noindent The unitstep is something we need to solve Signal
1870 % Processing problem classes. But in {{{\sisac}{}}} the rectangular
1871 % brackets have a different meaning. So we abuse them for our
1872 % requirements. We get something which is not defined, but usable. The
1873 % Result is syntax only without semantic.
1878 % In different problems, symbols and letters have different meanings and
1879 % ask for different ways to get through. (Follow up \hbox{Example
1880 % \ref{eg:symbint2}})
1884 % \label{eg:symbint2}
1886 % \widehat{\ }\ \widehat{\ }\ \widehat{\ } \ \ldots \ exponent
1889 % \noindent For using exponents the three \texttt{widehat} symbols
1890 % are required. The reason for that is due the development of
1891 % {{{\sisac}{}}} the single \texttt{widehat} and also the double were
1892 % already in use for different operations.
1897 % \paragraph{Also the output} can be a problem. We are familiar with a
1898 % specified notations and style taught in university but a computer
1899 % program has no knowledge of the form proved by a professor and the
1900 % machines themselves also have not yet the possibilities to print every
1901 % symbol (correct) Recent developments provide proofs in a human
1902 % readable format but according to the fact that there is no money for
1903 % good working formal editors yet, the style is one thing we have to
1906 % \section{Problems rising out of the Development Environment}
1908 % fehlermeldungen! TODO
1910 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{verbatim}
1912 \section{Conclusion}\label{conclusion}
1916 %This paper gives a first experience report about programming with a
1917 %TP-based programming language.
1919 A brief re-introduction of the novel kind of programming
1920 language by example of the {\sisac}-prototype makes the paper
1921 self-contained. The main section describes all the main concepts
1922 involved in TP-based programming and all the sub-tasks concerning
1923 respective implementation: mechanisation of mathematics and domain
1924 modeling, implementation of term rewriting systems for the
1925 rewriting-engine, formal (implicit) specification of the problem to be
1926 (explicitly) described by the program, implementation of the many components
1927 required for Lucas-Interpretation and finally implementation of the
1930 The many concepts and sub-tasks involved in programming require a
1931 comprehensive work-flow; first experiences with the work-flow as
1932 supported by the present prototype are described as well: Isabelle +
1933 Isar + jEdit provide appropriate components for establishing an
1934 efficient development environment integrating computation and
1935 deduction. However, the present state of the prototype is far off a
1936 state appropriate for wide-spread use: the prototype of the program
1937 language lacks expressiveness and elegance, the prototype of the
1938 development environment is hardly usable: error messages still address
1939 the developer of the prototype's interpreter rather than the
1940 application programmer, implementation of the many settings for the
1941 Lucas-Interpreter is cumbersome.
1943 From these experiences a successful proof of concept can be concluded:
1944 programming arbitrary problems from engineering sciences is possible,
1945 in principle even in the prototype. Furthermore the experiences allow
1946 to conclude detailed requirements for further development:
1948 \item Clarify underlying logics such that programming is smoothly
1949 integrated with verification of the program; the post-condition should
1950 be proved more or less automatically, otherwise working engineers
1951 would not encounter such programming.
1952 \item Combine the prototype's programming language with Isabelle's
1953 powerful function package and probably with more of SML's
1954 pattern-matching features; include parallel execution on multi-core
1955 machines into the language design.
1956 \item Extend the prototype's Lucas-Interpreter such that it also
1957 handles functions defined by use of Isabelle's functions package; and
1958 generalize Isabelle's code generator such that efficient code for the
1959 whole definition of the programming language can be generated (for
1960 multi-core machines).
1961 \item Develop an efficient development environment with
1962 integration of programming and proving, with management not only of
1963 Isabelle theories, but also of large collections of specifications and
1966 Provided successful accomplishment, these points provide distinguished
1967 components for virtual workbenches appealing to practitioner of
1968 engineering in the near future.
1970 \medskip Interactive course material, as addressed by the title, then
1971 can comprise step-wise problem solving created as a side-effect of a
1972 TP-based program: Lucas-Interpretation not only provides an
1973 interactive programming environment, Lucas-Interpretation also can
1974 provide TP-based services for a flexible dialogue component with
1975 adaptive user guidance for independent and inquiry-based learning.
1978 \bibliographystyle{alpha}
1979 {\small\bibliography{references}}
1982 % LocalWords: TP IST SPSC Telematics Dialogues dialogue HOL bool nat Hindley
1983 % LocalWords: Milner tt Subproblem Formulae ruleset generalisation initialised
1984 % LocalWords: axiomatization LCF Simplifiers simplifiers Isar rew Thm Calc SML
1985 % LocalWords: recognised hoc Trueprop redexes Unsynchronized pre rhs ord erls
1986 % LocalWords: srls prls nrls lhs errpats InverseZTransform SubProblem IDE IDEs
1987 % LocalWords: univariate jEdit rls RealDef calclist familiarisation ons pos eq
1988 % LocalWords: mout ctree SignalProcessing frac ZZ Postcond Atools wiki SML's
1989 % LocalWords: mechanisation multi