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61 \ and Technology, Volume 1, Number 1, ISSN 1933-2823} %
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70 % Please place your own definitions here
72 \def\isac{${\cal I}\mkern-2mu{\cal S}\mkern-5mu{\cal AC}$}
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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 Technologie\\
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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. %TODO ... connect to prototype ...
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
213 Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\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 a novice to the
230 The goal of the case study was (1) 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 from the {\sisac}-team and (3) records
234 and comments for the main steps of development in an Isabelle theory;
235 this theory should provide guidelines for future programmers. An
236 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 The problem is from the domain of Signal Processing and requests to
250 determine the inverse ${\cal z}$-transform for a given term.
251 Fig.\ref{fig-interactive}
252 also shows the beginning of the interactive construction of a solution
253 for the problem. This construction is done in the right window named
256 User-interaction on the Worksheet is {\em checked} and {\em guided} by
259 \item Formulas input by the user are {\em checked} by TP: such a
260 formula establishes a proof situation --- the prover has to derive the
261 formula from the logical context. The context is built up from the
262 formal specification of the problem (here hidden from the user) by the
264 \item If the user gets stuck, the program developed below in this
265 paper ``knows the next step'' and Lucas-Interpretation provides services
266 featuring so-called ``next-step-guidance''; this is out of scope of this
267 paper and can be studied in~\cite{gdaroczy-EP-13}.
268 \end{enumerate} It should be noted that the programmer using the
269 TP-based language is not concerned with interaction at all; we will
270 see that the program contains neither input-statements nor
271 output-statements. Rather, interaction is handled by the interpreter
274 So there is a clear separation of concerns: Dialogues are adapted by
275 dialogue authors (in Java-based tools), using TP services provided by
276 Lucas-Interpretation. The latter acts on programs developed by
277 mathematics-authors (in Isabelle/ML); their task is concern of this
280 \paragraph{The paper is structured} as follows: The introduction
281 \S\ref{intro} is followed by a brief re-introduction of the TP-based
282 programming language in \S\ref{PL}, which extends the executable
283 fragment of Isabelle's language (\S\ref{PL-isab}) by tactics which
284 play a specific role in Lucas-Interpretation and in providing the TP
285 services (\S\ref{PL-tacs}). The main part \S\ref{trial} describes
286 the main steps in developing the program for the running example:
287 prepare domain knowledge, implement the formal specification of the
288 problem, prepare the environment for the interpreter, implement the
289 program in \S\ref{isabisac} to \S\ref{progr} respectively.
290 The work-flow of programming, debugging and testing is
291 described in \S\ref{workflow}. The conclusion \S\ref{conclusion} will
292 give directions identified for future development.
295 \section{\isac's Prototype for a Programming Language}\label{PL}
296 The prototype of the language and of the Lucas-Interpreter are briefly
297 described from the point of view of a programmer. The language extends
298 the executable fragment in the language of the theorem prover
299 Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}.
301 \subsection{The Executable Fragment of Isabelle's Language}\label{PL-isab}
302 The executable fragment consists of data-type and function
303 definitions. It's usability even suggests that fragment for
304 introductory courses \cite{nipkow-prog-prove}. HOL is a typed logic
305 whose type system resembles that of functional programming
306 languages. Thus there are
308 \item[base types,] in particular \textit{bool}, the type of truth
309 values, \textit{nat}, \textit{int}, \textit{complex}, and the types of
310 natural, integer and complex numbers respectively in mathematics.
311 \item[type constructors] allow to define arbitrary types, from
312 \textit{set}, \textit{list} to advanced data-structures like
313 \textit{trees}, red-black-trees etc.
314 \item[function types,] denoted by $\Rightarrow$.
315 \item[type variables,] denoted by $^\prime a, ^\prime b$ etc, provide
316 type polymorphism. Isabelle automatically computes the type of each
317 variable in a term by use of Hindley-Milner type inference
318 \cite{pl:hind97,Milner-78}.
321 \textbf{Terms} are formed as in functional programming by applying
322 functions to arguments. If $f$ is a function of type
323 $\tau_1\Rightarrow \tau_2$ and $t$ is a term of type $\tau_1$ then
324 $f\;t$ is a term of type~$\tau_2$. $t\;::\;\tau$ means that term $t$
325 has type $\tau$. There are many predefined infix symbols like $+$ and
326 $\leq$ most of which are overloaded for various types.
328 HOL also supports some basic constructs from functional programming:
329 {\footnotesize\it\label{isabelle-stmts}
330 \begin{tabbing} 123\=\kill
331 01\>$( \; {\tt if} \; b \; {\tt then} \; t_1 \; {\tt else} \; t_2 \;)$\\
332 02\>$( \; {\tt let} \; x=t \; {\tt in} \; u \; )$\\
333 03\>$( \; {\tt case} \; t \; {\tt of} \; {\it pat}_1
334 \Rightarrow t_1 \; |\dots| \; {\it pat}_n\Rightarrow t_n \; )$
336 \noindent The running example's program uses some of these elements
337 (marked by {\tt tt-font} on p.\pageref{s:impl}): for instance {\tt
338 let}\dots{\tt in} in lines {\rm 02} \dots {\rm 13}. In fact, the whole program
339 is an Isabelle term with specific function constants like {\tt
340 program}, {\tt Take}, {\tt Rewrite}, {\tt Subproblem} and {\tt
341 Rewrite\_Set} in lines {\rm 01, 03. 04, 07, 10} and {\rm 11, 12}
344 % Terms may also contain $\lambda$-abstractions. For example, $\lambda
345 % x. \; x$ is the identity function.
347 %JR warum auskommentiert? WN2...
348 %WN2 weil ein Punkt wie dieser in weiteren Zusammenh"angen innerhalb
349 %WN2 des Papers auftauchen m"usste; nachdem ich einen solchen
350 %WN2 Zusammenhang _noch_ nicht sehe, habe ich den Punkt _noch_ nicht
352 %WN2 Wenn der Punkt nicht weiter gebraucht wird, nimmt er nur wertvollen
353 %WN2 Platz f"ur Anderes weg.
355 \textbf{Formulae} are terms of type \textit{bool}. There are the basic
356 constants \textit{True} and \textit{False} and the usual logical
357 connectives (in decreasing order of precedence): $\neg, \land, \lor,
360 \textbf{Equality} is available in the form of the infix function $=$
361 of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for
362 formulas, where it means ``if and only if''.
364 \textbf{Quantifiers} are written $\forall x. \; P$ and $\exists x. \;
365 P$. Quantifiers lead to non-executable functions, so functions do not
366 always correspond to programs, for instance, if comprising \\$(
367 \;{\it if} \; \exists x.\;P \; {\it then} \; e_1 \; {\it else} \; e_2
370 \subsection{\isac's Tactics for Lucas-Interpretation}\label{PL-tacs}
371 The prototype extends Isabelle's language by specific statements
372 called tactics~\footnote{{\sisac}'s tactics are different from
373 Isabelle's tactics: the former concern steps in a calculation, the
374 latter concern proofs.} and tactics. For the programmer these
375 statements are functions with the following signatures:
378 \item[Rewrite:] ${\it theorem}\Rightarrow{\it term}\Rightarrow{\it
379 term} * {\it term}\;{\it list}$:
380 this tactic applies {\it theorem} to a {\it term} yielding a {\it
381 term} and a {\it term list}, the list are assumptions generated by
382 conditional rewriting. For instance, the {\it theorem}
383 $b\not=0\land c\not=0\Rightarrow\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$
384 applied to the {\it term} $\frac{2\cdot x}{3\cdot x}$ yields
385 $(\frac{2}{3}, [x\not=0])$.
387 \item[Rewrite\_Set:] ${\it ruleset}\Rightarrow{\it
388 term}\Rightarrow{\it term} * {\it term}\;{\it list}$:
389 this tactic applies {\it ruleset} to a {\it term}; {\it ruleset} is
390 a confluent and terminating term rewrite system, in general. If
391 none of the rules ({\it theorem}s) is applicable on interpretation
392 of this tactic, an exception is thrown.
394 % \item[Rewrite\_Inst:] ${\it substitution}\Rightarrow{\it
395 % theorem}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
398 % \item[Rewrite\_Set\_Inst:] ${\it substitution}\Rightarrow{\it
399 % ruleset}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it
403 \item[Substitute:] ${\it substitution}\Rightarrow{\it
404 term}\Rightarrow{\it term}$: allows to access sub-terms.
407 \item[Take:] ${\it term}\Rightarrow{\it term}$:
408 this tactic has no effect in the program; but it creates a side-effect
409 by Lucas-Interpretation (see below) and writes {\it term} to the
412 \item[Subproblem:] ${\it theory} * {\it specification} * {\it
413 method}\Rightarrow{\it argument}\;{\it list}\Rightarrow{\it term}$:
414 this tactic is a generalisation of a function call: it takes an
415 \textit{argument list} as usual, and additionally a triple consisting
416 of an Isabelle \textit{theory}, an implicit \textit{specification} of the
417 program and a \textit{method} containing data for Lucas-Interpretation,
418 last not least a program (as an explicit specification)~\footnote{In
419 interactive tutoring these three items can be determined explicitly
422 The tactics play a specific role in
423 Lucas-Interpretation~\cite{wn:lucas-interp-12}: they are treated as
424 break-points where, as a side-effect, a line is added to a calculation
425 as a protocol for proceeding towards a solution in step-wise problem
426 solving. At the same points Lucas-Interpretation serves interactive
427 tutoring and hands over control to the user. The user is free to
428 investigate underlying knowledge, applicable theorems, etc. And the
429 user can proceed constructing a solution by input of a tactic to be
430 applied or by input of a formula; in the latter case the
431 Lucas-Interpreter has built up a logical context (initialised with the
432 precondition of the formal specification) such that Isabelle can
433 derive the formula from this context --- or give feedback, that no
434 derivation can be found.
436 \subsection{Tactics as Control Flow Statements}
437 The flow of control in a program can be determined by {\tt if then else}
438 and {\tt case of} as mentioned on p.\pageref{isabelle-stmts} and also
439 by additional tactics:
441 \item[Repeat:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it
442 term}$: iterates over tactics which take a {\it term} as argument as
443 long as a tactic is applicable (for instance, {\tt Rewrite\_Set} might
446 \item[Try:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it term}$:
447 if {\it tactic} is applicable, then it is applied to {\it term},
448 otherwise {\it term} is passed on without changes.
450 \item[Or:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
451 term}\Rightarrow{\it term}$: If the first {\it tactic} is applicable,
452 it is applied to the first {\it term} yielding another {\it term},
453 otherwise the second {\it tactic} is applied; if none is applicable an
456 \item[@@:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it
457 term}\Rightarrow{\it term}$: applies the first {\it tactic} to the
458 first {\it term} yielding an intermediate term (not appearing in the
459 signature) to which the second {\it tactic} is applied.
461 \item[While:] ${\it term::bool}\Rightarrow{\it tactic}\Rightarrow{\it
462 term}\Rightarrow{\it term}$: if the first {\it term} is true, then the
463 {\it tactic} is applied to the first {\it term} yielding an
464 intermediate term (not appearing in the signature); the intermediate
465 term is added to the environment the first {\it term} is evaluated in
466 etc. as long as the first {\it term} is true.
468 The tactics are not treated as break-points by Lucas-Interpretation
469 and thus do neither contribute to the calculation nor to interaction.
471 \section{Concepts and Tasks in TP-based Programming}\label{trial}
472 %\section{Development of a Program on Trial}
474 This section presents all the concepts involved in TP-based
475 programming and all the tasks to be accomplished by programmers. The
476 presentation uses the running example from
477 Fig.\ref{fig-interactive} on p.\pageref{fig-interactive}.
479 \subsection{Mechanization of Math --- Domain Engineering}\label{isabisac}
481 %WN was Fachleute unter obigem Titel interessiert findet sich
482 %WN unterhalb des auskommentierten Textes.
484 %WN der Text unten spricht Benutzer-Aspekte anund ist nicht speziell
485 %WN auf Computer-Mathematiker fokussiert.
486 % \paragraph{As mentioned in the introduction,} a prototype of an
487 % educational math assistant called
488 % {{\sisac}}\footnote{{{\sisac}}=\textbf{Isa}belle for
489 % \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges
490 % the gap between (1) introducation and (2) application of mathematics:
491 % {{\sisac}} is based on Computer Theorem Proving (TP), a technology which
492 % requires each fact and each action justified by formal logic, so
493 % {{{\sisac}{}}} makes justifications transparent to students in
494 % interactive step-wise problem solving. By that way {{\sisac}} already
497 % \item Introduction of math stuff (in e.g. partial fraction
498 % decomposition) by stepwise explaining and exercising respective
499 % symbolic calculations with ``next step guidance (NSG)'' and rigorously
500 % checking steps freely input by students --- this also in context with
501 % advanced applications (where the stuff to be taught in higher
502 % semesters can be skimmed through by NSG), and
503 % \item Application of math stuff in advanced engineering courses
504 % (e.g. problems to be solved by inverse Z-transform in a Signal
505 % Processing Lab) and now without much ado about basic math techniques
506 % (like partial fraction decomposition): ``next step guidance'' supports
507 % students in independently (re-)adopting such techniques.
509 % Before the question is answers, how {{\sisac}}
510 % accomplishes this task from a technical point of view, some remarks on
511 % the state-of-the-art is given, therefor follow up Section~\ref{emas}.
513 % \subsection{Educational Mathematics Assistants (EMAs)}\label{emas}
515 % \paragraph{Educational software in mathematics} is, if at all, based
516 % on Computer Algebra Systems (CAS, for instance), Dynamic Geometry
517 % Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org}
518 % \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC
519 % http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These
520 % base technologies are used to program math lessons and sometimes even
521 % exercises. The latter are cumbersome: the steps towards a solution of
522 % such an interactive exercise need to be provided with feedback, where
523 % at each step a wide variety of possible input has to be foreseen by
524 % the programmer - so such interactive exercises either require high
525 % development efforts or the exercises constrain possible inputs.
527 % \subparagraph{A new generation} of educational math assistants (EMAs)
528 % is emerging presently, which is based on Theorem Proving (TP). TP, for
529 % instance Isabelle and Coq, is a technology which requires each fact
530 % and each action justified by formal logic. Pushed by demands for
531 % \textit{proven} correctness of safety-critical software TP advances
532 % into software engineering; from these advancements computer
533 % mathematics benefits in general, and math education in particular. Two
534 % features of TP are immediately beneficial for learning:
536 % \paragraph{TP have knowledge in human readable format,} that is in
537 % standard predicate calculus. TP following the LCF-tradition have that
538 % knowledge down to the basic definitions of set, equality,
539 % etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html};
540 % following the typical deductive development of math, natural numbers
541 % are defined and their properties
542 % proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html},
543 % etc. Present knowledge mechanized in TP exceeds high-school
544 % mathematics by far, however by knowledge required in software
545 % technology, and not in other engineering sciences.
547 % \paragraph{TP can model the whole problem solving process} in
548 % mathematical problem solving {\em within} a coherent logical
549 % framework. This is already being done by three projects, by
550 % Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.
552 % Having the whole problem solving process within a logical coherent
553 % system, such a design guarantees correctness of intermediate steps and
554 % of the result (which seems essential for math software); and the
555 % second advantage is that TP provides a wealth of theories which can be
556 % exploited for mechanizing other features essential for educational
559 % \subsubsection{Generation of User Guidance in EMAs}\label{user-guid}
561 % One essential feature for educational software is feedback to user
562 % input and assistance in coming to a solution.
564 % \paragraph{Checking user input} by ATP during stepwise problem solving
565 % is being accomplished by the three projects mentioned above
566 % exclusively. They model the whole problem solving process as mentioned
567 % above, so all what happens between formalized assumptions (or formal
568 % specification) and goal (or fulfilled postcondition) can be
569 % mechanized. Such mechanization promises to greatly extend the scope of
570 % educational software in stepwise problem solving.
572 % \paragraph{NSG (Next step guidance)} comprises the system's ability to
573 % propose a next step; this is a challenge for TP: either a radical
574 % restriction of the search space by restriction to very specific
575 % problem classes is required, or much care and effort is required in
576 % designing possible variants in the process of problem solving
577 % \cite{proof-strategies-11}.
579 % Another approach is restricted to problem solving in engineering
580 % domains, where a problem is specified by input, precondition, output
581 % and postcondition, and where the postcondition is proven by ATP behind
582 % the scenes: Here the possible variants in the process of problem
583 % solving are provided with feedback {\em automatically}, if the problem
584 % is described in a TP-based programing language: \cite{plmms10} the
585 % programmer only describes the math algorithm without caring about
586 % interaction (the respective program is functional and even has no
587 % input or output statements!); interaction is generated as a
588 % side-effect by the interpreter --- an efficient separation of concern
589 % between math programmers and dialog designers promising application
590 % all over engineering disciplines.
593 % \subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}
594 % Authoring new mathematics knowledge in {{\sisac}} can be compared with
595 % ``application programing'' of engineering problems; most of such
596 % programing uses CAS-based programing languages (CAS = Computer Algebra
597 % Systems; e.g. Mathematica's or Maple's programing language).
599 % \paragraph{A novel type of TP-based language} is used by {{\sisac}{}}
600 % \cite{plmms10} for describing how to construct a solution to an
601 % engineering problem and for calling equation solvers, integration,
602 % etc~\footnote{Implementation of CAS-like functionality in TP is not
603 % primarily concerned with efficiency, but with a didactic question:
604 % What to decide for: for high-brow algorithms at the state-of-the-art
605 % or for elementary algorithms comprehensible for students?} within TP;
606 % TP can ensure ``systems that never make a mistake'' \cite{casproto} -
607 % are impossible for CAS which have no logics underlying.
609 % \subparagraph{Authoring is perfect} by writing such TP based programs;
610 % the application programmer is not concerned with interaction or with
611 % user guidance: this is concern of a novel kind of program interpreter
612 % called Lucas-Interpreter. This interpreter hands over control to a
613 % dialog component at each step of calculation (like a debugger at
614 % breakpoints) and calls automated TP to check user input following
615 % personalized strategies according to a feedback module.
617 % However ``application programing with TP'' is not done with writing a
618 % program: according to the principles of TP, each step must be
619 % justified. Such justifications are given by theorems. So all steps
620 % must be related to some theorem, if there is no such theorem it must
621 % be added to the existing knowledge, which is organized in so-called
622 % \textbf{theories} in Isabelle. A theorem must be proven; fortunately
623 % Isabelle comprises a mechanism (called ``axiomatization''), which
624 % allows to omit proofs. Such a theorem is shown in
625 % Example~\ref{eg:neuper1}.
627 The running example requires to determine the inverse $\cal
628 Z$-transform for a class of functions. The domain of Signal Processing
629 is accustomed to specific notation for the resulting functions, which
630 are absolutely capable of being totalled and are called step-response: $u[n]$, where $u$ is the
631 function, $n$ is the argument and the brackets indicate that the
632 arguments are discrete. Surprisingly, Isabelle accepts the rules for
633 $z^{-1}$ in this traditional notation~\footnote{Isabelle
634 experts might be particularly surprised, that the brackets do not
635 cause errors in typing (as lists).}:
639 {\footnotesize\begin{tabbing}
640 123\=123\=123\=123\=\kill
642 01\>axiomatization where \\
643 02\>\> rule1: ``$z^{-1}\;1 = \delta [n]$'' and\\
644 03\>\> rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow z^{-1}\;z / (z - 1) = u [n]$'' and\\
645 04\>\> rule3: ``$\vert\vert z \vert\vert < 1 \Rightarrow z / (z - 1) = -u [-n - 1]$'' and \\
646 05\>\> rule4: ``$\vert\vert z \vert\vert > \vert\vert$ $\alpha$ $\vert\vert \Rightarrow z / (z - \alpha) = \alpha^n \cdot u [n]$'' and\\
647 06\>\> rule5: ``$\vert\vert z \vert\vert < \vert\vert \alpha \vert\vert \Rightarrow z / (z - \alpha) = -(\alpha^n) \cdot u [-n - 1]$'' and\\
648 07\>\> rule6: ``$\vert\vert z \vert\vert > 1 \Rightarrow z/(z - 1)^2 = n \cdot u [n]$''
652 These 6 rules can be used as conditional rewrite rules, depending on
653 the respective convergence radius. Satisfaction from accordance with traditional notation
654 contrasts with the above word {\em axiomatization}: As TP-based, the
655 programming language expects these rules as {\em proved} theorems, and
656 not as axioms implemented in the above brute force manner; otherwise
657 all the verification efforts envisaged (like proof of the
658 post-condition, see below) would be meaningless.
660 Isabelle provides a large body of knowledge, rigorously proven from
661 the basic axioms of mathematics~\footnote{This way of rigorously
662 deriving all knowledge from first principles is called the
663 LCF-paradigm in TP.}. In the case of the ${\cal z}$-Transform the most advanced
664 knowledge can be found in the theories on Multivariate
665 Analysis~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate\_Analysis}. However,
666 building up knowledge such that a proof for the above rules would be
667 reasonably short and easily comprehensible, still requires lots of
668 work (and is definitely out of scope of our case study).
670 %REMOVED DUE TO SPACE CONSTRAINTS
671 %At the state-of-the-art in mechanization of knowledge in engineering
672 %sciences, the process does not stop with the mechanization of
673 %mathematics traditionally used in these sciences. Rather, ``Formal
674 %Methods''~\cite{ fm-03} are expected to proceed to formal and explicit
675 %description of physical items. Signal Processing, for instance is
676 %concerned with physical devices for signal acquisition and
677 %reconstruction, which involve measuring a physical signal, storing it,
678 %and possibly later rebuilding the original signal or an approximation
679 %thereof. For digital systems, this typically includes sampling and
680 %quantization; devices for signal compression, including audio
681 %compression, image compression, and video compression, etc. ``Domain
682 %engineering''\cite{db:dom-eng} is concerned with {\em specification}
683 %of these devices' components and features; this part in the process of
684 %mechanization is only at the beginning in domains like Signal
687 %TP-based programming, concern of this paper, is determined to
688 %add ``algorithmic knowledge'' to the mechanised body of knowledge.
689 %% in Fig.\ref{fig:mathuni} on
690 %% p.\pageref{fig:mathuni}. As we shall see below, TP-based programming
691 %% starts with a formal {\em specification} of the problem to be solved.
694 %% \includegraphics[width=110mm]{../../fig/jrocnik/math-universe-small}
695 %% \caption{The three-dimensional universe of mathematics knowledge}
696 %% \label{fig:mathuni}
699 %% The language for both axes is defined in the axis at the bottom, deductive
700 %% knowledge, in {\sisac} represented by Isabelle's theories.
702 \subsection{Preparation of Simplifiers for the Program}\label{simp}
704 All evaluation in the prototype's Lucas-Interpreter is done by term rewriting on
705 Isabelle's terms, see \S\ref{meth} below; in this section some of respective
706 preparations are described. In order to work reliably with term rewriting, the
707 respective rule-sets must be confluent and terminating~\cite{nipk:rew-all-that},
708 then they are called (canonical) simplifiers. These properties do not go without
709 saying, their establishment is a difficult task for the programmer; this task is
710 not yet supported in the prototype.
712 The prototype rewrites using theorems only. Axioms which are theorems as well
713 have been already shown in \S\ref{eg:neuper1} on p.\pageref{eg:neuper1} , we
714 assemble them in a rule-set and apply them in ML as follows:
718 01 val inverse_z = Rls
719 02 {id = "inverse_z",
720 03 rew_ord = dummy_ord,
722 05 rules = [Thm ("rule1", @{thm rule1}), Thm ("rule2", @{thm rule1}),
723 06 Thm ("rule3", @{thm rule3}), Thm ("rule4", @{thm rule4}),
724 07 Thm ("rule5", @{thm rule5}), Thm ("rule6", @{thm rule6})],
729 \noindent The items, line by line, in the above record have the following purpose:
731 \item[01..02] the ML-value \textit{inverse\_z} stores it's identifier
732 as a string for ``reflection'' when switching between the language
733 layers of Isabelle/ML (like in the Lucas-Interpreter) and
734 Isabelle/Isar (like in the example program on p.\pageref{s:impl} on
737 \item[03..04] both, (a) the rewrite-order~\cite{nipk:rew-all-that}
738 \textit{rew\_ord} and (b) the rule-set \textit{erls} are trivial here:
739 (a) the \textit{rules} in {\rm 07..12} don't need ordered rewriting
740 and (b) the assumptions of the \textit{rules} need not be evaluated
741 (they just go into the context during rewriting).
743 \item[05..07] the \textit{rules} are the axioms from p.\pageref{eg:neuper1};
744 also ML-functions (\S\ref{funs}) can come into this list as shown in
745 \S\ref{flow-prep}; so they are distinguished by type-constructors \textit{Thm}
746 and \textit{Calc} respectively; for the purpose of reflection both
747 contain their identifiers.
749 \item[08..09] are error-patterns not discussed here and \textit{scr}
750 is prepared to get a program, automatically generated by {\sisac} for
751 producing intermediate rewrites when requested by the user.
755 \noindent It is advisable to immediately test rule-sets; for that
756 purpose an appropriate term has to be created; \textit{parse} takes a
757 context \textit{ctxt} and a string (with \textit{ZZ\_1} denoting ${\cal
758 Z}^{-1}$) and creates a term:
763 02 val t = parse ctxt "ZZ_1 (z / (z - 1) + z / (z - </alpha>) + 1)";
765 04 val t = Const ("Build_Inverse_Z_Transform.ZZ_1",
766 05 "RealDef.real => RealDef.real => RealDef.real") $
767 06 (Const (...) $ (Const (...) $ Free (...) $ (Const (...) $ Free (...)
770 \noindent The internal representation of the term, as required for
771 rewriting, consists of \textit{Const}ants, a pair of a string
772 \textit{"Groups.plus\_class.plus"} for $+$ and a type, variables
773 \textit{Free} and the respective constructor \textit{\$}. Now the
774 term can be rewritten by the rule-set \textit{inverse\_z}:
779 02 val SOME (t', asm) = rewrite_set_ @{theory} inverse\_z t;
783 06 val it = "u[n] + </alpha> ^ n * u[n] + </delta>[n]" : string
784 07 val it = "|| z || > 1 & || z || > </alpha>" : string
787 \noindent The resulting term \textit{t} and the assumptions
788 \textit{asm} are converted to readable strings by \textit{term2str}
789 and \textit{terms2str}.
791 \subsection{Preparation of ML-Functions}\label{funs}
792 Some functionality required in programming, cannot be accomplished by
793 rewriting. So the prototype has a mechanism to call functions within
794 the rewrite-engine: certain regexes in Isabelle terms call these
795 functions written in SML~\cite{pl:milner97}, the implementation {\em
796 and} meta-language of Isabelle. The programmer has to use this
799 In the running example's program on p.\pageref{s:impl} the lines {\rm
800 05} and {\rm 06} contain such functions; we go into the details with
801 \textit{argument\_in X\_z;}. This function fetches the argument from a
802 function application: Line {\rm 03} in the example calculation on
803 p.\pageref{exp-calc} is created by line {\rm 06} of the example
804 program on p.\pageref{s:impl} where the program's environment assigns
805 the value \textit{X z} to the variable \textit{X\_z}; so the function
806 shall extract the argument \textit{z}.
808 \medskip In order to be recognised as a function constant in the
809 program source the constant needs to be declared in a theory, here in
810 \textit{Build\_Inverse\_Z\_Transform.thy}; then it can be parsed in
811 the context \textit{ctxt} of that theory:
816 02 argument'_in :: "real => real" ("argument'_in _" 10)
819 %^3.2^ ML {* val SOME t = parse ctxt "argument_in (X z)"; *}
820 %^3.2^ val t = Const ("Build_Inverse_Z_Transform.argument'_in", "RealDef.real ⇒ RealDef.real")
821 %^3.2^ $ (Free ("X", "RealDef.real ⇒ RealDef.real") $ Free ("z", "RealDef.real")): term
822 %^3.2^ \end{verbatim}}
824 %^3.2^ \noindent Parsing produces a term \texttt{t} in internal
825 %^3.2^ representation~\footnote{The attentive reader realizes the
826 %^3.2^ differences between interal and extermal representation even in the
827 %^3.2^ strings, i.e \texttt{'\_}}, consisting of \texttt{Const
828 %^3.2^ ("argument'\_in", type)} and the two variables \texttt{Free ("X",
829 %^3.2^ type)} and \texttt{Free ("z", type)}, \texttt{\$} is the term
831 The function body below is implemented directly in SML,
832 i.e in an \texttt{ML \{* *\}} block; the function definition provides
833 a unique prefix \texttt{eval\_} to the function name:
838 02 fun eval_argument_in _
839 03 "Build_Inverse_Z_Transform.argument'_in"
840 04 (t as (Const ("Build_Inverse_Z_Transform.argument'_in", _) $(f $arg))) _ =
841 05 if is_Free arg (*could be something to be simplified before*)
842 06 then SOME (term2str t ^"="^ term2str arg, Trueprop $(mk_equality (t, arg)))
844 08 | eval_argument_in _ _ _ _ = NONE;
848 \noindent The function body creates either creates \texttt{NONE}
849 telling the rewrite-engine to search for the next regex, or creates an
850 ad-hoc theorem for rewriting, thus the programmer needs to adopt many
851 technicalities of Isabelle, for instance, the \textit{Trueprop}
854 \bigskip This sub-task particularly sheds light on basic issues in the
855 design of a programming language, the integration of differential language
856 layers, the layer of Isabelle/Isar and Isabelle/ML.
858 Another point of improvement for the prototype is the rewrite-engine: The
859 program on p.\pageref{s:impl} would not allow to contract the two lines {\rm 05}
862 {\small\it\label{s:impl}
864 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
865 \>{\rm 05/06}\>\>\> (z::real) = argument\_in (lhs X\_eq) ;
868 \noindent because nested function calls would require creating redexes
869 inside-out; however, the prototype's rewrite-engine only works top down
870 from the root of a term down to the leaves.
872 How all these technicalities are to be checked in the prototype is
873 shown in \S\ref{flow-prep} below.
875 % \paragraph{Explicit Problems} require explicit methods to solve them, and within
876 % this methods we have some explicit steps to do. This steps can be unique for
877 % a special problem or refindable in other problems. No mather what case, such
878 % steps often require some technical functions behind. For the solving process
879 % of the Inverse Z Transformation and the corresponding partial fraction it was
880 % neccessary to build helping functions like \texttt{get\_denominator},
881 % \texttt{get\_numerator} or \texttt{argument\_in}. First two functions help us
882 % to filter the denominator or numerator out of a fraction, last one helps us to
883 % get to know the bound variable in a equation.
885 % By taking \texttt{get\_denominator} as an example, we want to explain how to
886 % implement new functions into the existing system and how we can later use them
889 % \subsubsection{Find a place to Store the Function}
891 % The whole system builds up on a well defined structure of Knowledge. This
892 % Knowledge sets up at the Path:
893 % \begin{center}\ttfamily src/Tools/isac/Knowledge\normalfont\end{center}
894 % For implementing the Function \texttt{get\_denominator} (which let us extract
895 % the denominator out of a fraction) we have choosen the Theory (file)
896 % \texttt{Rational.thy}.
898 % \subsubsection{Write down the new Function}
900 % In upper Theory we now define the new function and its purpose:
902 % get_denominator :: "real => real"
904 % This command tells the machine that a function with the name
905 % \texttt{get\_denominator} exists which gets a real expression as argument and
906 % returns once again a real expression. Now we are able to implement the function
907 % itself, upcoming example now shows the implementation of
908 % \texttt{get\_denominator}.
911 % \label{eg:getdenom}
915 % 02 *("get_denominator",
916 % 03 * ("Rational.get_denominator", eval_get_denominator ""))
918 % 05 fun eval_get_denominator (thmid:string) _
919 % 06 (t as Const ("Rational.get_denominator", _) $
920 % 07 (Const ("Rings.inverse_class.divide", _) $num
922 % 09 SOME (mk_thmid thmid ""
923 % 10 (Print_Mode.setmp []
924 % 11 (Syntax.string_of_term (thy2ctxt thy)) denom) "",
925 % 12 Trueprop $ (mk_equality (t, denom)))
926 % 13 | eval_get_denominator _ _ _ _ = NONE;\end{verbatim}
929 % Line \texttt{07} and \texttt{08} are describing the mode of operation the best -
930 % there is a fraction\\ (\ttfamily Rings.inverse\_class.divide\normalfont)
932 % into its two parts (\texttt{\$num \$denom}). The lines before are additionals
933 % commands for declaring the function and the lines after are modeling and
934 % returning a real variable out of \texttt{\$denom}.
936 % \subsubsection{Add a test for the new Function}
938 % \paragraph{Everytime when adding} a new function it is essential also to add
939 % a test for it. Tests for all functions are sorted in the same structure as the
940 % knowledge it self and can be found up from the path:
941 % \begin{center}\ttfamily test/Tools/isac/Knowledge\normalfont\end{center}
942 % This tests are nothing very special, as a first prototype the functionallity
943 % of a function can be checked by evaluating the result of a simple expression
944 % passed to the function. Example~\ref{eg:getdenomtest} shows the test for our
945 % \textit{just} created function \texttt{get\_denominator}.
948 % \label{eg:getdenomtest}
951 % 01 val thy = @{theory Isac};
952 % 02 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
953 % 03 val SOME (_, t') = eval_get_denominator "" 0 t thy;
954 % 04 if term2str t' = "get_denominator ((a + x) / b) = b" then ()
955 % 05 else error "get_denominator ((a + x) / b) = b" \end{verbatim}
958 % \begin{description}
959 % \item[01] checks if the proofer set up on our {\sisac{}} System.
960 % \item[02] passes a simple expression (fraction) to our suddenly created
962 % \item[04] checks if the resulting variable is the correct one (in this case
963 % ``b'' the denominator) and returns.
964 % \item[05] handels the error case and reports that the function is not able to
965 % solve the given problem.
968 \subsection{Specification of the Problem}\label{spec}
969 %WN <--> \chapter 7 der Thesis
970 %WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..
972 Mechanical treatment requires to translate a textual problem
973 description like in Fig.\ref{fig-interactive} on
974 p.\pageref{fig-interactive} into a {\em formal} specification. The
975 formal specification of the running example could look like is this:
977 %WN Hier brauchen wir die Spezifikation des 'running example' ...
978 %JR Habe input, output und precond vom Beispiel eingefügt brauche aber Hilfe bei
979 %JR der post condition - die existiert für uns ja eigentlich nicht aka
980 %JR haben sie bis jetzt nicht beachtet WN...
981 %WN2 Mein Vorschlag ist, das TODO zu lassen und deutlich zu kommentieren.
985 {\small\begin{tabbing}
986 123\=123\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill
989 \> \>input \>: ${\it filterExpression} \;\;X\;z=\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}, \;{\it domain}\;\mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$\\
990 \>\>precond \>: $\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}\;\; {\it continuous\_on}\; \mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$ \\
991 \>\>output \>: stepResponse $x[n]$ \\
992 \>\>postcond \>: TODO
995 %JR wie besprochen, kein remark, keine begründung, nur simples "nicht behandelt"
998 % Defining the postcondition requires a high amount mathematical
999 % knowledge, the difficult part in our case is not to set up this condition
1000 % nor it is more to define it in a way the interpreter is able to handle it.
1001 % Due the fact that implementing that mechanisms is quite the same amount as
1002 % creating the programm itself, it is not avaible in our prototype.
1003 % \label{rm:postcond}
1006 The implementation of the formal specification in the present
1007 prototype, still bar-bones without support for authoring, is done
1009 %WN Kopie von Inverse_Z_Transform.thy, leicht versch"onert:
1011 {\footnotesize\label{exp-spec}
1014 01 store_specification
1015 02 (prepare_specification
1016 03 "pbl_SP_Ztrans_inv"
1019 06 ( ["Inverse", "Z_Transform", "SignalProcessing"],
1020 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1021 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1022 09 ("#Find" , ["stepResponse n_eq"]),
1023 10 ("#Post" , [" TODO "])])
1026 13 [["SignalProcessing","Z_Transform","Inverse"]]);
1030 Although the above details are partly very technical, we explain them
1031 in order to document some intricacies of TP-based programming in the
1032 present state of the {\sisac} prototype:
1034 \item[01..02]\textit{store\_specification:} stores the result of the
1035 function \textit{prep\_specification} in a global reference
1036 \textit{Unsynchronized.ref}, which causes principal conflicts with
1037 Isabelle's asynchronous document model~\cite{Wenzel-11:doc-orient} and
1038 parallel execution~\cite{Makarius-09:parall-proof} and is under
1039 reconstruction already.
1041 \textit{prep\_specification:} translates the specification to an internal format
1042 which allows efficient processing; see for instance line {\rm 07}
1044 \item[03..04] are a unique identifier for the specification within {\sisac}
1045 and the ``mathematics author'' holding the copy-rights.
1046 \item[05] is the Isabelle \textit{theory} required to parse the
1047 specification in lines {\rm 07..10}.
1048 \item[06] is a key into the tree of all specifications as presented to
1049 the user (where some branches might be hidden by the dialogue
1051 \item[07..10] are the specification with input, pre-condition, output
1052 and post-condition respectively; note that the specification contains
1053 variables to be instantiated with concrete values for a concrete problem ---
1054 thus the specification actually captures a class of problems. The post-condition is not handled in
1055 the prototype presently.
1056 \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
1057 rewriting determined by rule-sets.
1058 \item[12]\textit{NONE:} could be \textit{SOME ``solve ...''} for a
1059 problem associated to a function from Computer Algebra (like an
1060 equation solver) which is not the case here.
1061 \item[13] is a list of methods solving the specified problem (here
1062 only one list item) represented analogously to {\rm 06}.
1066 %WN die folgenden Erkl"arungen finden sich durch "grep -r 'datatype pbt' *"
1069 % {guh : guh, (*unique within this isac-knowledge*)
1070 % mathauthors: string list, (*copyright*)
1071 % init : pblID, (*to start refinement with*)
1072 % thy : theory, (* which allows to compile that pbt
1073 % TODO: search generalized for subthy (ref.p.69*)
1074 % (*^^^ WN050912 NOT used during application of the problem,
1075 % because applied terms may be from 'subthy' as well as from super;
1076 % thus we take 'maxthy'; see match_ags !*)
1077 % cas : term option,(*'CAS-command'*)
1078 % prls : rls, (* for preds in where_*)
1079 % where_: term list, (* where - predicates*)
1081 % (*this is the model-pattern;
1082 % it contains "#Given","#Where","#Find","#Relate"-patterns
1083 % for constraints on identifiers see "fun cpy_nam"*)
1084 % met : metID list}; (* methods solving the pbt*)
1086 %WN weil dieser Code sehr unaufger"aumt ist, habe ich die Erkl"arungen
1087 %WN oben selbst geschrieben.
1092 %WN das w"urde ich in \sec\label{progr} verschieben und
1093 %WN das SubProblem partial fractions zum Erkl"aren verwenden.
1094 % Such a specification is checked before the execution of a program is
1095 % started, the same applies for sub-programs. In the following example
1096 % (Example~\ref{eg:subprob}) shows the call of such a subproblem:
1100 % \label{eg:subprob}
1102 % {\ttfamily \begin{tabbing}
1103 % ``(L\_L::bool list) = (\=SubProblem (\=Test','' \\
1104 % ``\>\>[linear,univariate,equation,test],'' \\
1105 % ``\>\>[Test,solve\_linear])'' \\
1106 % ``\>[BOOL equ, REAL z])'' \\
1110 % \noindent If a program requires a result which has to be
1111 % calculated first we can use a subproblem to do so. In our specific
1112 % case we wanted to calculate the zeros of a fraction and used a
1113 % subproblem to calculate the zeros of the denominator polynom.
1118 \subsection{Implementation of the Method}\label{meth}
1119 A method collects all data required to interpret a certain program by
1120 Lucas-Interpretation. The \texttt{program} from p.\pageref{s:impl} of
1121 the running example is embedded on the last line in the following method:
1122 %The methods represent the different ways a problem can be solved. This can
1123 %include mathematical tactics as well as tactics taught in different courses.
1124 %Declaring the Method itself gives us the possibilities to describe the way of
1125 %calculation in deep, as well we get the oppertunities to build in different
1133 03 "SP_InverseZTransformation_classic"
1136 06 ( ["SignalProcessing", "Z_Transform", "Inverse"],
1137 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1138 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1139 09 ("#Find" , ["stepResponse n_eq"]),
1147 \noindent The above code stores the whole structure analogously to a
1148 specification as described above:
1150 \item[01..06] are identical to those for the example specification on
1151 p.\pageref{exp-spec}.
1153 \item[07..09] show something looking like the specification; this is a
1154 {\em guard}: as long as not all \textit{Given} items are present and
1155 the \textit{Pre}-conditions is not true, interpretation of the program
1158 \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
1159 \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
1160 \textit{srls, prls, nrls} feature evaluating (a) the ML-functions in the program (e.g.
1161 \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}
1162 and (c) is required for the derivation-machinery checking user-input formulas.
1164 \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}.
1166 The many rule-sets above cause considerable efforts for the
1167 programmers, in particular, because there are no tools for checking
1168 essential features of rule-sets.
1170 % is again very technical and goes hard in detail. Unfortunataly
1171 % most declerations are not essential for a basic programm but leads us to a huge
1172 % range of powerful possibilities.
1174 % \begin{description}
1175 % \item[01..02] stores the method with the given name into the system under a global
1177 % \item[03] specifies the topic within which context the method can be found.
1178 % \item[04..05] as the requirements for different methods can be deviant we
1179 % declare what is \emph{given} and and what to \emph{find} for this specific method.
1180 % The code again helds on the topic of the case studie, where the inverse
1181 % z-transformation does a switch between a term describing a electrical filter into
1182 % its step response. Also the datatype has to be declared (bool - due the fact that
1183 % we handle equations).
1184 % \item[06] \emph{rewrite order} is the order of this rls (ruleset), where one
1185 % theorem of it is used for rewriting one single step.
1186 % \item[07] \texttt{rls} is the currently used ruleset for this method. This set
1187 % has already been defined before.
1188 % \item[08] we would have the possiblitiy to add this method to a predefined tree of
1189 % calculations, i.eg. if it would be a sub of a bigger problem, here we leave it
1191 % \item[09] The \emph{source ruleset}, can be used to evaluate list expressions in
1193 % \item[10] \emph{predicates ruleset} can be used to indicates predicates within
1195 % \item[11] The \emph{check ruleset} summarizes rules for checking formulas
1197 % \item[12] \emph{error patterns} which are expected in this kind of method can be
1198 % pre-specified to recognize them during the method.
1199 % \item[13] finally the \emph{canonical ruleset}, declares the canonical simplifier
1200 % of the specific method.
1201 % \item[14] for this code snipset we don't specify the programm itself and keep it
1202 % empty. Follow up \S\ref{progr} for informations on how to implement this
1203 % \textit{main} part.
1206 \subsection{Implementation of the TP-based Program}\label{progr}
1207 So finally all the prerequisites are described and the final task can
1208 be addressed. The program below comes back to the running example: it
1209 computes a solution for the problem from Fig.\ref{fig-interactive} on
1210 p.\pageref{fig-interactive}. The reader is reminded of
1211 \S\ref{PL-isab}, the introduction of the programming language:
1213 {\footnotesize\it\label{s:impl}
1215 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
1216 \>{\rm 00}\>ML \{*\\
1217 \>{\rm 00}\>val program =\\
1218 \>{\rm 01}\> "{\tt Program} InverseZTransform (X\_eq::bool) = \\
1219 \>{\rm 02}\>\> {\tt let} \\
1220 \>{\rm 03}\>\>\> X\_eq = {\tt Take} X\_eq ; \\
1221 \>{\rm 04}\>\>\> X\_eq = {\tt Rewrite} prep\_for\_part\_frac X\_eq ; \\
1222 \>{\rm 05}\>\>\> (X\_z::real) = lhs X\_eq ; \\ %no inside-out evaluation
1223 \>{\rm 06}\>\>\> (z::real) = argument\_in X\_z; \\
1224 \>{\rm 07}\>\>\> (part\_frac::real) = {\tt SubProblem} \\
1225 \>{\rm 08}\>\>\>\>\>\>\>\> ( Isac, [partial\_fraction, rational, simplification], [] )\\
1226 %\>{\rm 10}\>\>\>\>\>\>\>\>\> [simplification, of\_rationals, to\_partial\_fraction] ) \\
1227 \>{\rm 09}\>\>\>\>\>\>\>\> [ (rhs X\_eq)::real, z::real ]; \\
1228 \>{\rm 10}\>\>\> (X'\_eq::bool) = {\tt Take} ((X'::real =$>$ bool) z = ZZ\_1 part\_frac) ; \\
1229 \>{\rm 11}\>\>\> X'\_eq = (({\tt Rewrite\_Set} prep\_for\_inverse\_z) @@ \\
1230 \>{\rm 12}\>\>\>\>\> $\;\;$ ({\tt Rewrite\_Set} inverse\_z)) X'\_eq \\
1231 \>{\rm 13}\>\> {\tt in } \\
1232 \>{\rm 14}\>\>\> X'\_eq"\\
1235 % ORIGINAL FROM Inverse_Z_Transform.thy
1236 % "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1237 % "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1238 % " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1239 % " (X'_z::real) = lhs X'; "^(* ?X' z*)
1240 % " (zzz::real) = argument_in X'_z; "^(* z *)
1241 % " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1243 % " (pbz::real) = (SubProblem (Isac', "^(**)
1244 % " [partial_fraction,rational,simplification], "^
1245 % " [simplification,of_rationals,to_partial_fraction]) "^
1246 % " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1248 % " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1249 % " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1250 % " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1251 % " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1252 % " 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]*)
1253 % " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1254 % "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1255 The program is represented as a string and part of the method in
1256 \S\ref{meth}. As mentioned in \S\ref{PL} the program is purely
1257 functional and lacks any input statements and output statements. So
1258 the steps of calculation towards a solution (and interactive tutoring
1259 in step-wise problem solving) are created as a side-effect by
1260 Lucas-Interpretation. The side-effects are triggered by the tactics
1261 \texttt{Take}, \texttt{Rewrite}, \texttt{SubProblem} and
1262 \texttt{Rewrite\_Set} in the above lines {\rm 03, 04, 07, 10, 11} and
1263 {\rm 12} respectively. These tactics produce the respective lines in the
1264 calculation on p.\pageref{flow-impl}.
1266 The above lines {\rm 05, 06} do not contain a tactics, so they do not
1267 immediately contribute to the calculation on p.\pageref{flow-impl};
1268 rather, they compute actual arguments for the \texttt{SubProblem} in
1269 line {\rm 09}~\footnote{The tactics also are break-points for the
1270 interpreter, where control is handed over to the user in interactive
1271 tutoring.}. Line {\rm 11} contains tactical \textit{@@}.
1273 \medskip The above program also indicates the dominant role of interactive
1274 selection of knowledge in the three-dimensional universe of
1275 mathematics as depicted in Fig.\ref{fig:mathuni} on
1276 p.\pageref{fig:mathuni}, The \texttt{SubProblem} in the above lines
1277 {\rm 07..09} is more than a function call with the actual arguments
1278 \textit{[ (rhs X\_eq)::real, z::real ]}. The programmer has to determine
1282 \item the theory, in the example \textit{Isac} because different
1283 methods can be selected in Pt.3 below, which are defined in different
1284 theories with \textit{Isac} collecting them.
1285 \item the specification identified by \textit{[partial\_fraction,
1286 rational, simplification]} in the tree of specifications; this
1287 specification is analogous to the specification of the main program
1288 described in \S\ref{spec}; the problem is to find a ``partial fraction
1289 decomposition'' for a univariate rational polynomial.
1290 \item the method in the above example is \textit{[ ]}, i.e. empty,
1291 which supposes the interpreter to select one of the methods predefined
1292 in the specification, for instance in line {\rm 13} in the running
1293 example's specification on p.\pageref{exp-spec}~\footnote{The freedom
1294 (or obligation) for selection carries over to the student in
1295 interactive tutoring.}.
1298 The program code, above presented as a string, is parsed by Isabelle's
1299 parser --- the program is an Isabelle term. This fact is expected to
1300 simplify verification tasks in the future; on the other hand, this
1301 fact causes troubles in error detection which are discussed as part
1302 of the workflow in the subsequent section.
1304 \section{Workflow of Programming in the Prototype}\label{workflow}
1305 The new prover IDE Isabelle/jEdit~\cite{makar-jedit-12} is a great
1306 step forward for interactive theory and proof development. The
1307 {\sisac}-prototype re-uses this IDE as a programming environment. The
1308 experiences from this re-use show, that the essential components are
1309 available from Isabelle/jEdit. However, additional tools and features
1310 are required to achieve acceptable usability.
1312 So notable experiences are reported here, also as a requirement
1313 capture for further development of TP-based languages and respective
1316 \subsection{Preparations and Trials}\label{flow-prep}
1317 The many sub-tasks to be accomplished {\em before} the first line of
1318 program code can be written and tested suggest an approach which
1319 step-wise establishes the prerequisites. The case study underlying
1320 this paper~\cite{jrocnik-bakk} documents the approach in a separate
1322 \textit{Build\_Inverse\_Z\_Transform.thy}~\footnote{http://www.ist.tugraz.at/projects/isac/publ/Build\_Inverse\_Z\_Transform.thy}. Part
1323 II in the study comprises this theory, \LaTeX ed from the theory by
1324 use of Isabelle's document preparation system. This paper resembles
1325 the approach in \S\ref{isabisac} to \S\ref{meth}, which in actual
1326 implementation work involves several iterations.
1328 \bigskip For instance, only the last step, implementing the program
1329 described in \S\ref{meth}, reveals details required. Let us assume,
1330 this is the ML-function \textit{argument\_in} required in line {\rm 06}
1331 of the example program on p.\pageref{s:impl}; how this function needs
1332 to be implemented in the prototype has been discussed in \S\ref{funs}
1335 Now let us assume, that calling this function from the program code
1336 does not work; so testing this function is required in order to find out
1337 the reason: type errors, a missing entry of the function somewhere or
1338 even more nasty technicalities \dots
1343 02 val SOME t = parseNEW ctxt "argument_in (X (z::real))";
1344 03 val SOME (str, t') = eval_argument_in ""
1345 04 "Build_Inverse_Z_Transform.argument'_in" t 0;
1348 07 val it = "(argument_in X z) = z": string\end{verbatim}}
1350 \noindent So, this works: we get an ad-hoc theorem, which used in
1351 rewriting would reduce \texttt{argument\_in X z} to \texttt{z}. Now we check this
1352 reduction and create a rule-set \texttt{rls} for that purpose:
1357 02 val rls = append_rls "test" e_rls
1358 03 [Calc ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")]
1359 04 val SOME (t', asm) = rewrite_set_ @{theory} rls t;
1361 06 val t' = Free ("z", "RealDef.real"): term
1362 07 val asm = []: term list\end{verbatim}}
1364 \noindent The resulting term \texttt{t'} is \texttt{Free ("z",
1365 "RealDef.real")}, i.e the variable \texttt{z}, so all is
1366 perfect. Probably we have forgotten to store this function correctly~?
1367 We review the respective \texttt{calclist} (again an
1368 \textit{Unsynchronized.ref} to be removed in order to adjust to
1369 IsabelleIsar's asynchronous document model):
1373 01 calclist:= overwritel (! calclist,
1375 03 ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")),
1377 05 ]);\end{verbatim}}
1379 \noindent The entry is perfect. So what is the reason~? Ah, probably there
1380 is something messed up with the many rule-sets in the method, see \S\ref{meth} ---
1381 right, the function \texttt{argument\_in} is not contained in the respective
1382 rule-set \textit{srls} \dots this just as an example of the intricacies in
1383 debugging a program in the present state of the prototype.
1385 \subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}
1386 Given all the prerequisites from \S\ref{isabisac} to \S\ref{meth},
1387 usually developed within several iterations, the program can be
1388 assembled; on p.\pageref{s:impl} there is the complete program of the
1391 The completion of this program required efforts for several weeks
1392 (after some months of familiarisation with {\sisac}), caused by the
1393 abundance of intricacies indicated above. Also writing the program is
1394 not pleasant, given Isabelle/Isar/ without add-ons for
1395 programming. Already writing and parsing a few lines of program code
1396 is a challenge: the program is an Isabelle term; Isabelle's parser,
1397 however, is not meant for huge terms like the program of the running
1398 example. So reading out the specific error (usually type errors) from
1399 Isabelle's message is difficult.
1401 \medskip Testing the evaluation of the program has to rely on very
1402 simple tools. Step-wise execution is modelled by a function
1403 \texttt{me}, short for mathematics-engine~\footnote{The interface used
1404 by the fron-end which created the calculation on
1405 p.\pageref{fig-interactive} is different from this function}:
1406 %the following is a simplification of the actual function
1411 02 val it = tac -> ctree * pos -> mout * tac * ctree * pos\end{verbatim}}
1413 \noindent This function takes as arguments a tactic \texttt{tac} which
1414 determines the next step, the step applied to the interpreter-state
1415 \texttt{ctree * pos} as last argument taken. The interpreter-state is
1416 a pair of a tree \texttt{ctree} representing the calculation created
1417 (see the example below) and a position \texttt{pos} in the
1418 calculation. The function delivers a quadruple, beginning with the new
1419 formula \texttt{mout} and the next tactic followed by the new
1422 This function allows to stepwise check the program:
1428 03 ["filterExpression (X z = 3 / ((z::real) + 1/10 - 1/50*(1/z)))",
1429 04 "stepResponse (x[n::real]::bool)"];
1432 07 ["Inverse", "Z_Transform", "SignalProcessing"],
1433 08 ["SignalProcessing","Z_Transform","Inverse"]);
1434 09 val (mout, tac, ctree, pos) = CalcTreeTEST [(fmz, (dI, pI, mI))];
1435 10 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1436 11 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1437 12 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1438 13 ...\end{verbatim}}
1440 \noindent Several dozens of calls for \texttt{me} are required to
1441 create the lines in the calculation below (including the sub-problems
1442 not shown). When an error occurs, the reason might be located
1443 many steps before: if evaluation by rewriting, as done by the prototype,
1444 fails, then first nothing happens --- the effects come later and
1445 cause unpleasant checks.
1447 The checks comprise watching the rewrite-engine for many different
1448 kinds of rule-sets (see \S\ref{meth}), the interpreter-state, in
1449 particular the environment and the context at the states position ---
1450 all checks have to rely on simple functions accessing the
1451 \texttt{ctree}. So getting the calculation below (which resembles the
1452 calculation in Fig.\ref{fig-interactive} on p.\pageref{fig-interactive})
1453 is the result of several weeks of development:
1455 {\small\it\label{exp-calc}
1457 123l\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=\kill
1458 \>{\rm 01}\> $\bullet$ \> {\tt Problem } (Inverse\_Z\_Transform, [Inverse, Z\_Transform, SignalProcessing]) \`\\
1459 \>{\rm 02}\>\> $\vdash\;\;X z = \frac{3}{z - \frac{1}{4} - \frac{1}{8} \cdot z^{-1}}$ \`{\footnotesize {\tt Take} X\_eq}\\
1460 \>{\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}\\
1461 \>{\rm 04}\>\> $\bullet$\> {\tt Problem } [partial\_fraction,rational,simplification] \`{\footnotesize {\tt SubProblem} \dots}\\
1462 \>{\rm 05}\>\>\> $\vdash\;\;\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=$ \`- - -\\
1463 \>{\rm 06}\>\>\> $\frac{24}{-1 + -2 \cdot z + 8 \cdot z^2}$ \`- - -\\
1464 \>{\rm 07}\>\>\> $\bullet$\> solve ($-1 + -2 \cdot z + 8 \cdot z^2,\;z$ ) \`- - -\\
1465 \>{\rm 08}\>\>\>\> $\vdash$ \> $\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=0$ \`- - -\\
1466 \>{\rm 09}\>\>\>\> $z = \frac{2+\sqrt{-4+8}}{16}\;\lor\;z = \frac{2-\sqrt{-4+8}}{16}$ \`- - -\\
1467 \>{\rm 10}\>\>\>\> $z = \frac{1}{2}\;\lor\;z =$ \_\_\_ \`- - -\\
1468 \> \>\>\>\> \_\_\_ \`- - -\\
1469 \>{\rm 11}\>\> \dots\> $\frac{4}{z - \frac{1}{2}} + \frac{-4}{z - \frac{-1}{4}}$ \`\\
1470 \>{\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)}\\
1471 \>{\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 }\\
1472 \>{\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}\\
1473 \>{\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}}
1475 The tactics on the right margin of the above calculation are those in
1476 the program on p.\pageref{s:impl} which create the respective formulas
1478 % ORIGINAL FROM Inverse_Z_Transform.thy
1479 % "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1480 % "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1481 % " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1482 % " (X'_z::real) = lhs X'; "^(* ?X' z*)
1483 % " (zzz::real) = argument_in X'_z; "^(* z *)
1484 % " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1486 % " (pbz::real) = (SubProblem (Isac', "^(**)
1487 % " [partial_fraction,rational,simplification], "^
1488 % " [simplification,of_rationals,to_partial_fraction]) "^
1489 % " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1491 % " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1492 % " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1493 % " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1494 % " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1495 % " 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]*)
1496 % " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1497 % "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1499 \subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}
1500 Finally \textit{Build\_Inverse\_Z\_Transform.thy} has got the job done
1501 and the knowledge accumulated in it can be distributed to appropriate
1502 theories: the program to \textit{Inverse\_Z\_Transform.thy}, the
1503 sub-problem accomplishing the partial fraction decomposition to
1504 \textit{Partial\_Fractions.thy}. Since there are hacks into Isabelle's
1505 internals, this kind of distribution is not trivial. For instance, the
1506 function \texttt{argument\_in} in \S\ref{funs} explicitly contains a
1507 string with the theory it has been defined in, so this string needs to
1508 be updated from \texttt{Build\_Inverse\_Z\_Transform} to
1509 \texttt{Atools} if that function is transferred to theory
1510 \textit{Atools.thy}.
1512 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.
1513 This process is also rather bare-bones without authoring tools and is
1514 described in detail in the {\sisac} wiki~\footnote{http://www.ist.tugraz.at/isac/index.php/Generate\_representations\_for\_ISAC\_Knowledge}.
1517 % -------------------------------------------------------------------
1519 % Material, falls noch Platz bleibt ...
1521 % -------------------------------------------------------------------
1524 % \subsubsection{Trials on Notation and Termination}
1526 % \paragraph{Technical notations} are a big problem for our piece of software,
1527 % but the reason for that isn't a fault of the software itself, one of the
1528 % troubles comes out of the fact that different technical subtopics use different
1529 % symbols and notations for a different purpose. The most famous example for such
1530 % a symbol is the complex number $i$ (in cassique math) or $j$ (in technical
1531 % math). In the specific part of signal processing one of this notation issues is
1532 % the use of brackets --- we use round brackets for analoge signals and squared
1533 % brackets for digital samples. Also if there is no problem for us to handle this
1534 % fact, we have to tell the machine what notation leads to wich meaning and that
1535 % this purpose seperation is only valid for this special topic - signal
1537 % \subparagraph{In the programming language} itself it is not possible to declare
1538 % fractions, exponents, absolutes and other operators or remarks in a way to make
1539 % them pretty to read; our only posssiblilty were ASCII characters and a handfull
1540 % greek symbols like: $\alpha, \beta, \gamma, \phi,\ldots$.
1542 % With the upper collected knowledge it is possible to check if we were able to
1543 % donate all required terms and expressions.
1545 % \subsubsection{Definition and Usage of Rules}
1547 % \paragraph{The core} of our implemented problem is the Z-Transformation, due
1548 % the fact that the transformation itself would require higher math which isn't
1549 % yet avaible in our system we decided to choose the way like it is applied in
1550 % labratory and problem classes at our university - by applying transformation
1551 % rules (collected in transformation tables).
1552 % \paragraph{Rules,} in {\sisac{}}'s programming language can be designed by the
1553 % use of axiomatizations like shown in Example~\ref{eg:ruledef}
1556 % \label{eg:ruledef}
1559 % axiomatization where
1560 % rule1: ``1 = $\delta$[n]'' and
1561 % rule2: ``|| z || > 1 ==> z / (z - 1) = u [n]'' and
1562 % rule3: ``|| z || < 1 ==> z / (z - 1) = -u [-n - 1]''
1566 % This rules can be collected in a ruleset and applied to a given expression as
1567 % follows in Example~\ref{eg:ruleapp}.
1571 % \label{eg:ruleapp}
1573 % \item Store rules in ruleset:
1575 % val inverse_Z = append_rls "inverse_Z" e_rls
1576 % [ Thm ("rule1",num_str @{thm rule1}),
1577 % Thm ("rule2",num_str @{thm rule2}),
1578 % Thm ("rule3",num_str @{thm rule3})
1580 % \item Define exression:
1582 % val sample_term = str2term "z/(z-1)+z/(z-</delta>)+1";\end{verbatim}
1583 % \item Apply ruleset:
1585 % val SOME (sample_term', asm) =
1586 % rewrite_set_ thy true inverse_Z sample_term;\end{verbatim}
1590 % The use of rulesets makes it much easier to develop our designated applications,
1591 % but the programmer has to be careful and patient. When applying rulesets
1592 % two important issues have to be mentionend:
1593 % \subparagraph{How often} the rules have to be applied? In case of
1594 % transformations it is quite clear that we use them once but other fields
1595 % reuqire to apply rules until a special condition is reached (e.g.
1596 % a simplification is finished when there is nothing to be done left).
1597 % \subparagraph{The order} in which rules are applied often takes a big effect
1598 % and has to be evaluated for each purpose once again.
1600 % In our special case of Signal Processing and the rules defined in
1601 % Example~\ref{eg:ruledef} we have to apply rule~1 first of all to transform all
1602 % constants. After this step has been done it no mather which rule fit's next.
1604 % \subsubsection{Helping Functions}
1606 % \paragraph{New Programms require,} often new ways to get through. This new ways
1607 % means that we handle functions that have not been in use yet, they can be
1608 % something special and unique for a programm or something famous but unneeded in
1609 % the system yet. In our dedicated example it was for example neccessary to split
1610 % a fraction into numerator and denominator; the creation of such function and
1611 % even others is described in upper Sections~\ref{simp} and \ref{funs}.
1613 % \subsubsection{Trials on equation solving}
1614 % %simple eq and problem with double fractions/negative exponents
1615 % \paragraph{The Inverse Z-Transformation} makes it neccessary to solve
1616 % equations degree one and two. Solving equations in the first degree is no
1617 % problem, wether for a student nor for our machine; but even second degree
1618 % equations can lead to big troubles. The origin of this troubles leads from
1619 % the build up process of our equation solving functions; they have been
1620 % implemented some time ago and of course they are not as good as we want them to
1621 % be. Wether or not following we only want to show how cruel it is to build up new
1622 % work on not well fundamentials.
1623 % \subparagraph{A simple equation solving,} can be set up as shown in the next
1630 % ["equality (-1 + -2 * z + 8 * z ^^^ 2 = (0::real))",
1634 % val (dI',pI',mI') =
1636 % ["abcFormula","degree_2","polynomial","univariate","equation"],
1637 % ["no_met"]);\end{verbatim}
1640 % Here we want to solve the equation: $-1+-2\cdot z+8\cdot z^{2}=0$. (To give
1641 % a short overview on the commands; at first we set up the equation and tell the
1642 % machine what's the bound variable and where to store the solution. Second step
1643 % is to define the equation type and determine if we want to use a special method
1644 % to solve this type.) Simple checks tell us that the we will get two results for
1645 % this equation and this results will be real.
1646 % So far it is easy for us and for our machine to solve, but
1647 % mentioned that a unvariate equation second order can have three different types
1648 % of solutions it is getting worth.
1649 % \subparagraph{The solving of} all this types of solutions is not yet supported.
1650 % Luckily it was needed for us; but something which has been needed in this
1651 % context, would have been the solving of an euation looking like:
1652 % $-z^{-2}+-2\cdot z^{-1}+8=0$ which is basically the same equation as mentioned
1653 % before (remember that befor it was no problem to handle for the machine) but
1654 % now, after a simple equivalent transformation, we are not able to solve
1656 % \subparagraph{Error messages} we get when we try to solve something like upside
1657 % were very confusing and also leads us to no special hint about a problem.
1658 % \par The fault behind is, that we have no well error handling on one side and
1659 % no sufficient formed equation solving on the other side. This two facts are
1660 % making the implemention of new material very difficult.
1662 % \subsection{Formalization of missing knowledge in Isabelle}
1664 % \paragraph{A problem} behind is the mechanization of mathematic
1665 % theories in TP-bases languages. There is still a huge gap between
1666 % these algorithms and this what we want as a solution - in Example
1667 % Signal Processing.
1673 % X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY
1676 % \noindent A very simple example on this what we call gap is the
1677 % simplification above. It is needles to say that it is correct and also
1678 % Isabelle for fills it correct - \emph{always}. But sometimes we don't
1679 % want expand such terms, sometimes we want another structure of
1680 % them. Think of a problem were we now would need only the coefficients
1681 % of $X$ and $Y$. This is what we call the gap between mechanical
1682 % simplification and the solution.
1687 % \paragraph{We are not able to fill this gap,} until we have to live
1688 % with it but first have a look on the meaning of this statement:
1689 % Mechanized math starts from mathematical models and \emph{hopefully}
1690 % proceeds to match physics. Academic engineering starts from physics
1691 % (experimentation, measurement) and then proceeds to mathematical
1692 % modeling and formalization. The process from a physical observance to
1693 % a mathematical theory is unavoidable bound of setting up a big
1694 % collection of standards, rules, definition but also exceptions. These
1695 % are the things making mechanization that difficult.
1704 % \noindent Think about some units like that one's above. Behind
1705 % each unit there is a discerning and very accurate definition: One
1706 % Meter is the distance the light travels, in a vacuum, through the time
1707 % of 1 / 299.792.458 second; one kilogram is the weight of a
1708 % platinum-iridium cylinder in paris; and so on. But are these
1709 % definitions usable in a computer mechanized world?!
1714 % \paragraph{A computer} or a TP-System builds on programs with
1715 % predefined logical rules and does not know any mathematical trick
1716 % (follow up example \ref{eg:trick}) or recipe to walk around difficult
1722 % \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]
1723 % \[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)=
1724 % \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]
1725 % \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]
1727 % \noindent Sometimes it is also useful to be able to apply some
1728 % \emph{tricks} to get a beautiful and particularly meaningful result,
1729 % which we are able to interpret. But as seen in this example it can be
1730 % hard to find out what operations have to be done to transform a result
1731 % into a meaningful one.
1736 % \paragraph{The only possibility,} for such a system, is to work
1737 % through its known definitions and stops if none of these
1738 % fits. Specified on Signal Processing or any other application it is
1739 % often possible to walk through by doing simple creases. This creases
1740 % are in general based on simple math operational but the challenge is
1741 % to teach the machine \emph{all}\footnote{Its pride to call it
1742 % \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to
1743 % reach a high level of \emph{all} but it in real it will still be a
1744 % survey of knowledge which links to other knowledge and {{\sisac}{}} a
1745 % trainer and helper but no human compensating calculator.
1747 % {{{\sisac}{}}} itself aims to adds \emph{Algorithmic Knowledge} (formal
1748 % specifications of problems out of topics from Signal Processing, etc.)
1749 % and \emph{Application-oriented Knowledge} to the \emph{deductive} axis of
1750 % physical knowledge. The result is a three-dimensional universe of
1751 % mathematics seen in Figure~\ref{fig:mathuni}.
1755 % \includegraphics{fig/universe}
1756 % \caption{Didactic ``Math-Universe'': Algorithmic Knowledge (Programs) is
1757 % combined with Application-oriented Knowledge (Specifications) and Deductive Knowledge (Axioms, Definitions, Theorems). The Result
1758 % leads to a three dimensional math universe.\label{fig:mathuni}}
1762 % %WN Deine aktuelle Benennung oben wird Dir kein Fachmann abnehmen;
1763 % %WN bitte folgende Bezeichnungen nehmen:
1765 % %WN axis 1: Algorithmic Knowledge (Programs)
1766 % %WN axis 2: Application-oriented Knowledge (Specifications)
1767 % %WN axis 3: Deductive Knowledge (Axioms, Definitions, Theorems)
1769 % %WN und bitte die R"ander von der Grafik wegschneiden (was ich f"ur *.pdf
1770 % %WN nicht hinkriege --- weshalb ich auch die eJMT-Forderung nicht ganz
1771 % %WN verstehe, separierte PDFs zu schicken; ich w"urde *.png schicken)
1773 % %JR Ränder und beschriftung geändert. Keine Ahnung warum eJMT sich pdf's
1774 % %JR wünschen, würde ebenfalls png oder ähnliches verwenden, aber wenn pdf's
1775 % %JR gefordert werden WN2...
1776 % %WN2 meiner Meinung nach hat sich eJMT unklar ausgedr"uckt (z.B. kann
1777 % %WN2 man meines Wissens pdf-figures nicht auf eine bestimmte Gr"osse
1778 % %WN2 zusammenschneiden um die R"ander weg zu bekommen)
1779 % %WN2 Mein Vorschlag ist, in umserem tex-file bei *.png zu bleiben und
1780 % %WN2 png + pdf figures mitzuschicken.
1782 % \subsection{Notes on Problems with Traditional Notation}
1784 % \paragraph{During research} on these topic severely problems on
1785 % traditional notations have been discovered. Some of them have been
1786 % known in computer science for many years now and are still unsolved,
1787 % one of them aggregates with the so called \emph{Lambda Calculus},
1788 % Example~\ref{eg:lamda} provides a look on the problem that embarrassed
1795 % \[ f(x)=\ldots\; \quad R \rightarrow \quad R \]
1798 % \[ f(p)=\ldots\; p \in \quad R \]
1801 % \noindent Above we see two equations. The first equation aims to
1802 % be a mapping of an function from the reel range to the reel one, but
1803 % when we change only one letter we get the second equation which
1804 % usually aims to insert a reel point $p$ into the reel function. In
1805 % computer science now we have the problem to tell the machine (TP) the
1806 % difference between this two notations. This Problem is called
1807 % \emph{Lambda Calculus}.
1812 % \paragraph{An other problem} is that terms are not full simplified in
1813 % traditional notations, in {{\sisac}} we have to simplify them complete
1814 % to check weather results are compatible or not. in e.g. the solutions
1815 % of an second order linear equation is an rational in {{\sisac}} but in
1816 % tradition we keep fractions as long as possible and as long as they
1817 % aim to be \textit{beautiful} (1/8, 5/16,...).
1818 % \subparagraph{The math} which should be mechanized in Computer Theorem
1819 % Provers (\emph{TP}) has (almost) a problem with traditional notations
1820 % (predicate calculus) for axioms, definitions, lemmas, theorems as a
1821 % computer program or script is not able to interpret every Greek or
1822 % Latin letter and every Greek, Latin or whatever calculations
1823 % symbol. Also if we would be able to handle these symbols we still have
1824 % a problem to interpret them at all. (Follow up \hbox{Example
1825 % \ref{eg:symbint1}})
1829 % \label{eg:symbint1}
1831 % u\left[n\right] \ \ldots \ unitstep
1834 % \noindent The unitstep is something we need to solve Signal
1835 % Processing problem classes. But in {{{\sisac}{}}} the rectangular
1836 % brackets have a different meaning. So we abuse them for our
1837 % requirements. We get something which is not defined, but usable. The
1838 % Result is syntax only without semantic.
1843 % In different problems, symbols and letters have different meanings and
1844 % ask for different ways to get through. (Follow up \hbox{Example
1845 % \ref{eg:symbint2}})
1849 % \label{eg:symbint2}
1851 % \widehat{\ }\ \widehat{\ }\ \widehat{\ } \ \ldots \ exponent
1854 % \noindent For using exponents the three \texttt{widehat} symbols
1855 % are required. The reason for that is due the development of
1856 % {{{\sisac}{}}} the single \texttt{widehat} and also the double were
1857 % already in use for different operations.
1862 % \paragraph{Also the output} can be a problem. We are familiar with a
1863 % specified notations and style taught in university but a computer
1864 % program has no knowledge of the form proved by a professor and the
1865 % machines themselves also have not yet the possibilities to print every
1866 % symbol (correct) Recent developments provide proofs in a human
1867 % readable format but according to the fact that there is no money for
1868 % good working formal editors yet, the style is one thing we have to
1871 % \section{Problems rising out of the Development Environment}
1873 % fehlermeldungen! TODO
1875 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{verbatim}
1877 \section{Conclusion}\label{conclusion}
1881 %This paper gives a first experience report about programming with a
1882 %TP-based programming language.
1884 A brief re-introduction of the novel kind of programming
1885 language by example of the {\sisac}-prototype makes the paper
1886 self-contained. The main section describes all the main concepts
1887 involved in TP-based programming and all the sub-tasks concerning
1888 respective implementation: mechanisation of mathematics and domain
1889 modelling, implementation of term rewriting systems for the
1890 rewriting-engine, formal (implicit) specification of the problem to be
1891 (explicitly) described by the program, implementation of the many components
1892 required for Lucas-Interpretation and finally implementation of the
1895 The many concepts and sub-tasks involved in programming require a
1896 comprehensive workflow; first experiences with the workflow as
1897 supported by the present prototype are described as well: Isabelle +
1898 Isar + jEdit provide appropriate components for establishing an
1899 efficient development environment integrating computation and
1900 deduction. However, the present state of the prototype is far off a
1901 state appropriate for wide-spread use: the prototype of the program
1902 language lacks expressiveness and elegance, the prototype of the
1903 development environment is hardly usable: error messages still address
1904 the developer of the prototype's interpreter rather than the
1905 application programmer, implementation of the many settings for the
1906 Lucas-Interpreter is cumbersome.
1908 From these experiences a successful proof of concept can be concluded:
1909 programming arbitrary problems from engineering sciences is possible,
1910 in principle even in the prototype. Furthermore the experiences allow
1911 to conclude detailed requirements for further development:
1913 \item Clarify underlying logics such that programming is smoothly
1914 integrated with verification of the program; the post-condition should
1915 be proved more or less automatically, otherwise working engineers
1916 would not encounter such programming.
1917 \item Combine the prototype's programming language with Isabelle's
1918 powerful function package and probably with more of SML's
1919 pattern-matching features; include parallel execution on multi-core
1920 machines into the language design.
1921 \item Extend the prototype's Lucas-Interpreter such that it also
1922 handles functions defined by use of Isabelle's functions package; and
1923 generalize Isabelle's code generator such that efficient code for the
1924 whole definition of the programming language can be generated (for
1925 multi-core machines).
1926 \item Develop an efficient development environment with
1927 integration of programming and proving, with management not only of
1928 Isabelle theories, but also of large collections of specifications and
1931 Provided successful accomplishment, these points provide distinguished
1932 components for virtual workbenches appealing to practitioner of
1933 engineering in the near future.
1935 \medskip Interactive course material, as addressed by the title, then
1936 can comprise step-wise problem solving created as a side-effect of a
1937 TP-based program: Lucas-Interpretation not only provides an
1938 interactive programming environment, Lucas-Interpretation also can
1939 provide TP-based services for a flexible dialogue component with
1940 adaptive user guidance for independent and inquiry-based learning.
1943 \bibliographystyle{alpha}
1944 {\small\bibliography{references}}