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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 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. %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 \>$( \; {\tt if} \; b \; {\tt then} \; t_1 \; {\tt else} \; t_2 \;)$\\
332 \>$( \; {\tt let} \; x=t \; {\tt in} \; u \; )$\\
333 \>$( \; {\tt case} \; t \; {\tt of} \; {\it pat}_1
334 \Rightarrow t_1 \; |\dots| \; {\it pat}_n\Rightarrow t_n \; )$
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 ${\cal z}^{-1}$ 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 \>axiomatization where \\
643 \>\> rule1: ``${\cal z}^{-1}\;1 = \delta [n]$'' and\\
644 \>\> rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow {\cal z}^{-1}\;z / (z - 1) = u [n]$'' and\\
645 \>\> rule3: ``$\vert\vert$ z $\vert\vert$ < 1 ==> z / (z - 1) = -u [-n - 1]'' and \\
646 \>\> rule4: ``$\vert\vert$ z $\vert\vert$ > $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = $\alpha^n$ $\cdot$ u [n]'' and\\
647 \>\> rule5: ``$\vert\vert$ z $\vert\vert$ < $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = -($\alpha^n$) $\cdot$ u [-n - 1]'' and\\
648 \>\> rule6: ``$\vert\vert$ z $\vert\vert$ > 1 ==> 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 proved 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 %OUTCOMMENTED DUE TO SPACE RESTRICTIONS
756 % \noindent It is advisable to immediately test rule-sets; for that
757 % purpose an appropriate term has to be created; \textit{parse} takes a
758 % context \textit{ctxt} and a string (with \textit{ZZ\_1} denoting ${\cal
759 % Z}^{-1}$) and creates a term:
764 % 02 val t = parse ctxt "ZZ_1 (z / (z - 1) + z / (z - </alpha>) + 1)";
766 % 04 val t = Const ("Build_Inverse_Z_Transform.ZZ_1",
767 % 05 "RealDef.real => RealDef.real => RealDef.real") $
768 % 06 (Const (...) $ (Const (...) $ Free (...) $ (Const (...) $ Free (...)
771 % \noindent The internal representation of the term, as required for
772 % rewriting, consists of \textit{Const}ants, a pair of a string
773 % \textit{"Groups.plus\_class.plus"} for $+$ and a type, variables
774 % \textit{Free} and the respective constructor \textit{\$}. Now the
775 % term can be rewritten by the rule-set \textit{inverse\_z}:
780 % 02 val SOME (t', asm) = rewrite_set_ @{theory} inverse\_z t;
784 % 06 val it = "u[n] + </alpha> ^ n * u[n] + </delta>[n]" : string
785 % 07 val it = "|| z || > 1 & || z || > </alpha>" : string
788 % \noindent The resulting term \textit{t} and the assumptions
789 % \textit{asm} are converted to readable strings by \textit{term2str}
790 % and \textit{terms2str}.
792 \subsection{Preparation of ML-Functions}\label{funs}
793 Some functionality required in programming, cannot be accomplished by
794 rewriting. So the prototype has a mechanism to call functions within
795 the rewrite-engine: certain redexes in Isabelle terms call these
796 functions written in SML~\cite{pl:milner97}, the implementation {\em
797 and} meta-language of Isabelle. The programmer has to use this
800 In the running example's program on p.\pageref{s:impl} the lines {\rm
801 05} and {\rm 06} contain such functions; we go into the details with
802 \textit{argument\_in X\_z;}. This function fetches the argument from a
803 function application: Line {\rm 03} in the example calculation on
804 p.\pageref{exp-calc} is created by line {\rm 06} of the example
805 program on p.\pageref{s:impl} where the program's environment assigns
806 the value \textit{X z} to the variable \textit{X\_z}; so the function
807 shall extract the argument \textit{z}.
809 \medskip In order to be recognised as a function constant in the
810 program source the constant needs to be declared in a theory, here in
811 \textit{Build\_Inverse\_Z\_Transform.thy}; then it can be parsed in
812 the context \textit{ctxt} of that theory:
817 argument'_in :: "real => real" ("argument'_in _" 10)
820 %^3.2^ ML {* val SOME t = parse ctxt "argument_in (X z)"; *}
821 %^3.2^ val t = Const ("Build_Inverse_Z_Transform.argument'_in", "RealDef.real ⇒ RealDef.real")
822 %^3.2^ $ (Free ("X", "RealDef.real ⇒ RealDef.real") $ Free ("z", "RealDef.real")): term
823 %^3.2^ \end{verbatim}}
825 %^3.2^ \noindent Parsing produces a term \texttt{t} in internal
826 %^3.2^ representation~\footnote{The attentive reader realizes the
827 %^3.2^ differences between interal and extermal representation even in the
828 %^3.2^ strings, i.e \texttt{'\_}}, consisting of \texttt{Const
829 %^3.2^ ("argument'\_in", type)} and the two variables \texttt{Free ("X",
830 %^3.2^ type)} and \texttt{Free ("z", type)}, \texttt{\$} is the term
832 The function body below is implemented directly in SML,
833 i.e in an \texttt{ML \{* *\}} block; the function definition provides
834 a unique prefix \texttt{eval\_} to the function name:
839 fun eval_argument_in _
840 "Build_Inverse_Z_Transform.argument'_in"
841 (t as (Const ("Build_Inverse_Z_Transform.argument'_in", _) $(f $arg))) _ =
842 if is_Free arg (*could be something to be simplified before*)
843 then SOME (term2str t ^"="^ term2str arg, Trueprop $(mk_equality (t, arg)))
845 | eval_argument_in _ _ _ _ = NONE;
849 \noindent The function body creates either creates \texttt{NONE}
850 telling the rewrite-engine to search for the next redex, or creates an
851 ad-hoc theorem for rewriting, thus the programmer needs to adopt many
852 technicalities of Isabelle, for instance, the \textit{Trueprop}
855 \bigskip This sub-task particularly sheds light on basic issues in the
856 design of a programming language, the integration of differential language
857 layers, the layer of Isabelle/Isar and Isabelle/ML.
859 Another point of improvement for the prototype is the rewrite-engine: The
860 program on p.\pageref{s:impl} would not allow to contract the two lines {\rm 05}
863 {\small\it\label{s:impl}
865 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
866 \>{\rm 05/06}\>\>\> (z::real) = argument\_in (lhs X\_eq) ;
869 \noindent because nested function calls would require creating redexes
870 inside-out; however, the prototype's rewrite-engine only works top down
871 from the root of a term down to the leaves.
873 How all these technicalities are to be checked in the prototype is
874 shown in \S\ref{flow-prep} below.
876 % \paragraph{Explicit Problems} require explicit methods to solve them, and within
877 % this methods we have some explicit steps to do. This steps can be unique for
878 % a special problem or refindable in other problems. No mather what case, such
879 % steps often require some technical functions behind. For the solving process
880 % of the Inverse Z Transformation and the corresponding partial fraction it was
881 % neccessary to build helping functions like \texttt{get\_denominator},
882 % \texttt{get\_numerator} or \texttt{argument\_in}. First two functions help us
883 % to filter the denominator or numerator out of a fraction, last one helps us to
884 % get to know the bound variable in a equation.
886 % By taking \texttt{get\_denominator} as an example, we want to explain how to
887 % implement new functions into the existing system and how we can later use them
890 % \subsubsection{Find a place to Store the Function}
892 % The whole system builds up on a well defined structure of Knowledge. This
893 % Knowledge sets up at the Path:
894 % \begin{center}\ttfamily src/Tools/isac/Knowledge\normalfont\end{center}
895 % For implementing the Function \texttt{get\_denominator} (which let us extract
896 % the denominator out of a fraction) we have choosen the Theory (file)
897 % \texttt{Rational.thy}.
899 % \subsubsection{Write down the new Function}
901 % In upper Theory we now define the new function and its purpose:
903 % get_denominator :: "real => real"
905 % This command tells the machine that a function with the name
906 % \texttt{get\_denominator} exists which gets a real expression as argument and
907 % returns once again a real expression. Now we are able to implement the function
908 % itself, upcoming example now shows the implementation of
909 % \texttt{get\_denominator}.
912 % \label{eg:getdenom}
916 % 02 *("get_denominator",
917 % 03 * ("Rational.get_denominator", eval_get_denominator ""))
919 % 05 fun eval_get_denominator (thmid:string) _
920 % 06 (t as Const ("Rational.get_denominator", _) $
921 % 07 (Const ("Rings.inverse_class.divide", _) $num
923 % 09 SOME (mk_thmid thmid ""
924 % 10 (Print_Mode.setmp []
925 % 11 (Syntax.string_of_term (thy2ctxt thy)) denom) "",
926 % 12 Trueprop $ (mk_equality (t, denom)))
927 % 13 | eval_get_denominator _ _ _ _ = NONE;\end{verbatim}
930 % Line \texttt{07} and \texttt{08} are describing the mode of operation the best -
931 % there is a fraction\\ (\ttfamily Rings.inverse\_class.divide\normalfont)
933 % into its two parts (\texttt{\$num \$denom}). The lines before are additionals
934 % commands for declaring the function and the lines after are modeling and
935 % returning a real variable out of \texttt{\$denom}.
937 % \subsubsection{Add a test for the new Function}
939 % \paragraph{Everytime when adding} a new function it is essential also to add
940 % a test for it. Tests for all functions are sorted in the same structure as the
941 % knowledge it self and can be found up from the path:
942 % \begin{center}\ttfamily test/Tools/isac/Knowledge\normalfont\end{center}
943 % This tests are nothing very special, as a first prototype the functionallity
944 % of a function can be checked by evaluating the result of a simple expression
945 % passed to the function. Example~\ref{eg:getdenomtest} shows the test for our
946 % \textit{just} created function \texttt{get\_denominator}.
949 % \label{eg:getdenomtest}
952 % 01 val thy = @{theory Isac};
953 % 02 val t = term_of (the (parse thy "get_denominator ((a +x)/b)"));
954 % 03 val SOME (_, t') = eval_get_denominator "" 0 t thy;
955 % 04 if term2str t' = "get_denominator ((a + x) / b) = b" then ()
956 % 05 else error "get_denominator ((a + x) / b) = b" \end{verbatim}
959 % \begin{description}
960 % \item[01] checks if the proofer set up on our {\sisac{}} System.
961 % \item[02] passes a simple expression (fraction) to our suddenly created
963 % \item[04] checks if the resulting variable is the correct one (in this case
964 % ``b'' the denominator) and returns.
965 % \item[05] handels the error case and reports that the function is not able to
966 % solve the given problem.
969 \subsection{Specification of the Problem}\label{spec}
970 %WN <--> \chapter 7 der Thesis
971 %WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..
973 Mechanical treatment requires to translate a textual problem
974 description like in Fig.\ref{fig-interactive} on
975 p.\pageref{fig-interactive} into a {\em formal} specification. The
976 formal specification of the running example could look like is this:
978 %WN Hier brauchen wir die Spezifikation des 'running example' ...
979 %JR Habe input, output und precond vom Beispiel eingefügt brauche aber Hilfe bei
980 %JR der post condition - die existiert für uns ja eigentlich nicht aka
981 %JR haben sie bis jetzt nicht beachtet WN...
982 %WN2 Mein Vorschlag ist, das TODO zu lassen und deutlich zu kommentieren.
986 {\small\begin{tabbing}
987 123\=123\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill
990 \> \>input \>: ${\it filterExpression} \;\;X\;z=\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}, \;{\it domain}\;\mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$\\
991 \>\>precond \>: $\frac{3}{z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}\;\; {\it continuous\_on}\; \mathbb{R}-\{\frac{1}{2}, \frac{-1}{4}\}$ \\
992 \>\>output \>: stepResponse $x[n]$ \\
993 \>\>postcond \>: TODO
996 %JR wie besprochen, kein remark, keine begründung, nur simples "nicht behandelt"
999 % Defining the postcondition requires a high amount mathematical
1000 % knowledge, the difficult part in our case is not to set up this condition
1001 % nor it is more to define it in a way the interpreter is able to handle it.
1002 % Due the fact that implementing that mechanisms is quite the same amount as
1003 % creating the programm itself, it is not avaible in our prototype.
1004 % \label{rm:postcond}
1007 The implementation of the formal specification in the present
1008 prototype, still bar-bones without support for authoring, is done
1010 %WN Kopie von Inverse_Z_Transform.thy, leicht versch"onert:
1012 {\footnotesize\label{exp-spec}
1015 01 store_specification
1016 02 (prepare_specification
1017 03 "pbl_SP_Ztrans_inv"
1020 06 ( ["Inverse", "Z_Transform", "SignalProcessing"],
1021 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1022 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1023 09 ("#Find" , ["stepResponse n_eq"]),
1024 10 ("#Post" , [" TODO "])])
1027 13 [["SignalProcessing","Z_Transform","Inverse"]]);
1031 Although the above details are partly very technical, we explain them
1032 in order to document some intricacies of TP-based programming in the
1033 present state of the {\sisac} prototype:
1035 \item[01..02]\textit{store\_specification:} stores the result of the
1036 function \textit{prep\_specification} in a global reference
1037 \textit{Unsynchronized.ref}, which causes principal conflicts with
1038 Isabelle's asynchronous document model~\cite{Wenzel-11:doc-orient} and
1039 parallel execution~\cite{Makarius-09:parall-proof} and is under
1040 reconstruction already.
1042 \textit{prep\_specification:} translates the specification to an internal format
1043 which allows efficient processing; see for instance line {\rm 07}
1045 \item[03..04] are a unique identifier for the specification within {\sisac}
1046 and the ``mathematics author'' holding the copy-rights.
1047 \item[05] is the Isabelle \textit{theory} required to parse the
1048 specification in lines {\rm 07..10}.
1049 \item[06] is a key into the tree of all specifications as presented to
1050 the user (where some branches might be hidden by the dialogue
1052 \item[07..10] are the specification with input, pre-condition, output
1053 and post-condition respectively; note that the specification contains
1054 variables to be instantiated with concrete values for a concrete problem ---
1055 thus the specification actually captures a class of problems. The post-condition is not handled in
1056 the prototype presently.
1057 \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
1058 rewriting determined by rule-sets.
1059 \item[12]\textit{NONE:} could be \textit{SOME ``solve ...''} for a
1060 problem associated to a function from Computer Algebra (like an
1061 equation solver) which is not the case here.
1062 \item[13] is a list of methods solving the specified problem (here
1063 only one list item) represented analogously to {\rm 06}.
1067 %WN die folgenden Erkl"arungen finden sich durch "grep -r 'datatype pbt' *"
1070 % {guh : guh, (*unique within this isac-knowledge*)
1071 % mathauthors: string list, (*copyright*)
1072 % init : pblID, (*to start refinement with*)
1073 % thy : theory, (* which allows to compile that pbt
1074 % TODO: search generalized for subthy (ref.p.69*)
1075 % (*^^^ WN050912 NOT used during application of the problem,
1076 % because applied terms may be from 'subthy' as well as from super;
1077 % thus we take 'maxthy'; see match_ags !*)
1078 % cas : term option,(*'CAS-command'*)
1079 % prls : rls, (* for preds in where_*)
1080 % where_: term list, (* where - predicates*)
1082 % (*this is the model-pattern;
1083 % it contains "#Given","#Where","#Find","#Relate"-patterns
1084 % for constraints on identifiers see "fun cpy_nam"*)
1085 % met : metID list}; (* methods solving the pbt*)
1087 %WN weil dieser Code sehr unaufger"aumt ist, habe ich die Erkl"arungen
1088 %WN oben selbst geschrieben.
1093 %WN das w"urde ich in \sec\label{progr} verschieben und
1094 %WN das SubProblem partial fractions zum Erkl"aren verwenden.
1095 % Such a specification is checked before the execution of a program is
1096 % started, the same applies for sub-programs. In the following example
1097 % (Example~\ref{eg:subprob}) shows the call of such a subproblem:
1101 % \label{eg:subprob}
1103 % {\ttfamily \begin{tabbing}
1104 % ``(L\_L::bool list) = (\=SubProblem (\=Test','' \\
1105 % ``\>\>[linear,univariate,equation,test],'' \\
1106 % ``\>\>[Test,solve\_linear])'' \\
1107 % ``\>[BOOL equ, REAL z])'' \\
1111 % \noindent If a program requires a result which has to be
1112 % calculated first we can use a subproblem to do so. In our specific
1113 % case we wanted to calculate the zeros of a fraction and used a
1114 % subproblem to calculate the zeros of the denominator polynom.
1119 \subsection{Implementation of the Method}\label{meth}
1120 A method collects all data required to interpret a certain program by
1121 Lucas-Interpretation. The \texttt{program} from p.\pageref{s:impl} of
1122 the running example is embedded on the last line in the following method:
1123 %The methods represent the different ways a problem can be solved. This can
1124 %include mathematical tactics as well as tactics taught in different courses.
1125 %Declaring the Method itself gives us the possibilities to describe the way of
1126 %calculation in deep, as well we get the oppertunities to build in different
1134 03 "SP_InverseZTransformation_classic"
1137 06 ( ["SignalProcessing", "Z_Transform", "Inverse"],
1138 07 [ ("#Given", ["filterExpression X_eq", "domain D"]),
1139 08 ("#Pre" , ["(rhs X_eq) is_continuous_in D"]),
1140 09 ("#Find" , ["stepResponse n_eq"]),
1148 \noindent The above code stores the whole structure analogously to a
1149 specification as described above:
1151 \item[01..06] are identical to those for the example specification on
1152 p.\pageref{exp-spec}.
1154 \item[07..09] show something looking like the specification; this is a
1155 {\em guard}: as long as not all \textit{Given} items are present and
1156 the \textit{Pre}-conditions is not true, interpretation of the program
1159 \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
1160 \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
1161 \textit{srls, prls, nrls} feature evaluating (a) the ML-functions in the program (e.g.
1162 \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}
1163 and (c) is required for the derivation-machinery checking user-input formulas.
1165 \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}.
1167 The many rule-sets above cause considerable efforts for the
1168 programmers, in particular, because there are no tools for checking
1169 essential features of rule-sets.
1171 % is again very technical and goes hard in detail. Unfortunataly
1172 % most declerations are not essential for a basic programm but leads us to a huge
1173 % range of powerful possibilities.
1175 % \begin{description}
1176 % \item[01..02] stores the method with the given name into the system under a global
1178 % \item[03] specifies the topic within which context the method can be found.
1179 % \item[04..05] as the requirements for different methods can be deviant we
1180 % declare what is \emph{given} and and what to \emph{find} for this specific method.
1181 % The code again helds on the topic of the case studie, where the inverse
1182 % z-transformation does a switch between a term describing a electrical filter into
1183 % its step response. Also the datatype has to be declared (bool - due the fact that
1184 % we handle equations).
1185 % \item[06] \emph{rewrite order} is the order of this rls (ruleset), where one
1186 % theorem of it is used for rewriting one single step.
1187 % \item[07] \texttt{rls} is the currently used ruleset for this method. This set
1188 % has already been defined before.
1189 % \item[08] we would have the possiblitiy to add this method to a predefined tree of
1190 % calculations, i.eg. if it would be a sub of a bigger problem, here we leave it
1192 % \item[09] The \emph{source ruleset}, can be used to evaluate list expressions in
1194 % \item[10] \emph{predicates ruleset} can be used to indicates predicates within
1196 % \item[11] The \emph{check ruleset} summarizes rules for checking formulas
1198 % \item[12] \emph{error patterns} which are expected in this kind of method can be
1199 % pre-specified to recognize them during the method.
1200 % \item[13] finally the \emph{canonical ruleset}, declares the canonical simplifier
1201 % of the specific method.
1202 % \item[14] for this code snipset we don't specify the programm itself and keep it
1203 % empty. Follow up \S\ref{progr} for informations on how to implement this
1204 % \textit{main} part.
1207 \subsection{Implementation of the TP-based Program}\label{progr}
1208 So finally all the prerequisites are described and the final task can
1209 be addressed. The program below comes back to the running example: it
1210 computes a solution for the problem from Fig.\ref{fig-interactive} on
1211 p.\pageref{fig-interactive}. The reader is reminded of
1212 \S\ref{PL-isab}, the introduction of the programming language:
1214 {\footnotesize\it\label{s:impl}
1216 123l\=123\=123\=123\=123\=123\=123\=((x\=123\=(x \=123\=123\=\kill
1217 \>{\rm 00}\>ML \{*\\
1218 \>{\rm 00}\>val program =\\
1219 \>{\rm 01}\> "{\tt Program} InverseZTransform (X\_eq::bool) = \\
1220 \>{\rm 02}\>\> {\tt let} \\
1221 \>{\rm 03}\>\>\> X\_eq = {\tt Take} X\_eq ; \\
1222 \>{\rm 04}\>\>\> X\_eq = {\tt Rewrite} prep\_for\_part\_frac X\_eq ; \\
1223 \>{\rm 05}\>\>\> (X\_z::real) = lhs X\_eq ; \\ %no inside-out evaluation
1224 \>{\rm 06}\>\>\> (z::real) = argument\_in X\_z; \\
1225 \>{\rm 07}\>\>\> (part\_frac::real) = {\tt SubProblem} \\
1226 \>{\rm 08}\>\>\>\>\>\>\>\> ( Isac, [partial\_fraction, rational, simplification], [] )\\
1227 %\>{\rm 10}\>\>\>\>\>\>\>\>\> [simplification, of\_rationals, to\_partial\_fraction] ) \\
1228 \>{\rm 09}\>\>\>\>\>\>\>\> [ (rhs X\_eq)::real, z::real ]; \\
1229 \>{\rm 10}\>\>\> (X'\_eq::bool) = {\tt Take} ((X'::real =$>$ bool) z = ZZ\_1 part\_frac) ; \\
1230 \>{\rm 11}\>\>\> X'\_eq = (({\tt Rewrite\_Set} prep\_for\_inverse\_z) @@ \\
1231 \>{\rm 12}\>\>\>\>\> $\;\;$ ({\tt Rewrite\_Set} inverse\_z)) X'\_eq \\
1232 \>{\rm 13}\>\> {\tt in } \\
1233 \>{\rm 14}\>\>\> X'\_eq"\\
1236 % ORIGINAL FROM Inverse_Z_Transform.thy
1237 % "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1238 % "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1239 % " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1240 % " (X'_z::real) = lhs X'; "^(* ?X' z*)
1241 % " (zzz::real) = argument_in X'_z; "^(* z *)
1242 % " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1244 % " (pbz::real) = (SubProblem (Isac', "^(**)
1245 % " [partial_fraction,rational,simplification], "^
1246 % " [simplification,of_rationals,to_partial_fraction]) "^
1247 % " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1249 % " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1250 % " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1251 % " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1252 % " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1253 % " 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]*)
1254 % " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1255 % "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1256 The program is represented as a string and part of the method in
1257 \S\ref{meth}. As mentioned in \S\ref{PL} the program is purely
1258 functional and lacks any input statements and output statements. So
1259 the steps of calculation towards a solution (and interactive tutoring
1260 in step-wise problem solving) are created as a side-effect by
1261 Lucas-Interpretation. The side-effects are triggered by the tactics
1262 \texttt{Take}, \texttt{Rewrite}, \texttt{SubProblem} and
1263 \texttt{Rewrite\_Set} in the above lines {\rm 03, 04, 07, 10, 11} and
1264 {\rm 12} respectively. These tactics produce the respective lines in the
1265 calculation on p.\pageref{flow-impl}.
1267 The above lines {\rm 05, 06} do not contain a tactics, so they do not
1268 immediately contribute to the calculation on p.\pageref{flow-impl};
1269 rather, they compute actual arguments for the \texttt{SubProblem} in
1270 line {\rm 09}~\footnote{The tactics also are break-points for the
1271 interpreter, where control is handed over to the user in interactive
1272 tutoring.}. Line {\rm 11} contains tactical \textit{@@}.
1274 \medskip The above program also indicates the dominant role of interactive
1275 selection of knowledge in the three-dimensional universe of
1276 mathematics as depicted in Fig.\ref{fig:mathuni} on
1277 p.\pageref{fig:mathuni}, The \texttt{SubProblem} in the above lines
1278 {\rm 07..09} is more than a function call with the actual arguments
1279 \textit{[ (rhs X\_eq)::real, z::real ]}. The programmer has to determine
1283 \item the theory, in the example \textit{Isac} because different
1284 methods can be selected in Pt.3 below, which are defined in different
1285 theories with \textit{Isac} collecting them.
1286 \item the specification identified by \textit{[partial\_fraction,
1287 rational, simplification]} in the tree of specifications; this
1288 specification is analogous to the specification of the main program
1289 described in \S\ref{spec}; the problem is to find a ``partial fraction
1290 decomposition'' for a univariate rational polynomial.
1291 \item the method in the above example is \textit{[ ]}, i.e. empty,
1292 which supposes the interpreter to select one of the methods predefined
1293 in the specification, for instance in line {\rm 13} in the running
1294 example's specification on p.\pageref{exp-spec}~\footnote{The freedom
1295 (or obligation) for selection carries over to the student in
1296 interactive tutoring.}.
1299 The program code, above presented as a string, is parsed by Isabelle's
1300 parser --- the program is an Isabelle term. This fact is expected to
1301 simplify verification tasks in the future; on the other hand, this
1302 fact causes troubles in error detection which are discussed as part
1303 of the work-flow in the subsequent section.
1305 \section{Work-flow of Programming in the Prototype}\label{workflow}
1306 The new prover IDE Isabelle/jEdit~\cite{makar-jedit-12} is a great
1307 step forward for interactive theory and proof development. The
1308 {\sisac}-prototype re-uses this IDE as a programming environment. The
1309 experiences from this re-use show, that the essential components are
1310 available from Isabelle/jEdit. However, additional tools and features
1311 are required to achieve acceptable usability.
1313 So notable experiences are reported here, also as a requirement
1314 capture for further development of TP-based languages and respective
1317 \subsection{Preparations and Trials}\label{flow-prep}
1318 The many sub-tasks to be accomplished {\em before} the first line of
1319 program code can be written and tested suggest an approach which
1320 step-wise establishes the prerequisites. The case study underlying
1321 this paper~\cite{jrocnik-bakk} documents the approach in a separate
1323 \textit{Build\_Inverse\_Z\_Transform.thy}~\footnote{http://www.ist.tugraz.at/projects/isac/publ/Build\_Inverse\_Z\_Transform.thy}. Part
1324 II in the study comprises this theory, \LaTeX ed from the theory by
1325 use of Isabelle's document preparation system. This paper resembles
1326 the approach in \S\ref{isabisac} to \S\ref{meth}, which in actual
1327 implementation work involves several iterations.
1329 \bigskip For instance, only the last step, implementing the program
1330 described in \S\ref{meth}, reveals details required. Let us assume,
1331 this is the ML-function \textit{argument\_in} required in line {\rm 06}
1332 of the example program on p.\pageref{s:impl}; how this function needs
1333 to be implemented in the prototype has been discussed in \S\ref{funs}
1336 Now let us assume, that calling this function from the program code
1337 does not work; so testing this function is required in order to find out
1338 the reason: type errors, a missing entry of the function somewhere or
1339 even more nasty technicalities \dots
1344 val SOME t = parseNEW ctxt "argument_in (X (z::real))";
1345 val SOME (str, t') = eval_argument_in ""
1346 "Build_Inverse_Z_Transform.argument'_in" t 0;
1349 val it = "(argument_in X z) = z": string
1352 \noindent So, this works: we get an ad-hoc theorem, which used in
1353 rewriting would reduce \texttt{argument\_in X z} to \texttt{z}. Now we check this
1354 reduction and create a rule-set \texttt{rls} for that purpose:
1359 val rls = append_rls "test" e_rls
1360 [Calc ("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")]
1361 val SOME (t', asm) = rewrite_set_ @{theory} rls t;
1363 val t' = Free ("z", "RealDef.real"): term
1364 val asm = []: term list
1367 \noindent The resulting term \texttt{t'} is \texttt{Free ("z",
1368 "RealDef.real")}, i.e the variable \texttt{z}, so all is
1369 perfect. Probably we have forgotten to store this function correctly~?
1370 We review the respective \texttt{calclist} (again an
1371 \textit{Unsynchronized.ref} to be removed in order to adjust to
1372 Isabelle/Isar's asynchronous document model):
1376 calclist:= overwritel (! calclist,
1377 [("argument_in",("Build_Inverse_Z_Transform.argument'_in", eval_argument_in "")),
1382 \noindent The entry is perfect. So what is the reason~? Ah, probably there
1383 is something messed up with the many rule-sets in the method, see \S\ref{meth} ---
1384 right, the function \texttt{argument\_in} is not contained in the respective
1385 rule-set \textit{srls} \dots this just as an example of the intricacies in
1386 debugging a program in the present state of the prototype.
1388 \subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}
1389 Given all the prerequisites from \S\ref{isabisac} to \S\ref{meth},
1390 usually developed within several iterations, the program can be
1391 assembled; on p.\pageref{s:impl} there is the complete program of the
1394 The completion of this program required efforts for several weeks
1395 (after some months of familiarisation with {\sisac}), caused by the
1396 abundance of intricacies indicated above. Also writing the program is
1397 not pleasant, given Isabelle/Isar/ without add-ons for
1398 programming. Already writing and parsing a few lines of program code
1399 is a challenge: the program is an Isabelle term; Isabelle's parser,
1400 however, is not meant for huge terms like the program of the running
1401 example. So reading out the specific error (usually type errors) from
1402 Isabelle's message is difficult.
1404 \medskip Testing the evaluation of the program has to rely on very
1405 simple tools. Step-wise execution is modeled by a function
1406 \texttt{me}, short for mathematics-engine~\footnote{The interface used
1407 by the front-end which created the calculation on
1408 p.\pageref{fig-interactive} is different from this function}:
1409 %the following is a simplification of the actual function
1414 val it = tac -> ctree * pos -> mout * tac * ctree * pos
1417 \noindent This function takes as arguments a tactic \texttt{tac} which
1418 determines the next step, the step applied to the interpreter-state
1419 \texttt{ctree * pos} as last argument taken. The interpreter-state is
1420 a pair of a tree \texttt{ctree} representing the calculation created
1421 (see the example below) and a position \texttt{pos} in the
1422 calculation. The function delivers a quadruple, beginning with the new
1423 formula \texttt{mout} and the next tactic followed by the new
1426 This function allows to stepwise check the program:
1432 ["filterExpression (X z = 3 / ((z::real) + 1/10 - 1/50*(1/z)))",
1433 "stepResponse (x[n::real]::bool)"];
1436 ["Inverse", "Z_Transform", "SignalProcessing"],
1437 ["SignalProcessing","Z_Transform","Inverse"]);
1438 val (mout, tac, ctree, pos) = CalcTreeTEST [(fmz, (dI, pI, mI))];
1439 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1440 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1441 val (mout, tac, ctree, pos) = me tac (ctree, pos);
1444 \noindent Several dozens of calls for \texttt{me} are required to
1445 create the lines in the calculation below (including the sub-problems
1446 not shown). When an error occurs, the reason might be located
1447 many steps before: if evaluation by rewriting, as done by the prototype,
1448 fails, then first nothing happens --- the effects come later and
1449 cause unpleasant checks.
1451 The checks comprise watching the rewrite-engine for many different
1452 kinds of rule-sets (see \S\ref{meth}), the interpreter-state, in
1453 particular the environment and the context at the states position ---
1454 all checks have to rely on simple functions accessing the
1455 \texttt{ctree}. So getting the calculation below (which resembles the
1456 calculation in Fig.\ref{fig-interactive} on p.\pageref{fig-interactive})
1457 is the result of several weeks of development:
1459 {\small\it\label{exp-calc}
1461 123l\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=123\=\kill
1462 \>{\rm 01}\> $\bullet$ \> {\tt Problem } (Inverse\_Z\_Transform, [Inverse, Z\_Transform, SignalProcessing]) \`\\
1463 \>{\rm 02}\>\> $\vdash\;\;X z = \frac{3}{z - \frac{1}{4} - \frac{1}{8} \cdot z^{-1}}$ \`{\footnotesize {\tt Take} X\_eq}\\
1464 \>{\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}\\
1465 \>{\rm 04}\>\> $\bullet$\> {\tt Problem } [partial\_fraction,rational,simplification] \`{\footnotesize {\tt SubProblem} \dots}\\
1466 \>{\rm 05}\>\>\> $\vdash\;\;\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=$ \`- - -\\
1467 \>{\rm 06}\>\>\> $\frac{24}{-1 + -2 \cdot z + 8 \cdot z^2}$ \`- - -\\
1468 \>{\rm 07}\>\>\> $\bullet$\> solve ($-1 + -2 \cdot z + 8 \cdot z^2,\;z$ ) \`- - -\\
1469 \>{\rm 08}\>\>\>\> $\vdash$ \> $\frac{3}{z + \frac{-1}{4} + \frac{-1}{8} \cdot \frac{1}{z}}=0$ \`- - -\\
1470 \>{\rm 09}\>\>\>\> $z = \frac{2+\sqrt{-4+8}}{16}\;\lor\;z = \frac{2-\sqrt{-4+8}}{16}$ \`- - -\\
1471 \>{\rm 10}\>\>\>\> $z = \frac{1}{2}\;\lor\;z =$ \_\_\_ \`- - -\\
1472 \> \>\>\>\> \_\_\_ \`- - -\\
1473 \>{\rm 11}\>\> \dots\> $\frac{4}{z - \frac{1}{2}} + \frac{-4}{z - \frac{-1}{4}}$ \`\\
1474 \>{\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)}\\
1475 \>{\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 }\\
1476 \>{\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}\\
1477 \>{\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}}
1479 The tactics on the right margin of the above calculation are those in
1480 the program on p.\pageref{s:impl} which create the respective formulas
1482 % ORIGINAL FROM Inverse_Z_Transform.thy
1483 % "Script InverseZTransform (X_eq::bool) = "^(*([], Frm), Problem (Isac, [Inverse, Z_Transform, SignalProcessing])*)
1484 % "(let X = Take X_eq; "^(*([1], Frm), X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))*)
1485 % " X' = Rewrite ruleZY False X; "^(*([1], Res), ?X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1486 % " (X'_z::real) = lhs X'; "^(* ?X' z*)
1487 % " (zzz::real) = argument_in X'_z; "^(* z *)
1488 % " (funterm::real) = rhs X'; "^(* 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
1490 % " (pbz::real) = (SubProblem (Isac', "^(**)
1491 % " [partial_fraction,rational,simplification], "^
1492 % " [simplification,of_rationals,to_partial_fraction]) "^
1493 % " [REAL funterm, REAL zzz]); "^(*([2], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1495 % " (pbz_eq::bool) = Take (X'_z = pbz); "^(*([3], Frm), ?X' z = 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
1496 % " pbz_eq = Rewrite ruleYZ False pbz_eq; "^(*([3], Res), ?X' z = 4 * (?z / (z - 1 / 2)) + -4 * (?z / (z - -1 / 4))*)
1497 % " pbz_eq = drop_questionmarks pbz_eq; "^(* 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1498 % " (X_zeq::bool) = Take (X_z = rhs pbz_eq); "^(*([4], Frm), X_z = 4 * (z / (z - 1 / 2)) + -4 * (z / (z - -1 / 4))*)
1499 % " 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]*)
1500 % " n_eq = drop_questionmarks n_eq "^(* X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1501 % "in n_eq)" (*([], Res), X_z = 4 * (1 / 2) ^^^ n * u [n] + -4 * (-1 / 4) ^^^ n * u [n]*)
1503 \subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}
1504 Finally \textit{Build\_Inverse\_Z\_Transform.thy} has got the job done
1505 and the knowledge accumulated in it can be distributed to appropriate
1506 theories: the program to \textit{Inverse\_Z\_Transform.thy}, the
1507 sub-problem accomplishing the partial fraction decomposition to
1508 \textit{Partial\_Fractions.thy}. Since there are hacks into Isabelle's
1509 internals, this kind of distribution is not trivial. For instance, the
1510 function \texttt{argument\_in} in \S\ref{funs} explicitly contains a
1511 string with the theory it has been defined in, so this string needs to
1512 be updated from \texttt{Build\_Inverse\_Z\_Transform} to
1513 \texttt{Atools} if that function is transferred to theory
1514 \textit{Atools.thy}.
1516 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.
1517 This process is also rather bare-bones without authoring tools and is
1518 described in detail in the {\sisac} wiki~\footnote{http://www.ist.tugraz.at/isac/index.php/Generate\_representations\_for\_ISAC\_Knowledge}.
1521 % -------------------------------------------------------------------
1523 % Material, falls noch Platz bleibt ...
1525 % -------------------------------------------------------------------
1528 % \subsubsection{Trials on Notation and Termination}
1530 % \paragraph{Technical notations} are a big problem for our piece of software,
1531 % but the reason for that isn't a fault of the software itself, one of the
1532 % troubles comes out of the fact that different technical subtopics use different
1533 % symbols and notations for a different purpose. The most famous example for such
1534 % a symbol is the complex number $i$ (in cassique math) or $j$ (in technical
1535 % math). In the specific part of signal processing one of this notation issues is
1536 % the use of brackets --- we use round brackets for analoge signals and squared
1537 % brackets for digital samples. Also if there is no problem for us to handle this
1538 % fact, we have to tell the machine what notation leads to wich meaning and that
1539 % this purpose seperation is only valid for this special topic - signal
1541 % \subparagraph{In the programming language} itself it is not possible to declare
1542 % fractions, exponents, absolutes and other operators or remarks in a way to make
1543 % them pretty to read; our only posssiblilty were ASCII characters and a handfull
1544 % greek symbols like: $\alpha, \beta, \gamma, \phi,\ldots$.
1546 % With the upper collected knowledge it is possible to check if we were able to
1547 % donate all required terms and expressions.
1549 % \subsubsection{Definition and Usage of Rules}
1551 % \paragraph{The core} of our implemented problem is the Z-Transformation, due
1552 % the fact that the transformation itself would require higher math which isn't
1553 % yet avaible in our system we decided to choose the way like it is applied in
1554 % labratory and problem classes at our university - by applying transformation
1555 % rules (collected in transformation tables).
1556 % \paragraph{Rules,} in {\sisac{}}'s programming language can be designed by the
1557 % use of axiomatizations like shown in Example~\ref{eg:ruledef}
1560 % \label{eg:ruledef}
1563 % axiomatization where
1564 % rule1: ``1 = $\delta$[n]'' and
1565 % rule2: ``|| z || > 1 ==> z / (z - 1) = u [n]'' and
1566 % rule3: ``|| z || < 1 ==> z / (z - 1) = -u [-n - 1]''
1570 % This rules can be collected in a ruleset and applied to a given expression as
1571 % follows in Example~\ref{eg:ruleapp}.
1575 % \label{eg:ruleapp}
1577 % \item Store rules in ruleset:
1579 % val inverse_Z = append_rls "inverse_Z" e_rls
1580 % [ Thm ("rule1",num_str @{thm rule1}),
1581 % Thm ("rule2",num_str @{thm rule2}),
1582 % Thm ("rule3",num_str @{thm rule3})
1584 % \item Define exression:
1586 % val sample_term = str2term "z/(z-1)+z/(z-</delta>)+1";\end{verbatim}
1587 % \item Apply ruleset:
1589 % val SOME (sample_term', asm) =
1590 % rewrite_set_ thy true inverse_Z sample_term;\end{verbatim}
1594 % The use of rulesets makes it much easier to develop our designated applications,
1595 % but the programmer has to be careful and patient. When applying rulesets
1596 % two important issues have to be mentionend:
1597 % \subparagraph{How often} the rules have to be applied? In case of
1598 % transformations it is quite clear that we use them once but other fields
1599 % reuqire to apply rules until a special condition is reached (e.g.
1600 % a simplification is finished when there is nothing to be done left).
1601 % \subparagraph{The order} in which rules are applied often takes a big effect
1602 % and has to be evaluated for each purpose once again.
1604 % In our special case of Signal Processing and the rules defined in
1605 % Example~\ref{eg:ruledef} we have to apply rule~1 first of all to transform all
1606 % constants. After this step has been done it no mather which rule fit's next.
1608 % \subsubsection{Helping Functions}
1610 % \paragraph{New Programms require,} often new ways to get through. This new ways
1611 % means that we handle functions that have not been in use yet, they can be
1612 % something special and unique for a programm or something famous but unneeded in
1613 % the system yet. In our dedicated example it was for example neccessary to split
1614 % a fraction into numerator and denominator; the creation of such function and
1615 % even others is described in upper Sections~\ref{simp} and \ref{funs}.
1617 % \subsubsection{Trials on equation solving}
1618 % %simple eq and problem with double fractions/negative exponents
1619 % \paragraph{The Inverse Z-Transformation} makes it neccessary to solve
1620 % equations degree one and two. Solving equations in the first degree is no
1621 % problem, wether for a student nor for our machine; but even second degree
1622 % equations can lead to big troubles. The origin of this troubles leads from
1623 % the build up process of our equation solving functions; they have been
1624 % implemented some time ago and of course they are not as good as we want them to
1625 % be. Wether or not following we only want to show how cruel it is to build up new
1626 % work on not well fundamentials.
1627 % \subparagraph{A simple equation solving,} can be set up as shown in the next
1634 % ["equality (-1 + -2 * z + 8 * z ^^^ 2 = (0::real))",
1638 % val (dI',pI',mI') =
1640 % ["abcFormula","degree_2","polynomial","univariate","equation"],
1641 % ["no_met"]);\end{verbatim}
1644 % Here we want to solve the equation: $-1+-2\cdot z+8\cdot z^{2}=0$. (To give
1645 % a short overview on the commands; at first we set up the equation and tell the
1646 % machine what's the bound variable and where to store the solution. Second step
1647 % is to define the equation type and determine if we want to use a special method
1648 % to solve this type.) Simple checks tell us that the we will get two results for
1649 % this equation and this results will be real.
1650 % So far it is easy for us and for our machine to solve, but
1651 % mentioned that a unvariate equation second order can have three different types
1652 % of solutions it is getting worth.
1653 % \subparagraph{The solving of} all this types of solutions is not yet supported.
1654 % Luckily it was needed for us; but something which has been needed in this
1655 % context, would have been the solving of an euation looking like:
1656 % $-z^{-2}+-2\cdot z^{-1}+8=0$ which is basically the same equation as mentioned
1657 % before (remember that befor it was no problem to handle for the machine) but
1658 % now, after a simple equivalent transformation, we are not able to solve
1660 % \subparagraph{Error messages} we get when we try to solve something like upside
1661 % were very confusing and also leads us to no special hint about a problem.
1662 % \par The fault behind is, that we have no well error handling on one side and
1663 % no sufficient formed equation solving on the other side. This two facts are
1664 % making the implemention of new material very difficult.
1666 % \subsection{Formalization of missing knowledge in Isabelle}
1668 % \paragraph{A problem} behind is the mechanization of mathematic
1669 % theories in TP-bases languages. There is still a huge gap between
1670 % these algorithms and this what we want as a solution - in Example
1671 % Signal Processing.
1677 % X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY
1680 % \noindent A very simple example on this what we call gap is the
1681 % simplification above. It is needles to say that it is correct and also
1682 % Isabelle for fills it correct - \emph{always}. But sometimes we don't
1683 % want expand such terms, sometimes we want another structure of
1684 % them. Think of a problem were we now would need only the coefficients
1685 % of $X$ and $Y$. This is what we call the gap between mechanical
1686 % simplification and the solution.
1691 % \paragraph{We are not able to fill this gap,} until we have to live
1692 % with it but first have a look on the meaning of this statement:
1693 % Mechanized math starts from mathematical models and \emph{hopefully}
1694 % proceeds to match physics. Academic engineering starts from physics
1695 % (experimentation, measurement) and then proceeds to mathematical
1696 % modeling and formalization. The process from a physical observance to
1697 % a mathematical theory is unavoidable bound of setting up a big
1698 % collection of standards, rules, definition but also exceptions. These
1699 % are the things making mechanization that difficult.
1708 % \noindent Think about some units like that one's above. Behind
1709 % each unit there is a discerning and very accurate definition: One
1710 % Meter is the distance the light travels, in a vacuum, through the time
1711 % of 1 / 299.792.458 second; one kilogram is the weight of a
1712 % platinum-iridium cylinder in paris; and so on. But are these
1713 % definitions usable in a computer mechanized world?!
1718 % \paragraph{A computer} or a TP-System builds on programs with
1719 % predefined logical rules and does not know any mathematical trick
1720 % (follow up example \ref{eg:trick}) or recipe to walk around difficult
1726 % \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]
1727 % \[ \frac{1}{j\omega}\cdot e^{-j2\omega}\cdot\left(e^{j\omega}-e^{-j\omega}\right)=
1728 % \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$\frac{1}{j}\,\left(e^{j\omega}-e^{-j\omega}\right)$}= \]
1729 % \[ \frac{1}{\omega}\, e^{-j2\omega}\cdot\colorbox{lgray}{$2\, sin(\omega)$} \]
1731 % \noindent Sometimes it is also useful to be able to apply some
1732 % \emph{tricks} to get a beautiful and particularly meaningful result,
1733 % which we are able to interpret. But as seen in this example it can be
1734 % hard to find out what operations have to be done to transform a result
1735 % into a meaningful one.
1740 % \paragraph{The only possibility,} for such a system, is to work
1741 % through its known definitions and stops if none of these
1742 % fits. Specified on Signal Processing or any other application it is
1743 % often possible to walk through by doing simple creases. This creases
1744 % are in general based on simple math operational but the challenge is
1745 % to teach the machine \emph{all}\footnote{Its pride to call it
1746 % \emph{all}.} of them. Unfortunately the goal of TP Isabelle is to
1747 % reach a high level of \emph{all} but it in real it will still be a
1748 % survey of knowledge which links to other knowledge and {{\sisac}{}} a
1749 % trainer and helper but no human compensating calculator.
1751 % {{{\sisac}{}}} itself aims to adds \emph{Algorithmic Knowledge} (formal
1752 % specifications of problems out of topics from Signal Processing, etc.)
1753 % and \emph{Application-oriented Knowledge} to the \emph{deductive} axis of
1754 % physical knowledge. The result is a three-dimensional universe of
1755 % mathematics seen in Figure~\ref{fig:mathuni}.
1759 % \includegraphics{fig/universe}
1760 % \caption{Didactic ``Math-Universe'': Algorithmic Knowledge (Programs) is
1761 % combined with Application-oriented Knowledge (Specifications) and Deductive Knowledge (Axioms, Definitions, Theorems). The Result
1762 % leads to a three dimensional math universe.\label{fig:mathuni}}
1766 % %WN Deine aktuelle Benennung oben wird Dir kein Fachmann abnehmen;
1767 % %WN bitte folgende Bezeichnungen nehmen:
1769 % %WN axis 1: Algorithmic Knowledge (Programs)
1770 % %WN axis 2: Application-oriented Knowledge (Specifications)
1771 % %WN axis 3: Deductive Knowledge (Axioms, Definitions, Theorems)
1773 % %WN und bitte die R"ander von der Grafik wegschneiden (was ich f"ur *.pdf
1774 % %WN nicht hinkriege --- weshalb ich auch die eJMT-Forderung nicht ganz
1775 % %WN verstehe, separierte PDFs zu schicken; ich w"urde *.png schicken)
1777 % %JR Ränder und beschriftung geändert. Keine Ahnung warum eJMT sich pdf's
1778 % %JR wünschen, würde ebenfalls png oder ähnliches verwenden, aber wenn pdf's
1779 % %JR gefordert werden WN2...
1780 % %WN2 meiner Meinung nach hat sich eJMT unklar ausgedr"uckt (z.B. kann
1781 % %WN2 man meines Wissens pdf-figures nicht auf eine bestimmte Gr"osse
1782 % %WN2 zusammenschneiden um die R"ander weg zu bekommen)
1783 % %WN2 Mein Vorschlag ist, in umserem tex-file bei *.png zu bleiben und
1784 % %WN2 png + pdf figures mitzuschicken.
1786 % \subsection{Notes on Problems with Traditional Notation}
1788 % \paragraph{During research} on these topic severely problems on
1789 % traditional notations have been discovered. Some of them have been
1790 % known in computer science for many years now and are still unsolved,
1791 % one of them aggregates with the so called \emph{Lambda Calculus},
1792 % Example~\ref{eg:lamda} provides a look on the problem that embarrassed
1799 % \[ f(x)=\ldots\; \quad R \rightarrow \quad R \]
1802 % \[ f(p)=\ldots\; p \in \quad R \]
1805 % \noindent Above we see two equations. The first equation aims to
1806 % be a mapping of an function from the reel range to the reel one, but
1807 % when we change only one letter we get the second equation which
1808 % usually aims to insert a reel point $p$ into the reel function. In
1809 % computer science now we have the problem to tell the machine (TP) the
1810 % difference between this two notations. This Problem is called
1811 % \emph{Lambda Calculus}.
1816 % \paragraph{An other problem} is that terms are not full simplified in
1817 % traditional notations, in {{\sisac}} we have to simplify them complete
1818 % to check weather results are compatible or not. in e.g. the solutions
1819 % of an second order linear equation is an rational in {{\sisac}} but in
1820 % tradition we keep fractions as long as possible and as long as they
1821 % aim to be \textit{beautiful} (1/8, 5/16,...).
1822 % \subparagraph{The math} which should be mechanized in Computer Theorem
1823 % Provers (\emph{TP}) has (almost) a problem with traditional notations
1824 % (predicate calculus) for axioms, definitions, lemmas, theorems as a
1825 % computer program or script is not able to interpret every Greek or
1826 % Latin letter and every Greek, Latin or whatever calculations
1827 % symbol. Also if we would be able to handle these symbols we still have
1828 % a problem to interpret them at all. (Follow up \hbox{Example
1829 % \ref{eg:symbint1}})
1833 % \label{eg:symbint1}
1835 % u\left[n\right] \ \ldots \ unitstep
1838 % \noindent The unitstep is something we need to solve Signal
1839 % Processing problem classes. But in {{{\sisac}{}}} the rectangular
1840 % brackets have a different meaning. So we abuse them for our
1841 % requirements. We get something which is not defined, but usable. The
1842 % Result is syntax only without semantic.
1847 % In different problems, symbols and letters have different meanings and
1848 % ask for different ways to get through. (Follow up \hbox{Example
1849 % \ref{eg:symbint2}})
1853 % \label{eg:symbint2}
1855 % \widehat{\ }\ \widehat{\ }\ \widehat{\ } \ \ldots \ exponent
1858 % \noindent For using exponents the three \texttt{widehat} symbols
1859 % are required. The reason for that is due the development of
1860 % {{{\sisac}{}}} the single \texttt{widehat} and also the double were
1861 % already in use for different operations.
1866 % \paragraph{Also the output} can be a problem. We are familiar with a
1867 % specified notations and style taught in university but a computer
1868 % program has no knowledge of the form proved by a professor and the
1869 % machines themselves also have not yet the possibilities to print every
1870 % symbol (correct) Recent developments provide proofs in a human
1871 % readable format but according to the fact that there is no money for
1872 % good working formal editors yet, the style is one thing we have to
1875 % \section{Problems rising out of the Development Environment}
1877 % fehlermeldungen! TODO
1879 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\end{verbatim}
1881 \section{Conclusion}\label{conclusion}
1885 %This paper gives a first experience report about programming with a
1886 %TP-based programming language.
1888 A brief re-introduction of the novel kind of programming
1889 language by example of the {\sisac}-prototype makes the paper
1890 self-contained. The main section describes all the main concepts
1891 involved in TP-based programming and all the sub-tasks concerning
1892 respective implementation: mechanisation of mathematics and domain
1893 modeling, implementation of term rewriting systems for the
1894 rewriting-engine, formal (implicit) specification of the problem to be
1895 (explicitly) described by the program, implementation of the many components
1896 required for Lucas-Interpretation and finally implementation of the
1899 The many concepts and sub-tasks involved in programming require a
1900 comprehensive work-flow; first experiences with the work-flow as
1901 supported by the present prototype are described as well: Isabelle +
1902 Isar + jEdit provide appropriate components for establishing an
1903 efficient development environment integrating computation and
1904 deduction. However, the present state of the prototype is far off a
1905 state appropriate for wide-spread use: the prototype of the program
1906 language lacks expressiveness and elegance, the prototype of the
1907 development environment is hardly usable: error messages still address
1908 the developer of the prototype's interpreter rather than the
1909 application programmer, implementation of the many settings for the
1910 Lucas-Interpreter is cumbersome.
1912 From these experiences a successful proof of concept can be concluded:
1913 programming arbitrary problems from engineering sciences is possible,
1914 in principle even in the prototype. Furthermore the experiences allow
1915 to conclude detailed requirements for further development:
1917 \item Clarify underlying logics such that programming is smoothly
1918 integrated with verification of the program; the post-condition should
1919 be proved more or less automatically, otherwise working engineers
1920 would not encounter such programming.
1921 \item Combine the prototype's programming language with Isabelle's
1922 powerful function package and probably with more of SML's
1923 pattern-matching features; include parallel execution on multi-core
1924 machines into the language design.
1925 \item Extend the prototype's Lucas-Interpreter such that it also
1926 handles functions defined by use of Isabelle's functions package; and
1927 generalize Isabelle's code generator such that efficient code for the
1928 whole definition of the programming language can be generated (for
1929 multi-core machines).
1930 \item Develop an efficient development environment with
1931 integration of programming and proving, with management not only of
1932 Isabelle theories, but also of large collections of specifications and
1935 Provided successful accomplishment, these points provide distinguished
1936 components for virtual workbenches appealing to practitioner of
1937 engineering in the near future.
1939 \medskip Interactive course material, as addressed by the title, then
1940 can comprise step-wise problem solving created as a side-effect of a
1941 TP-based program: Lucas-Interpretation not only provides an
1942 interactive programming environment, Lucas-Interpretation also can
1943 provide TP-based services for a flexible dialogue component with
1944 adaptive user guidance for independent and inquiry-based learning.
1947 \bibliographystyle{alpha}
1948 {\small\bibliography{references}}
1951 % LocalWords: TP IST SPSC Telematics Dialogues dialogue HOL bool nat Hindley
1952 % LocalWords: Milner tt Subproblem Formulae ruleset generalisation initialised
1953 % LocalWords: axiomatization LCF Simplifiers simplifiers Isar rew Thm Calc SML
1954 % LocalWords: recognised hoc Trueprop redexes Unsynchronized pre rhs ord erls
1955 % LocalWords: srls prls nrls lhs errpats InverseZTransform SubProblem IDE IDEs
1956 % LocalWords: univariate jEdit rls RealDef calclist familiarisation ons pos eq
1957 % LocalWords: mout ctree SignalProcessing frac ZZ Postcond Atools wiki SML's
1958 % LocalWords: mechanisation multi