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\title{Trials with TP-based Programming

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for Interactive Course Material}%

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\author{\begin{tabular}{c}

\textit{Jan Ro\v{c}nik} \\

jan.rocnik@student.tugraz.at \\

IST, SPSC\\

Graz University of Technologie\\

Austria\end{tabular}

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\begin{abstract}

Traditional course material in engineering disciplines lacks an

important component, interactive support for step-wise problem

solving. Theorem-Proving (TP) technology is appropriate for one part

of such support, in checking user-input. For the other part of such

support, guiding the learner towards a solution, another kind of

technology is required. %TODO ... connect to prototype ...

A prototype combines TP with a programming language, the latter

interpreted in a specific way: certain statements in a program, called

tactics, are treated as breakpoints where control is handed over to

the user. An input formula is checked by TP (using logical context

built up by the interpreter); and if a learner gets stuck, a program

describing the steps towards a solution of a problem ``knows the next

step''. This kind of interpretation is called Lucas-Interpretation for

\emph{TP-based programming languages}.

This paper describes the prototype's TP-based programming language

within a case study creating interactive material for an advanced

course in Signal Processing: implementation of definitions and

theorems in TP, formal specification of a problem and step-wise

development of the program solving the problem. Experiences with the

ork flow in iterative development with testing and identifying errors

are described, too. The description clarifies the components missing

in the prototype's language as well as deficiencies experienced during

programming.

\par

These experiences are particularly notable, because the author is the

first programmer using the language beyond the core team which

developed the prototype's TP-based language interpreter.

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\section{Introduction}\label{intro}

% \paragraph{Didactics of mathematics} 

%WN: wenn man in einem high-quality paper von 'didactics' spricht, 

%WN muss man am state-of-the-art ankn"upfen -- siehe

%WN W.Neuper, On the Emergence of TP-based Educational Math Assistants

% faces a specific issue, a gap

% between (1) introduction of math concepts and skills and (2)

% application of these concepts and skills, which usually are separated

% into different units in curricula (for good reasons). For instance,

% (1) teaching partial fraction decomposition is separated from (2)

% application for inverse Z-transform in signal processing.

% 

% \par This gap is an obstacle for applying math as an fundamental

% thinking technology in engineering: In (1) motivation is lacking

% because the question ``What is this stuff good for?'' cannot be

% treated sufficiently, and in (2) the ``stuff'' is not available to

% students in higher semesters as widespread experience shows.

% 

% \paragraph{Motivation} taken by this didactic issue on the one hand,

% and ongoing research and development on a novel kind of educational

% mathematics assistant at Graz University of

% Technology~\footnote{http://www.ist.tugraz.at/isac/} promising to

% scope with this issue on the other hand, several institutes are

% planning to join their expertise: the Institute for Information

% Systems and Computer Media (IICM), the Institute for Software

% Technology (IST), the Institutes for Mathematics, the Institute for

% Signal Processing and Speech Communication (SPSC), the Institute for

% Structural Analysis and the Institute of Electrical Measurement and

% Measurement Signal Processing.

%WN diese Information ist f"ur das Paper zu spezielle, zu aktuell 

%WN und damit zu verg"anglich.

% \par This thesis is the first attempt to tackle the above mentioned

% issue, it focuses on Telematics, because these specific studies focus

% on mathematics in \emph{STEOP}, the introductory orientation phase in

% Austria. \emph{STEOP} is considered an opportunity to investigate the

% impact of {\sisac}'s prototype on the issue and others.

% 

\paragraph{Traditional course material} in engineering disciplines lacks an

important component, interactive support for step-wise problem

solving. Theorem-Proving (TP) technology can provide such support by

specific services. An important part of such services is called

``next-step-guidance'', generated by a specific kind of ``TP-based

programming language''. In the

{\sisac}-project~\footnote{http://www.ist.tugraz.at/projects/isac/} such

a language is prototyped in line with~\cite{plmms10} and built upon

the theorem prover

Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}.

The TP services are coordinated by a specific interpreter for the

programming language, called

Lucas-Interpreter~\cite{wn:lucas-interp-12}. The language and the

interpreter will be briefly re-introduced in order to make the paper

self-contained.

\subparagraph{The main part} of the paper is an account of first experiences

with programming in this TP-based language. The experience was gained

in a case study by the author. The author was considered an ideal

candidate for this study for the following reasons: as a student in

Telematics (computer science with focus on Signal Processing) he had

general knowledge in programming as well as specific domain knowledge

in Signal Processing; and he was not involved in the development of

{\sisac}'s programming language and interpeter, thus a novice to the

language.

\subparagraph{The goal} of the case study was (1) some TP-based programs for

interactive course material for a specific ``Adavanced Signal

Processing Lab'' in a higher semester, (2) respective program

development with as little advice from the {\sisac}-team and (3) records

and comments for the main steps of development in an Isabelle theory;

this theory should provide guidelines for future programmers. An

excerpt from this theory is the main part of this paper.

\par

The paper will use the problem in Fig.\ref{fig-interactive} as a

running example:

\begin{figure} [htb]

\begin{center}

\includegraphics[width=140mm]{fig/isac-Ztrans-math}

\caption{Step-wise problem solving guided by the TP-based program}

\label{fig-interactive}

\end{center}

\end{figure}

\paragraph{The problem is} from the domain of Signal Processing and requests to

determine the inverse Z-transform for a given term. Fig.\ref{fig-interactive}

also shows the beginning of the interactive construction of a solution

for the problem. This construction is done in the right window named

``Worksheet''.

\par

User-interaction on the Worksheet is {\em checked} and {\em guided} by

TP services:

\begin{enumerate}

\item Formulas input by the user are {\em checked} by TP: such a

formula establishes a proof situation --- the prover has to derive the

formula from the logical context. The context is built up from the

formal specification of the problem (here hidden from the user) by the

Lucas-Interpreter.

\item If the user gets stuck, the program developed below in this

paper ``knows the next step'' from behind the scenes. How the latter

TP-service is exploited by dialogue authoring is out of scope of this

paper and can be studied in~\cite{gdaroczy-EP-13}.

\end{enumerate} It should be noted that the programmer using the

TP-based language is not concerned with interaction at all; we will

see that the program contains neither input-statements nor

output-statements. Rather, interaction is handled by services

generated automatically.

\par

So there is a clear separation of concerns: Dialogues are

adapted by dialogue authors (in Java-based tools), using automatically

generated TP services, while the TP-based program is written by

mathematics experts (in Isabelle/ML). The latter is concern of this

paper.

\paragraph{The paper is structed} as follows: The introduction

\S\ref{intro} is followed by a brief re-introduction of the TP-based

programming language in \S\ref{PL}, which extends the executable

fragment of Isabelle's language (\S\ref{PL-isab}) by tactics which

play a specific role in Lucas-Interpretation and in providing the TP

services (\S\ref{PL-tacs}). The main part in \S\ref{trial} describes

the main steps in developing the program for the running example:

prepare domain knowledge, implement the formal specification of the

problem, prepare the environment for the program, implement the

program. The workflow of programming, debugging and testing is

described in \S\ref{workflow}. The conclusion \S\ref{conclusion} will

give directions identified for future development. 

\section{\isac's Prototype for a Programming Language}\label{PL} 

The prototype's language extends the executable fragment in the

language of the theorem prover

Isabelle~\cite{Nipkow-Paulson-Wenzel:2002}\footnote{http://isabelle.in.tum.de/}

by tactics which have a specific role in Lucas-Interpretation.

\subsection{The Executable Fragment of Isabelle's Language}\label{PL-isab}

The executable fragment consists of data-type and function

definitions.  It's usability even suggests that fragment for

introductory courses \cite{nipkow-prog-prove}. HOL is a typed logic

whose type system resembles that of functional programming

languages. Thus there are

\begin{description}

\item[base types,] in particular \textit{bool}, the type of truth

values, \textit{nat}, \textit{int}, \textit{complex}, and the types of

natural, integer and complex numbers respectively in mathematics.

\item[type constructors] allow to define arbitrary types, from

\textit{set}, \textit{list} to advanced data-structures like

\textit{trees}, red-black-trees etc.

\item[function types,] denoted by $\Rightarrow$.

\item[type variables,] denoted by $^\prime a, ^\prime b$ etc, provide

type polymorphism. Isabelle automatically computes the type of each

variable in a term by use of Hindley-Milner type inference

\cite{pl:hind97,Milner-78}.

\end{description}

\textbf{Terms} are formed as in functional programming by applying

functions to arguments. If $f$ is a function of type

$\tau_1\Rightarrow \tau_2$ and $t$ is a term of type $\tau_1$ then

$f\;t$ is a term of type~$\tau_2$. $t\;::\;\tau$ means that term $t$

has type $\tau$. There are many predefined infix symbols like $+$ and

$\leq$ most of which are overloaded for various types.

HOL also supports some basic constructs from functional programming:

{\it\label{isabelle-stmts}

\begin{tabbing} 123\=\kill

\>$( \; {\tt if} \; b \; {\tt then} \; t_1 \; {\tt else} \; t_2 \;)$\\

\>$( \; {\tt let} \; x=t \; {\tt in} \; u \; )$\\

\>$( \; {\tt case} \; t \; {\tt of} \; {\it pat}_1

  \Rightarrow t_1 \; |\dots| \; {\it pat}_n\Rightarrow t_n \; )$

\end{tabbing} }

\noindent \textit{The running example's program uses some of these elements

(marked by {\tt tt-font} on p.\pageref{expl-program}): ${\tt

let}\dots{\tt in}$ in lines $02 \dots 11$, as well as {\tt last} for

lists and {\tt o} for functional (forward) composition in line

$10$. In fact, the whole program is an Isabelle term with specific

function constants like {\sc program}, {\sc Substitute} and {\sc

Rewrite\_Set\_Inst} in lines $01$ and $10$ respectively.}

% Terms may also contain $\lambda$-abstractions. For example, $\lambda

% x. \; x$ is the identity function.

%JR warum auskommentiert?

\textbf{Formulae} are terms of type \textit{bool}. There are the basic

constants \textit{True} and \textit{False} and the usual logical

connectives (in decreasing order of precedence): $\neg, \land, \lor,

\rightarrow$.

\textbf{Equality} is available in the form of the infix function $=$

of type $a \Rightarrow a \Rightarrow {\it bool}$. It also works for

formulas, where it means ``if and only if''.

\textbf{Quantifiers} are written $\forall x. \; P$ and $\exists x. \;

P$.  Quantifiers lead to non-executable functions, so functions do not

always correspond to programs, for instance, if comprising \\$(

\;{\it if} \; \exists x.\;P \; {\it then} \; e_1 \; {\it else} \; e_2

\;)$.

\subsection{\isac's Tactics for Lucas-Interpretation}\label{PL-tacs}

The prototype extends Isabelle's language by specific statements

called tactics~\footnote{{\sisac}'s tactics are different from

Isabelle's tactics: the former concern steps in a calculation, the

latter concern proof steps.}  and tacticals. For the programmer these

statements are functions with the following signatures:

\begin{description}

\item[Rewrite:] ${\it theorem}\Rightarrow{\it term}\Rightarrow{\it

term} * {\it term}\;{\it list}$:

this tactic appplies {\it theorem} to a {\it term} yielding a {\it

term} and a {\it term list}, the list are assumptions generated by

conditional rewriting. For instance, the {\it theorem}

$b\not=0\land c\not=0\Rightarrow\frac{a\cdot c}{b\cdot c}=\frac{a}{b}$

applied to the {\it term} $\frac{2\cdot x}{3\cdot x}$ yields

$(\frac{2}{3}, [x\not=0])$.

\item[Rewrite\_Set:] ${\it ruleset}\Rightarrow{\it

term}\Rightarrow{\it term} * {\it term}\;{\it list}$:

this tactic appplies {\it ruleset} to a {\it term}; {\it ruleset} is

a confluent and terminating term rewrite system, in general. If

none of the rules ({\it theorem}s) is applicable on interpretation

of this tactic, an exception is thrown.

% \item[Rewrite\_Inst:] ${\it substitution}\Rightarrow{\it

% theorem}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it

% list}$:

% 

% \item[Rewrite\_Set\_Inst:] ${\it substitution}\Rightarrow{\it

% ruleset}\Rightarrow{\it term}\Rightarrow{\it term} * {\it term}\;{\it

% list}$:

\item[Substitute:] ${\it substitution}\Rightarrow{\it

term}\Rightarrow{\it term}$:

\item[Take:] ${\it term}\Rightarrow{\it term}$:

this tactic has no effect in the program; but it creates a side-effect

by Lucas-Interpretation (see below) and writes {\it term} to the

Worksheet.

\item[Subproblem:] ${\it theory} * {\it specification} * {\it

method}\Rightarrow{\it argument}\;{\it list}\Rightarrow{\it term}$:

this tactic allows to enter a phase of interactive specification

of a theory ($\Re$, $\cal C$, etc), a formal specification (for instance,

a specific type of equation) and a method (for instance, solving an

equation symbolically or numerically).

\end{description}

The tactics play a specific role in

Lucas-Interpretation~\cite{wn:lucas-interp-12}: they are treated as

break-points and control is handed over to the user. The user is free

to investigate underlying knowledge, applicable theorems, etc.  And

the user can proceed constructing a solution by input of a tactic to

be applied or by input of a formula; in the latter case the

Lucas-Interpreter has built up a logical context (initialised with the

precondition of the formal specification) such that Isabelle can

derive the formula from this context --- or give feedback, that no

derivation can be found.

\subsection{Tacticals for Control of Interpretation}

The flow of control in a program can be determined by {\tt if then else}

and {\tt case of} as mentioned on p.\pageref{isabelle-stmts} and also

by additional tacticals:

\begin{description}

\item[Repeat:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it

term}$: iterates over tactics which take a {\it term} as argument as

long as a tactic is applicable (for instance, {\it Rewrite\_Set} might

not be applicable).

\item[Try:] ${\it tactic}\Rightarrow{\it term}\Rightarrow{\it term}$:

if {\it tactic} is applicable, then it is applied to {\it term},

otherwise {\it term} is passed on unchanged.

\item[Or:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it

term}\Rightarrow{\it term}$:

\item[@@:] ${\it tactic}\Rightarrow{\it tactic}\Rightarrow{\it

term}\Rightarrow{\it term}$:

\item[While:] ${\it term::bool}\Rightarrow{\it tactic}\Rightarrow{\it

term}\Rightarrow{\it term}$:

\end{description}

no input / output --- Lucas-Interpretation

.\\.\\.\\TODO\\.\\.\\

\section{Development of a Program on Trial}\label{trial} 

As mentioned above, {\sisac} is an experimental system for a proof of

concept for Lucas-Interpretation~\cite{wn:lucas-interp-12}.  The

latter interprets a specific kind of TP-based programming language,

which is as experimental as the whole prototype --- so programming in

this language can be only ``on trial'', presently.  However, as a

prototype, the language addresses essentials described below.

\subsection{Mechanization of Math --- Domain Engineering}\label{isabisac}

%WN was Fachleute unter obigem Titel interessiert findet

%WN unterhalb des auskommentierten Textes.

%WN der Text unten spricht Benutzer-Aspekte anund ist nicht speziell

%WN auf Computer-Mathematiker fokussiert.

% \paragraph{As mentioned in the introduction,} a prototype of an

% educational math assistant called

% {{\sisac}}\footnote{{{\sisac}}=\textbf{Isa}belle for

% \textbf{C}alculations, see http://www.ist.tugraz.at/isac/.} bridges

% the gap between (1) introducation and (2) application of mathematics:

% {{\sisac}} is based on Computer Theorem Proving (TP), a technology which

% requires each fact and each action justified by formal logic, so

% {{{\sisac}{}}} makes justifications transparent to students in

% interactive step-wise problem solving. By that way {{\sisac}} already

% can serve both:

% \begin{enumerate}

%   \item Introduction of math stuff (in e.g. partial fraction

% decomposition) by stepwise explaining and exercising respective

% symbolic calculations with ``next step guidance (NSG)'' and rigorously

% checking steps freely input by students --- this also in context with

% advanced applications (where the stuff to be taught in higher

% semesters can be skimmed through by NSG), and

%   \item Application of math stuff in advanced engineering courses

% (e.g. problems to be solved by inverse Z-transform in a Signal

% Processing Lab) and now without much ado about basic math techniques

% (like partial fraction decomposition): ``next step guidance'' supports

% students in independently (re-)adopting such techniques.

% \end{enumerate} 

% Before the question is answers, how {{\sisac}}

% accomplishes this task from a technical point of view, some remarks on

% the state-of-the-art is given, therefor follow up Section~\ref{emas}.

% 

% \subsection{Educational Mathematics Assistants (EMAs)}\label{emas}

% 

% \paragraph{Educational software in mathematics} is, if at all, based

% on Computer Algebra Systems (CAS, for instance), Dynamic Geometry

% Systems (DGS, for instance \footnote{GeoGebra http://www.geogebra.org}

% \footnote{Cinderella http://www.cinderella.de/}\footnote{GCLC

% http://poincare.matf.bg.ac.rs/~janicic/gclc/}) or spread-sheets. These

% base technologies are used to program math lessons and sometimes even

% exercises. The latter are cumbersome: the steps towards a solution of

% such an interactive exercise need to be provided with feedback, where

% at each step a wide variety of possible input has to be foreseen by

% the programmer - so such interactive exercises either require high

% development efforts or the exercises constrain possible inputs.

% 

% \subparagraph{A new generation} of educational math assistants (EMAs)

% is emerging presently, which is based on Theorem Proving (TP). TP, for

% instance Isabelle and Coq, is a technology which requires each fact

% and each action justified by formal logic. Pushed by demands for

% \textit{proven} correctness of safety-critical software TP advances

% into software engineering; from these advancements computer

% mathematics benefits in general, and math education in particular. Two

% features of TP are immediately beneficial for learning:

% 

% \paragraph{TP have knowledge in human readable format,} that is in

% standard predicate calculus. TP following the LCF-tradition have that

% knowledge down to the basic definitions of set, equality,

% etc~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL.html};

% following the typical deductive development of math, natural numbers

% are defined and their properties

% proven~\footnote{http://isabelle.in.tum.de/dist/library/HOL/Number\_Theory/Primes.html},

% etc. Present knowledge mechanized in TP exceeds high-school

% mathematics by far, however by knowledge required in software

% technology, and not in other engineering sciences.

% 

% \paragraph{TP can model the whole problem solving process} in

% mathematical problem solving {\em within} a coherent logical

% framework. This is already being done by three projects, by

% Ralph-Johan Back, by ActiveMath and by Carnegie Mellon Tutor.

% \par

% Having the whole problem solving process within a logical coherent

% system, such a design guarantees correctness of intermediate steps and

% of the result (which seems essential for math software); and the

% second advantage is that TP provides a wealth of theories which can be

% exploited for mechanizing other features essential for educational

% software.

% 

% \subsubsection{Generation of User Guidance in EMAs}\label{user-guid}

% 

% One essential feature for educational software is feedback to user

% input and assistance in coming to a solution.

% 

% \paragraph{Checking user input} by ATP during stepwise problem solving

% is being accomplished by the three projects mentioned above

% exclusively. They model the whole problem solving process as mentioned

% above, so all what happens between formalized assumptions (or formal

% specification) and goal (or fulfilled postcondition) can be

% mechanized. Such mechanization promises to greatly extend the scope of

% educational software in stepwise problem solving.

% 

% \paragraph{NSG (Next step guidance)} comprises the system's ability to

% propose a next step; this is a challenge for TP: either a radical

% restriction of the search space by restriction to very specific

% problem classes is required, or much care and effort is required in

% designing possible variants in the process of problem solving

% \cite{proof-strategies-11}.

% \par

% Another approach is restricted to problem solving in engineering

% domains, where a problem is specified by input, precondition, output

% and postcondition, and where the postcondition is proven by ATP behind

% the scenes: Here the possible variants in the process of problem

% solving are provided with feedback {\em automatically}, if the problem

% is described in a TP-based programing language: \cite{plmms10} the

% programmer only describes the math algorithm without caring about

% interaction (the respective program is functional and even has no

% input or output statements!); interaction is generated as a

% side-effect by the interpreter --- an efficient separation of concern

% between math programmers and dialog designers promising application

% all over engineering disciplines.

% 

% 

% \subsubsection{Math Authoring in Isabelle/ISAC\label{math-auth}}

% Authoring new mathematics knowledge in {{\sisac}} can be compared with

% ``application programing'' of engineering problems; most of such

% programing uses CAS-based programing languages (CAS = Computer Algebra

% Systems; e.g. Mathematica's or Maple's programing language).

% 

% \paragraph{A novel type of TP-based language} is used by {{\sisac}{}}

% \cite{plmms10} for describing how to construct a solution to an

% engineering problem and for calling equation solvers, integration,

% etc~\footnote{Implementation of CAS-like functionality in TP is not

% primarily concerned with efficiency, but with a didactic question:

% What to decide for: for high-brow algorithms at the state-of-the-art

% or for elementary algorithms comprehensible for students?} within TP;

% TP can ensure ``systems that never make a mistake'' \cite{casproto} -

% are impossible for CAS which have no logics underlying.

% 

% \subparagraph{Authoring is perfect} by writing such TP based programs;

% the application programmer is not concerned with interaction or with

% user guidance: this is concern of a novel kind of program interpreter

% called Lucas-Interpreter. This interpreter hands over control to a

% dialog component at each step of calculation (like a debugger at

% breakpoints) and calls automated TP to check user input following

% personalized strategies according to a feedback module.

% \par

% However ``application programing with TP'' is not done with writing a

% program: according to the principles of TP, each step must be

% justified. Such justifications are given by theorems. So all steps

% must be related to some theorem, if there is no such theorem it must

% be added to the existing knowledge, which is organized in so-called

% \textbf{theories} in Isabelle. A theorem must be proven; fortunately

% Isabelle comprises a mechanism (called ``axiomatization''), which

% allows to omit proofs. Such a theorem is shown in

% Example~\ref{eg:neuper1}.

The running example, introduced by Fig.\ref{fig-interactive} on

p.\pageref{fig-interactive}, requires to determine the inverse $\cal

Z$-transform for a class of functions. The domain of Signal Processing

is accustomed to specific notation for the resulting functions, which

are absolutely summable and are called TODO: $u[n]$, where $u$ is the

function, $n$ is the argument and the brackets indicate that the

arguments are TODO. Surprisingly, Isabelle accepts the rules for

${\cal Z}^{-1}$ in this traditional notation~\footnote{Isabelle

experts might be particularly surprised, that the brackets do not

cause errors in typing (as lists).}:

%\vbox{

% \begin{example}

  \label{eg:neuper1}

  {\small\begin{tabbing}

  123\=123\=123\=123\=\kill

  \hfill \\

  \>axiomatization where \\

  \>\>  rule1: ``${\cal Z}^{-1}\;1 = \delta [n]$'' and\\

  \>\>  rule2: ``$\vert\vert z \vert\vert > 1 \Rightarrow {\cal Z}^{-1}\;z / (z - 1) = u [n]$'' and\\

  \>\>  rule3: ``$\vert\vert$ z $\vert\vert$ < 1 ==> z / (z - 1) = -u [-n - 1]'' and \\

%TODO

  \>\>  rule4: ``$\vert\vert$ z $\vert\vert$ > $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = $\alpha^n$ $\cdot$ u [n]'' and\\

%TODO

  \>\>  rule5: ``$\vert\vert$ z $\vert\vert$ < $\vert\vert$ $\alpha$ $\vert\vert$ ==> z / (z - $\alpha$) = -($\alpha^n$) $\cdot$ u [-n - 1]'' and\\

%TODO

  \>\>  rule6: ``$\vert\vert$ z $\vert\vert$ > 1 ==> z/(z - 1)$^2$ = n $\cdot$ u [n]''\\

%TODO

  \end{tabbing}

  }

% \end{example}

%}

These 6 rules can be used as conditional rewrite rules, depending on

the respective convergence radius. Satisfaction from accordance with traditional notation

contrasts with the above word {\em axiomatization}: As TP-based, the

programming language expects these rules as {\em proved} theorems, and

not as axioms implemented in the above brute force manner; otherwise

all the verification efforts envisaged (like proof of the

post-condition, see below) would be meaningless.

Isabelle provides a large body of knowledge, rigorously proven from

the basic axioms of mathematics~\footnote{This way of rigorously

deriving all knowledge from first principles is called the

LCF-paradigm in TP.}. In the case of the Z-Transform the most advanced

knowledge can be found in the theoris on Multivariate

Analysis~\footnote{http://isabelle.in.tum.de/dist/library/HOL/HOL-Multivariate\_Analysis}. However,

building up knowledge such that a proof for the above rules would be

reasonably short and easily comprehensible, still requires lots of

work (and is definitely out of scope of our case study).

\paragraph{At the state-of-the-art in mechanization of knowledge} in

engineering sciences, the process does not stop with the mechanization of

mathematics traditionally used in these sciences. Rather, ``Formal Methods''~\cite{TODO-formal-methods}

are expected to proceed to formal and explicit description of physical items.  Signal Processing,

for instance is concerned with physical devices for signal acquisition

and reconstruction, which involve measuring a physical signal, storing

it, and possibly later rebuilding the original signal or an

approximation thereof. For digital systems, this typically includes

sampling and quantization; devices for signal compression, including

audio compression, image compression, and video compression, etc.

``Domain engineering''\cite{db-domain-engineering} is concerned with

{\em specification} of these devices' components and features; this

part in the process of mechanization is only at the beginning in domains

like Signal Processing.

\subparagraph{TP-based programming, concern of this paper,} is determined to

add ``algorithmic knowledge'' in Fig.\ref{fig:mathuni} on

p.\pageref{fig:mathuni}.  As we shall see below, TP-based programming

starts with a formal {\em specification} of the problem to be solved.

\subsection{Specification of the Problem}\label{spec}

%WN <--> \chapter 7 der Thesis

%WN die Argumentation unten sollte sich NUR auf Verifikation beziehen..

The problem of the running example is textually described in

Fig.\ref{fig-interactive} on p.\pageref{fig-interactive}. The {\em

formal} specification of this problem, in traditional mathematical

notation, could look lik is this:

%WN Hier brauchen wir die Spezifikation des 'running example' ...

%JR Habe input, output und precond vom Beispiel eingefügt brauche aber Hilfe bei

%JR der post condition - die existiert für uns ja eigentlich nicht aka

%JR haben sie bis jetzt nicht beachtet

  \label{eg:neuper2}

  {\small\begin{tabbing}

  123,\=postcond \=: \= $\forall \,A^\prime\, u^\prime \,v^\prime.\,$\=\kill

  \hfill \\

  Specification:\\

    \>input    \>: filterExpression $X=\frac{3}{(z-\frac{1}{4}+-\frac{1}{8}*\frac{1}{z}}$\\

  \>precond  \>: filterExpression continius on $\mathbb{R}$ \\

  \>output   \>: stepResponse $x[n]$ \\

  \>postcond \>:{\small  $\;A=2uv-u^2 \;\land\; (\frac{u}{2})^2+(\frac{v}{2})^2=r^2 \;\land$}\\

  \>     \>\>{\small $\;\forall \;A^\prime\; u^\prime \;v^\prime.\;(A^\prime=2u^\prime v^\prime-(u^\prime)^2 \land

  (\frac{u^\prime}{2})^2+(\frac{v^\prime}{2})^2=r^2) \Longrightarrow A^\prime \leq A$} \\

  \end{tabbing}

  }

And this is the implementation of the formal specification in the present

prototype, still bar-bones without support for authoring:

%WN Kopie von Inverse_Z_Transform.thy, leicht versch"onert:

{\footnotesize

\begin{verbatim}

   01  store_specification

   02    (prepare_specification

   03      ["Jan Rocnik"]

   04      "pbl_SP_Ztrans_inv"

   05      thy

   06      ( ["Inverse", "Z_Transform", "SignalProcessing"],

   07        [ ("#Given", ["filterExpression X_eq"]),

   08          ("#Pre"  , ["X_eq is_continuous"]),

   19          ("#Find" , ["stepResponse n_eq"]),

   10          ("#Post" , [" TODO "])],

   11        append_rls Erls [Calc ("Atools.is_continuous", eval_is_continuous)], 

   12        NONE, 

   13        [["SignalProcessing","Z_Transform","Inverse"]]));

\end{verbatim}}

Although the above details are partly very technical, we explain them

in order to document some intricacies of TP-based programming in the

present state of the {\sisac} prototype:

\begin{description}

\item[01..02]\textit{store\_specification:} stores the result of the

function \textit{prep\_specification} in a global reference

\textit{Unsynchronized.ref}, which causes principal conflicts with

Isabelle's asyncronous document model~\cite{Wenzel-11:doc-orient} and

parallel execution~\cite{Makarius-09:parall-proof} and is under

reconstruction already.

\textit{prep\_pbt:} translates the specification to an internal format

which allows efficient processing; see for instance line {\rm 07}

below.

\item[03..04] are the ``mathematics author'' holding the copy-rights

and a unique identifier for the specification within {\sisac},

complare line {\rm 06}.

\item[05] is the Isabelle \textit{theory} required to parse the

specification in lines {\rm 07..10}.

\item[06] is a key into the tree of all specifications as presented to

the user (where some branches might be hidden by the dialog

component).

\item[07..10] are the specification with input, pre-condition, output

and post-condition respectively; the post-condition is not handled in

the prototype presently.

\item[11] creates a term rewriting system (abbreviated \textit{rls} in

{\sisac}) which evaluates the pre-condition for the actual input data.

Only if the evaluation yields \textit{True}, a program con be started.

\item[12]\textit{NONE:} could be \textit{SOME ``solve ...''} for a

problem associated to a function from Computer Algebra (like an

equation solver) which is not the case here.

\item[13] is the specific key into the tree of programs addressing a

method which is able to find a solution which satiesfies the

post-condition of the specification.

\end{description}

%WN die folgenden Erkl"arungen finden sich durch "grep -r 'datatype pbt' *"

%WN ...

%  type pbt = 

%     {guh  : guh,         (*unique within this isac-knowledge*)

%      mathauthors: string list, (*copyright*)

%      init  : pblID,      (*to start refinement with*)

%      thy   : theory,     (* which allows to compile that pbt

%			  TODO: search generalized for subthy (ref.p.69*)

%      (*^^^ WN050912 NOT used during application of the problem,

%       because applied terms may be from 'subthy' as well as from super;

%       thus we take 'maxthy'; see match_ags !*)

%      cas   : term option,(*'CAS-command'*)

%      prls  : rls,        (* for preds in where_*)

%      where_: term list,  (* where - predicates*)

%      ppc   : pat list,

%      (*this is the model-pattern; 

%       it contains "#Given","#Where","#Find","#Relate"-patterns

%       for constraints on identifiers see "fun cpy_nam"*)

%      met   : metID list}; (* methods solving the pbt*)

%

%WN weil dieser Code sehr unaufger"aumt ist, habe ich die Erkl"arungen

%WN oben selbst geschrieben.

%WN das w"urde ich in \sec\label{progr} verschieben und

%WN das SubProblem partial fractions zum Erkl"aren verwenden.

% Such a specification is checked before the execution of a program is

% started, the same applies for sub-programs. In the following example

% (Example~\ref{eg:subprob}) shows the call of such a subproblem:

% 

% \vbox{

%   \begin{example}

%   \label{eg:subprob}

%   \hfill \\

%   {\ttfamily \begin{tabbing}

%   ``(L\_L::bool list) = (\=SubProblem (\=Test','' \\

%   ``\>\>[linear,univariate,equation,test],'' \\

%   ``\>\>[Test,solve\_linear])'' \\

%   ``\>[BOOL equ, REAL z])'' \\

%   \end{tabbing}

%   }

%   {\small\textit{

%     \noindent If a program requires a result which has to be

% calculated first we can use a subproblem to do so. In our specific

% case we wanted to calculate the zeros of a fraction and used a

% subproblem to calculate the zeros of the denominator polynom.

%     }}

%   \end{example}

% }

\subsection{Implementation of the Method}\label{meth}

%WN <--> \chapter 7 der Thesis

TODO

\subsection{Preparation of Simplifiers for the Program}\label{simp}

TODO

\subsection{Preparation of ML-Functions}\label{funs}

%WN <--> Thesis 6.1 -- 6.3: jene ausw"ahlen, die Du f"ur \label{progr}

%WN brauchst

TODO

\subsection{Implementation of the TP-based Program}\label{progr}

%WN <--> \chapter 8 der Thesis

TODO

\section{Workflow of Programming in the Prototype}\label{workflow}

%WN ``workflow'' heisst: das mache ich zuerst, dann das ...

\subsection{Preparations and Trials}\label{flow-prep}

\subsubsection{Trials on Notation and Termination}

\paragraph{Technical notations} are a big problem for our piece of software,

but the reason for that isn't a fault of the software itself, one of the

troubles comes out of the fact that different technical subtopics use different

symbols and notations for a different purpose. The most famous example for such

a symbol is the complex number $i$ (in cassique math) or $j$ (in technical

math). In the specific part of signal processing one of this notation issues is

the use of brackets --- we use round brackets for analoge signals and squared

brackets for digital samples. Also if there is no problem for us to handle this

fact, we have to tell the machine what notation leads to wich meaning and that

this purpose seperation is only valid for this special topic - signal

processing.

\subparagraph{In the programming language} itself it is not possible to declare

fractions, exponents, absolutes and other operators or remarks in a way to make

them pretty to read; our only posssiblilty were ASCII characters and a handfull

greek symbols like: $\alpha, \beta, \gamma, \phi,\ldots$.

\par

With the upper collected knowledge it is possible to check if we were able to

donate all required terms and expressions.

\subsubsection{Definition and Usage of Rules}

\paragraph{The core} of our implemented problem is the Z-Transformation, due

the fact that the transformation itself would require higher math which isn't

yet avaible in our system we decided to choose the way like it is applied in

labratory and problem classes at our university - by applying transformation

rules (collected in transformation tables).

\paragraph{Rules,} in {\sisac{}}'s programming language can be designed by the

use of axiomatizations like shown in Example~\ref{eg:ruledef}

\begin{example}

  \label{eg:ruledef}

  \hfill\\

  \begin{verbatim}

  axiomatization where

    rule1: ``1 = $\delta$[n]'' and

    rule2: ``|| z || > 1 ==> z / (z - 1) = u [n]'' and

    rule3: ``|| z || < 1 ==> z / (z - 1) = -u [-n - 1]''

  \end{verbatim}

\end{example}

This rules can be collected in a ruleset and applied to a given expression as

follows in Example~\ref{eg:ruleapp}.

\begin{example}

  \hfill\\

  \label{eg:ruleapp}

  \begin{enumerate}

  \item Store rules in ruleset:

  \begin{verbatim}

  val inverse_Z = append_rls "inverse_Z" e_rls

    [ Thm ("rule1",num_str @{thm rule1}),

      Thm ("rule2",num_str @{thm rule2}),

      Thm ("rule3",num_str @{thm rule3})

    ];\end{verbatim}

  \item Define exression:

  \begin{verbatim}

  val sample_term = str2term "z/(z-1)+z/(z-</delta>)+1";\end{verbatim}

  \item Apply ruleset:

  \begin{verbatim}

  val SOME (sample_term', asm) = 

    rewrite_set_ thy true inverse_Z sample_term;\end{verbatim}

  \end{enumerate}

\end{example}

The use of rulesets makes it much easier to develop our designated applications,

but the programmer has to be careful and patient. When applying rulesets

two important issues have to be mentionend:

\subparagraph{How often} the rules have to be applied? In case of

transformations it is quite clear that we use them once but other fields

reuqire to apply rules until a special condition is reached (e.g.

a simplification is finished when there is nothing to be done left).

\subparagraph{The order} in which rules are applied often takes a big effect

and has to be evaluated for each purpose once again.

\par

In our special case of Signal Processing and the rules defined in

Example~\ref{eg:ruledef} we have to apply rule~1 first of all to transform all

constants. After this step has been done it no mather which rule fit's next.

\subsubsection{Helping Functions}

%get denom, argument in

\subsubsection{Trials on equation solving}

%simple eq and problem with double fractions/negative exponents

\subsection{Implementation in Isabelle/{\isac}}\label{flow-impl}

TODO Build\_Inverse\_Z\_Transform.thy ... ``imports Isac''

\subsection{Transfer into the Isabelle/{\isac} Knowledge}\label{flow-trans}

TODO http://www.ist.tugraz.at/isac/index.php/Extend\_ISAC\_Knowledge\#Add\_an\_example ?

-------------------------------------------------------------------

Das unterhalb hab' ich noch nicht durchgearbeitet; einiges w\"are 

vermutlich auf andere sections aufzuteilen.

-------------------------------------------------------------------

\subsection{Formalization of missing knowledge in Isabelle}

\paragraph{A problem} behind is the mechanization of mathematic

theories in TP-bases languages. There is still a huge gap between

these algorithms and this what we want as a solution - in Example

Signal Processing. 

\vbox{

  \begin{example}

    \label{eg:gap}

    \[

      X\cdot(a+b)+Y\cdot(c+d)=aX+bX+cY+dY

    \]

    {\small\textit{

      \noindent A very simple example on this what we call gap is the

simplification above. It is needles to say that it is correct and also

Isabelle for fills it correct - \emph{always}. But sometimes we don't

want expand such terms, sometimes we want another structure of

them. Think of a problem were we now would need only the coefficients

of $X$ and $Y$. This is what we call the gap between mechanical

simplification and the solution.

    }}

  \end{example}

}

\paragraph{We are not able to fill this gap,} until we have to live

with it but first have a look on the meaning of this statement:

Mechanized math starts from mathematical models and \emph{hopefully}

proceeds to match physics. Academic engineering starts from physics

(experimentation, measurement) and then proceeds to mathematical

modeling and formalization. The process from a physical observance to

a mathematical theory is unavoidable bound of setting up a big

collection of standards, rules, definition but also exceptions. These

are the things making mechanization that difficult.

\vbox{

  \begin{example}

    \label{eg:units}

    \[

      m,\ kg,\ s,\ldots

    \]

    {\small\textit{

      \noindent Think about some units like that one's above. Behind

each unit there is a discerning and very accurate definition: One

Meter is the distance the light travels, in a vacuum, through the time

of 1 / 299.792.458 second; one kilogram is the weight of a

platinum-iridium cylinder in paris; and so on. But are these

definitions usable in a computer mechanized world?!

    }}

  \end{example}

}

\paragraph{A computer} or a TP-System builds on programs with

predefined logical rules and does not know any mathematical trick

(follow up example \ref{eg:trick}) or recipe to walk around difficult

expressions. 

\vbox{

  \begin{example}

    \label{eg:trick}

  \[ \frac{1}{j\omega}\cdot\left(e^{-j\omega}-e^{j3\omega}\right)= \]