1.1 --- a/doc-src/isac/dmeindl/proposal.tex Fri Oct 28 19:31:41 2011 +0200
1.2 +++ b/doc-src/isac/dmeindl/proposal.tex Mon Nov 07 14:49:40 2011 +0100
1.3 @@ -1,4 +1,5 @@
1.4 -%WN mit diesen 3 Zeichen beginnen meine Kommentare
1.5 +%WN mit diesen 3 Zeichen beginnen meine Kommentare
1.6 +%WN111107: bitte spellchecker dr"uberlaufen lassen !!!
1.7
1.8 \documentclass[12pt,a4paper]{article}
1.9 \bibliographystyle{alpha}
1.10 @@ -172,9 +173,9 @@
1.11
1.12 Calculation with fractions is an important part of Computer Algebra Systems (CAS). This proposal aims at a specific part of such calculations, the greatest common divisor (GCD) used for cancellation, but in the very general context of multivariate polynomials. Cancellation of multivariate polynomials is a settled topic in Computer Algebra, respective algorithms well documented and implementations available in all CASs.
1.13
1.14 -This proposal claims for novelty with respect to the context of implementation in Computer Theorem Proving (CTP). On CTP's present development towards industrial use in software and systems verification, specific domain models involve demand on more and more mathematics, and within mathematics involve demand for more and more features. The proposed implementation of GCD and cancellation follows an actual demand.
1.15 +This proposal claims for novelty with respect to the context of implementation, an implementation as a CAS-feature in Computer Theorem Proving (CTP). On CTP's present development towards industrial use in software and systems verification, specific domain models involve demand on more and more mathematics, and within mathematics involve demand for more and more features. Thus the proposed implementation of GCD and cancellation follows an actual demand.
1.16
1.17 -If the implementation is successful, it might be included into the distribution of Isabelle, one of the two dominating CTPs in Europe.
1.18 +If the implementation is successful, it is planned to be included into the distribution of Isabelle, one of the two dominating CTPs in Europe. As part of the Isabelle distribution it will also serve the {\sisac} project aiming at an educational math assistant under development at RISC Linz and Graz University of Technology.
1.19 }
1.20
1.21 \newpage
1.22 @@ -182,24 +183,28 @@
1.23 \tableofcontents
1.24
1.25 \section{Background}
1.26 -The \sisac-project is a research and development project at the Institute for Software Technology of the Graz University of Technology. It is an educational mathematics assistant, a single-stepping system for applied mathematics based on the computer theorem prover Isabelle. The special is an easy readable knowledge base including Isabelles HOL-theories and a transparently working knowledge interpreter (a generalization of 'single stepping' algebra systems).
1.27 -The background to both, development and research, is given by actual needs in math education as well as by foundamental questions about 'the mechanization of thinking' as an essential aspect in mathematics and in technology.
1.28 -The \sisac-system under construction comprises a tutoring-system and an authoring-system. The latter provides for adaption to various needs of individual users and educational institutions and for extensions to arbitrary fields of applied mathematics.
1.29 +The \sisac-project is a research and development project launched at the Institute for Software Technology of the Graz University of Technology (TUG) and now continued at the Research Institute for Symbolic Computation (RISC) of University of Linz and at the Institute for Information Systems and Computer Media (IICM) of TUG. The resulting \sisac{} prototype is a ``transparent single-stepping system for applied mathematics'' based on the computer theorem prover Isabelle. The prototype has been proven useful in field tests at Austrain schools \cite{imst-htl06-SH,imst-htl07-SH,imst-hpts08-SH} and is now extended for wider use.
1.30 +
1.31 +Authoring knowledge in \sisac{} provides a strict separation of concerns between authoring math knowledge and authoring dialogues. The latter is pursued at IICM, the former is concern of this thesis. Math authoring is done by use of a CTP-based programming language \cite{plmms10} or by use of SML \cite{pl:milner97} as the meta language and implementation language of Isabelle. Since the code resulting from this thesis shall serve Isabelle, it will be written in SML. Via Isabelle distribution this thesis shall also serve \sisac; a re-implementation in \sisac's CTP-based language is planned as a subsequent project -- this will make cancellation transparent for singe-stepping.
1.32 +
1.33 +%The special is an easy readable knowledge base including Isabelles HOL-theories and a transparently working knowledge interpreter (a generalization of 'single stepping' algebra systems).
1.34 +%The background to both, development and research, is given by actual needs in math education as well as by foundamental questions about 'the mechanization of thinking' as an essential aspect in mathematics and in technology.
1.35 +%The \sisac-system under construction comprises a tutoring-system and an authoring-system. The latter provides for adaption to various needs of individual users and educational institutions and for extensions to arbitrary fields of applied mathematics.
1.36
1.37 -TODO:\\
1.38 +TODO.WN111107 bitte googeln und je einen Absatz kopieren + zitieren woher (PLAGIATsgefahr):\\
1.39 European provers: Isabelle \cite{Nipkow-Paulson-Wenzel:2002}, Coq \cite{Huet_all:94}\\
1.40 American provers: PVS~\cite{pvs}, ACL2~\footnote{http://userweb.cs.utexas.edu/~moore/acl2/}\\
1.41
1.42 \section{Goal of the thesis}
1.43 \subsection{Current situation}
1.44 -At the time there is no good implimentation for the problem of canceling fractions in \sisac and or in Isabelle. But because canceling is important for calculating with fractions a new implimentation is necessary.
1.45 -
1.46 +At the presetn time there is no implimentation for the problem of canceling fractions in Isabelle, and a deficient one in \sisac. But because canceling is important for calculating with fractions a new implimentation is necessary.
1.47 +
1.48 \subsection{Problem}
1.49 -The wish is to handle fractions in \sisac not only in one variable also in more. So the goal of this thesis ist to find, assess and evaluate the existing algorithms and methods for finding the GCD. This will be an functional programm with the posibility to include it into Isabelle.
1.50 +The wish is to handle fractions in \sisac{} not only in one variable also in more. So the goal of this thesis ist to find, assess and evaluate the existing algorithms and methods for finding the GCD. This will be an functional programm with the posibility to include it into Isabelle, where it will be used by \sisac{} as well.
1.51
1.52 %WN eine pr"azisere Beschreibung des Problems kann ich mir nicht vorstellen (englische Version der Mail haben wir auch, aber sie passt nicht zur deutschen Antwort von Prof.Nipkow) ...
1.53 \bigskip
1.54 -TODO Mail to Prof. Nipkow, leader of the development of Isabelle \cite{Nipkow-Paulson-Wenzel:2002} at TU M\"unchen, Mon, 23 May 2011 08:58:14 +0200:
1.55 +A mail to Prof. Nipkow, leader of the development of Isabelle \cite{Nipkow-Paulson-Wenzel:2002} at TU M\"unchen, Mon, 23 May 2011 08:58:14 +0200 describes the problem as follows:
1.56 \begin{verbatim}
1.57 Eine erste Idee, wie die Integration der Diplomarbeit f"ur
1.58 einen Benutzer von Isabelle aussehen k"onnte, w"are zum
1.59 @@ -237,14 +242,14 @@
1.60
1.61
1.62 \subsection{Expected results}
1.63 -Find good algorithms for the different problems, and find out which one will be the best for the special problem.\\
1.64 -The program should handling:
1.65 +Implementation of algorithms for the different problems, and find out which one will be the best for the specific requirements in Isabelle.\\
1.66 +The program should accomplish:
1.67 \begin{itemize}
1.68 -\item[]real and rational coefficients. Maybe also imaginary coefficients.
1.69 -\item[]Multi variable polynomials canceling and adding, when they are in normal form.
1.70 +\item Real and rational coefficients. Maybe also imaginary coefficients.
1.71 +\item Canceling and adding multivariate polynomials, when they are in normal form.
1.72 \end{itemize}
1.73 -For the program should be used a functional programming language with good commentaries. And it should be based on Isabelle and works correctly in \sisac.
1.74 -
1.75 +The program will be written in the functional programming language SML with appropriate comments. The resulting code shall meet the coding standards of Isabelle \cite{isar-impl} p.3-10. The integration of the code into the Isabelle distribution will be done by the Isabelle developer team.
1.76 +
1.77 \section{State of the art}
1.78 In a broad view the context of this thesis can be seen as ``computation and deduction'': simplification and in particular cancellation of rational terms is concern of \textbf{computation} implemented in Computer Algebra Systems (CAS) --- whereas the novelty within the thesis is given by an implementation of cancellation in a computer theorem prover (CTP), i.e. in the domain of \textbf{deduction} with respective logical rigor not addressed in the realm of CAS.
1.79
1.80 @@ -297,7 +302,7 @@
1.81
1.82 1994 was an important year for CTP: the Pentium Bug caused excitement in the IT community all around the world and motivated INTEL to invest greatly into formal verification of circuits (which carried over to verification of software). Not much later John Harrison mechanized real numbers as Dedekind Cuts in HOL Light \footnote{http://www.cl.cam.ac.uk/~jrh13/hol-light/} and derived calculus, derivative and integral from that definition \cite{harr:thesis}, an implementation which has been transferred to Isabelle very soon after that~\footnote{In the directory src/HOL/Multivariate\_Analysis/ see the files Gauge\_Measure.thy, Integration.thy, Derivative.thy, Real\_Integration.thy, Brouwer\_Fixpoint.thy, Fashoda.thy}.
1.83
1.84 -Harrison also says that ``CAS are ill-defined'' and gives, among others, an example relevant for this thesis on cancellation: TODO ... meromorphic functions ...
1.85 +Harrison also says that ``CAS are ill-defined'' and gives, among others, an example relevant for this thesis on cancellation: TODO.WN111104 search for ... meromorphic functions in http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-428.ps.gz
1.86
1.87 \medskip
1.88 The main motivation for further introduction of CAS-functionality to CTP is also technology-driven: In this decade domain engineering is becoming an academic discipline with industrial relevance \cite{db:dom-eng}: vigorous efforts extend the scope of formal specifications even beyond software technology, and thus respective domains of mathematical knowledge are being mechanized in CTP. The Archive of Formal Proofs~\footnote{http://afp.sourceforge.net/} is Isabelle's repository for such work.
1.89 @@ -352,7 +357,7 @@
1.90
1.91 \subsection{Open Issues with CAS-functionality in CTP}\label{cas-funct}
1.92 There is at least one effort explicitly dedicated to implement CAS-functionality in CTP \cite{cezary-phd}. %WN bitte unbedingt lesen (kann von mir in Papierform ausgeborgt werden) !!!
1.93 -In this work three issues has been identified: partiality conditions, multi-valued functions and real numbers. These issues are addressed in the subsequent paragraphs, followed by a forth issue raised by \sisac.
1.94 +In this work three issues has been identified: partiality conditions, multi-valued functions and real numbers. These issues are addressed in the subsequent paragraphs, followed by a forth issue raised by \sisac{}.
1.95
1.96 \paragraph{Partiality conditions}\label{part-cond} are introduced by partial functions or by conditional rewriting. An example of how the CAS-functionality \cite{cezary-phd} looks like is given on p.\pageref{fig:casproto}.
1.97 \cite{cezary-phd} gives an introductory example (floated to p.\pageref{fig:casproto}) which will be referred to in the sequel.
1.98 @@ -383,16 +388,15 @@
1.99 lines together with HOL Light theorems that state the equality
1.100 between the input and the output.}
1.101 \end{figure}
1.102 -In lines {\tt In6, Out6} this examples shows how to reliably simplify $\sqrt{x}$. \cite{caspartial} %TODO
1.103 -gives more details on handling side conditions in formalized partial functions.
1.104 +In lines {\tt In6, Out6} this examples shows how to reliably simplify $\sqrt{x}$. \cite{caspartial} gives more details on handling side conditions in formalized partial functions.
1.105
1.106 Analoguous to this example, cancellations (this thesis is concerned with) like
1.107 $$\frac{x^2-y^2}{x^2-x\cdot y}=\frac{x+y}{x}\;\;\;\;{\it assuming}\;x-y\not=0\land x\not=0$$
1.108 -produce assumptions, $x-y\not=0, x\not=0$ here. Since the code produced in the framework of this thesis will be implemented in Isabelle's simplifier (outside this thesis), the presentation to the user will be determined by Isabelle and \sisac{} using the respective component of Isabelle. Also reliable handling of assumptions like $x-y\not=0, x\not=0$ is up to these systems.
1.109 +produce assumptions, $x-y\not=0, x\not=0$ here. Since the code produced in the framework of this thesis will be implemented in Isabelle's simplifier (outside this thesis), the presentation to the user will be determined by Isabelle and \sisac{}{} using the respective component of Isabelle. Also reliable handling of assumptions like $x-y\not=0, x\not=0$ is up to these systems.
1.110
1.111 \paragraph{Multi-valued functions:}\label{multi-valued}
1.112 \cite{seeingroots,davenp-multival-10} discuss cases where CAS are error prone when dropping a branch of a multi-valued function~\footnote{``Multivalued \textit{function}'' is a misnomer, since the value of a function applied to a certain argument is unique by definition of function.}. Familiar examples are ...
1.113 -%WN ... zur Erkl"arung ein paar Beispiele von http://en.wikipedia.org/wiki/Multivalued_function
1.114 +%TODO.WN111104 ... zur Erkl"arung ein paar Beispiele von http://en.wikipedia.org/wiki/Multivalued_function
1.115
1.116 \paragraph{Real numbers} cannot be represented by numerals. In engineering applications, however, approximation by floating-point numbers are frequently useful. In CTP floating-point numbers must be handled rigorously as approximations. Already \cite{harr:thesis} introduced operations on real numerals accompanied by rigorous calculation of precision. \cite{russellphd} describes efficient implementation of infinite precision real numbers in Coq.
1.117
1.118 @@ -401,25 +405,28 @@
1.119 \exists x_s.\;x_s\in S &\Rightarrow& f(x_s) = 0 \\\label{is-solut}
1.120 x_s\in S &\Leftarrow& \exists x_s.\;f(x_s) = 0 \label{all-solut}
1.121 \end{eqnarray}
1.122 -where (\ref{all-solut}) ensures that $S$ contains {\em all} solutions of the equation. The \sisac-project has implemented a prototype of an equation solver~\footnote{See \textit{equations} in the hierarchy of specifications at http://www.ist.tugraz.at/projects/isac/www/kbase/pbl/index\_pbl.html}.
1.123 +where (\ref{all-solut}) ensures that $S$ contains {\em all} solutions of the equation. The \sisac{}-project has implemented a prototype of an equation solver~\footnote{See \textit{equations} in the hierarchy of specifications at http://www.ist.tugraz.at/projects/isac/www/kbase/pbl/index\_pbl.html}.
1.124
1.125 -There is demand for fullfledged equation solving in CTP, including equational systems and differential equations, because \sisac{} has a prototype of a CTP-based programming language calling CAS functions; and Lucas-Interpretation \cite{wn:lucas-interp-12} makes these functions accessible by single-stepping and ``next step guidance'', which would automatically generate a learning system for equation solving.
1.126 -
1.127 +There is demand for fullfledged equation solving in CTP, including equational systems and differential equations, because \sisac{}{} has a prototype of a CTP-based programming language calling CAS functions; and Lucas-Interpretation \cite{wn:lucas-interp-12} makes these functions accessible by single-stepping and ``next step guidance'', which would automatically generate a learning system for equation solving.
1.128 +
1.129 +\subsection{Algorithms for cancellation of multivariate polynomials}
1.130 +The most appropriate book for implementing the required algorithms in this thesis is \cite{Winkler:96}. TODO.WN111104 welche noch ?
1.131 +
1.132 \section{Thesis structure}
1.133 The proposed table of contents of the thesis on the chapter level is as follows:
1.134 \begin{enumerate}
1.135 \item Introduction (2-3 pages)
1.136 \item Computer Algebra Systems (CAS) (5 - 7 pages)\\
1.137 Which different CAS exists and whats the focus of them.
1.138 - \item The \sisac-Project (5 - 7 pages)\\
1.139 - This chapter will describe the \sisac-Project and the goals of the project.
1.140 + \item The \sisac{}-Project (5 - 7 pages)\\
1.141 + This chapter will describe the \sisac{}-Project and the goals of the project.
1.142 \item Univariate Polynomials (15-20 pages)\\
1.143 This chapter will describe different Algorithms for univariate polynomials, with different coefficients.
1.144 \item Multivariate Polynomials (20-25 pages)\\
1.145 This chapter will describe different Algorithms for multivariate polynomials, with different coefficients
1.146 \item Functional programming and SML(2-5 pages)\\
1.147 The basic idea of this programming languages.
1.148 - \item Implimentation in \sisac-Project (15-20 pages)
1.149 + \item Implimentation in \sisac{}-Project (15-20 pages)
1.150 \item Conclusion (2-3 pages)
1.151 \end{enumerate}
1.152 %\newpage
1.153 @@ -447,10 +454,10 @@
1.154 \end{center}
1.155
1.156 %WN oben an passender stelle einf"ugen
1.157 -\cite{einf-funct-progr} \cite{Winkler:96}
1.158 +\cite{einf-funct-progr}
1.159
1.160
1.161 -\bibliography{references}
1.162 +\bibliography{bib/math-eng,bib/didact,bib/bk,bib/RISC_2,bib/isac,bib/pl,bib/math,references}
1.163 %\section{Bibliography}
1.164 %%mindestens 10
1.165 %\begin{enumerate}