| version 1.1, 2001/10/03 08:32:58 |
version 1.6, 2001/10/12 07:45:11 |
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% $OpenXM: OpenXM/doc/Papers/dag-noro.tex,v 1.5 2001/10/12 05:11:36 noro Exp $ |
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| \title{Introduction of a computer algebra system/library Risa/Asir} |
\title{\tc A computer algebra system Risa/Asir and OpenXM} |
| |
|
| \author{Masayuki Noro\\ Kobe University} |
\author{Masayuki Noro\\ Kobe University, Japan} |
| \begin{document} |
\begin{document} |
| \setlength{\parskip}{10pt} |
\setlength{\parskip}{10pt} |
| \maketitle |
\maketitle |
| \blackandwhite{dagb-noro.tex} |
|
| \end{document} |
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|
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%\begin{slide}{} |
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%\begin{center} |
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%\fbox{\fbc \large Part I : OpenXM and Risa/Asir --- overview and history} |
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%\end{center} |
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%\end{slide} |
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|
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%\begin{slide}{} |
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%\fbox{\fbc Integration of mathematical software systems} |
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% |
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%\begin{itemize} |
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%\item Data integration |
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% |
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%\begin{itemize} |
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%\item OpenMath ({\urlc \tt http://www.openmath.org}) , MP [GRAY98] |
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%\end{itemize} |
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% |
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%Standards for representing mathematical objects |
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% |
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%\item Control integration |
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% |
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%\begin{itemize} |
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%\item MCP [WANG99], OMEI [LIAO01] |
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%\end{itemize} |
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% |
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%Protocols for remote subroutine calls or session management |
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% |
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%\item Combination of two integrations |
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% |
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%\begin{itemize} |
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%\item MathLink, OpenMath+MCP, MP+MCP |
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% |
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%and OpenXM ({\urlc \tt http://www.openxm.org}) |
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%\end{itemize} |
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% |
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%Both are necessary for practical implementation |
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% |
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%\end{itemize} |
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%\end{slide} |
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\begin{slide}{} |
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\fbox{\fbc A computer algebra system Risa/Asir} |
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|
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\begin{itemize} |
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\item {\itc Software mainly for polynomial computation} |
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|
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Polynomial factorization, Groebner basis computation |
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\item {\itc User language with C-like syntax} |
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|
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C language without type declaration, with list processing |
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|
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\item {\itc Builtin {\tt gdb}-like debugger for user programs} |
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|
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\item {\itc Open source} ({\urlc \tt http://www.math.kobe-u.ac.jp/Asir/asir.html}) |
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|
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The source and binaries are available via ftp or CVS |
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|
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See {\urlc \tt http://www.openxm.org} to get the latest version |
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|
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\item {\itc OpenXM interface} |
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|
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\begin{itemize} |
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\item OpenXM ({\urlc \tt http://www.openxm.org}) |
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|
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An infrastructure for exchanging mathematical data |
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\item Risa/Asir is a main client in OpenXM package |
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\item {\tt ox\_asir} is an OpenXM server |
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\item {\tt libasir.a} provides OpenXM interface via function call |
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\end{itemize} |
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\end{itemize} |
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\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc Goal of developing Risa/Asir} |
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|
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\begin{itemize} |
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\item {\itc Testing new algorithms} |
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|
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\begin{itemize} |
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\item Development started in Fujitsu labs |
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|
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Polynomial factorization, Groebner basis related computation, |
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cryptosystems , quantifier elimination , $\ldots$ |
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\end{itemize} |
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|
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\item {\itc To be a general purpose, open system} |
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|
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Since 1997, we have been developing OpenXM package |
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containing various servers and clients |
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|
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Risa/Asir is a component of OpenXM |
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|
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\item {\itc Environment for parallel and distributed computation} |
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\end{itemize} |
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\end{slide} |
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|
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%\begin{slide}{} |
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%\fbox{\fbc Capability for polynomial computation} |
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% |
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%\begin{itemize} |
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%\item Fundamental polynomial arithmetics |
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% |
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%recursive representation and distributed representation |
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% |
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%\item Polynomial factorization |
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% |
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%\begin{itemize} |
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%\item Univariate : over {\bf Q}, algebraic number fields and finite fields |
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% |
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%\item Multivariate : over {\bf Q} |
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%\end{itemize} |
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% |
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%\item Groebner basis computation |
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% |
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%\begin{itemize} |
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%\item Buchberger and $F_4$ [FAUG99] algorithm |
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% |
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%\item Change of ordering/RUR [ROUI96] of 0-dimensional ideals |
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% |
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%\item Primary ideal decomposition |
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% |
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%\item Computation of $b$-function (in Weyl Algebra) |
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%\end{itemize} |
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%\end{itemize} |
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%\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc History of development : Polynomial factorization} |
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|
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\begin{itemize} |
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\item {\itc 1989} |
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|
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Start of Risa/Asir with Boehm's conservative GC |
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({\urlc \tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc}) |
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|
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\item {\itc 1989-1992} |
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|
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Univariate and multivariate factorizers over {\bf Q} |
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\item {\itc 1992-1994} |
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Univariate factorization over algebraic number fields |
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Intensive use of successive extension, non-squarefree norms |
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Application to splitting field and Galois group computation |
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|
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\item {\itc 1996-1998} |
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|
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Univariate factorization over large finite fields |
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|
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Motivated by a reseach project in Fujitsu on cryptography |
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|
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\item {\itc 2000-current} |
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|
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Multivariate factorization over small finite fields (in progress) |
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\end{itemize} |
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\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc History of development : Groebner basis} |
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|
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\begin{itemize} |
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\item {\itc 1992-1994} |
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|
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User language $\Rightarrow$ C version; trace lifting [TRAV88] |
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|
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\item {\itc 1994-1996} |
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|
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Trace lifting with homogenization |
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|
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Omitting GB check by compatible prime [NOYO99] |
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Modular change of ordering/RUR[ROUI96] [NOYO99] |
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|
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Primary ideal decomposition [SHYO96] |
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|
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\item {\itc 1996-1998} |
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|
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Efficient content reduction during NF computation [NORO97] |
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Solved {\it McKay} system for the first time |
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|
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\item {\itc 1998-2000} |
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|
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Test implementation of $F_4$ [FAUG99] |
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|
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\item {\itc 2000-current} |
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|
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Buchberger algorithm in Weyl algebra |
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|
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Efficient $b$-function computation[OAKU97] by a modular method |
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\end{itemize} |
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\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc Timing data --- Factorization} |
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|
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\underline{\itc Univariate; over {\bf Q}} (on Pentium III, 1GHz; unit : second) |
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$N_{i,j}$ : a norm of a polynomial, $\deg(N_i) = i$ with $j$ modular factor |
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\begin{center} |
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\begin{tabular}{|c||c|c|c|c|} \hline |
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& $N_{105,23}$ & $N_{120,20}$ & $N_{168,24}$ & $N_{210,54}$ \\ \hline |
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{\tc Asir} & {\tc 0.86} & {\tc 59} & {\tc 840} & {\tc hard} \\ \hline |
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Asir NormFactor & 1.6 & 2.2& 6.1& hard \\ \hline |
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%Singular& hard? & hard?& hard? & hard? \\ \hline |
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CoCoA 4 & 0.2 & 7.1 & 16 & 0.5 \\ \hline\hline |
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NTL-5.2 & 0.16 & 0.9 & 1.4 & 0.4 \\ \hline |
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\end{tabular} |
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\end{center} |
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|
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\underline{\itc Multivariate; over {\bf Q}} |
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|
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$W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\tt asir2000/lib/fctrdata} |
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\begin{center} |
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\begin{tabular}{|c||c|c|c|c|c|} \hline |
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& $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline |
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variables & 3 & 5 & 5 & 5 & 4 \\ \hline |
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monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline |
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{\tc Asir} & {\tc 0.2} & {\tc 4.7} & {\tc 14} & {\tc 17} & {\tc 0.4} \\ \hline |
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%Singular& $>$15min & --- & ---& ---& ---\\ \hline |
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CoCoA 4 & 5.2 & $>$15min & $>$15min & $>$15min & 117 \\ \hline\hline |
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Mathematica 4& 0.2 & 16 & 23 & 36 & 1.1 \\ \hline |
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Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline |
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\end{tabular} |
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\end{center} |
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|
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%--- : not tested |
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\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc Timing data --- DRL Groebner basis computation} |
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|
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\underline{\itc Over $GF(32003)$} |
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\begin{center} |
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\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline |
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& $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline |
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{\tc Asir $Buchberger$} & {\tc 31} & {\tc 1687} & {\tc 2.6} & {\tc 27} & {\tc 294} & {\tc 4309} & --- \\ \hline |
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Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline |
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CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021 & --- \\ \hline\hline |
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{\tc Asir $F_4$} & {\tc 5.3} & {\tc 129} & {\tc 0.5} & {\tc 4.5} & {\tc 31} & {\tc 273} & {\tc 2641} \\ \hline |
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FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline |
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\end{tabular} |
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\end{center} |
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|
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\underline{\itc Over {\bf Q}} |
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|
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\begin{center} |
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\begin{tabular}{|c||c|c|c|c|c|} \hline |
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& $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline |
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{\tc Asir $Buchberger$} & {\tc 389} & {\tc 594} & {\tc 29} & {\tc 299} & {\tc 34950} \\ \hline |
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Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline |
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CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline |
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{\tc Asir $F_4$} & {\tc 989} & {\tc 456} & {\tc 90} & {\tc 991} & {\tc 4939} \\ \hline |
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FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline |
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\end{tabular} |
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\end{center} |
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--- : not tested |
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\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc Summary of performance} |
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|
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\begin{itemize} |
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\item {\itc Factorizer} |
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|
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\begin{itemize} |
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\item Multivariate : reasonable performance |
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\item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00] |
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\end{itemize} |
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\item {\itc Groebner basis computation} |
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|
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\begin{itemize} |
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\item Buchberger |
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Singular shows nice perfomance |
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Trace lifting is efficient in some cases over {\bf Q} |
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\item $F_4$ |
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FGb is much faster than Risa/Asir |
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But we observe that {\it McKay} is computed efficiently by $F_4$ |
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\end{itemize} |
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\end{itemize} |
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\end{slide} |
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\begin{slide}{} |
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\fbox{\fbc What is the merit to use Risa/Asir?} |
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|
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\begin{itemize} |
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\item {\itc Total performance is not excellent, but not bad} |
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|
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\item {\itc A completely open system} |
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The whole source is available |
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\item {\itc It serves as a test bench to try new ideas} |
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Interactive debugger is very useful |
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\item {\itc Interface compliant to OpenXM RFC-100} |
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The interface is fully documented |
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\end{itemize} |
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\end{slide} |
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|
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%\begin{slide}{} |
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%\fbox{\fbc CMO = Serialized representation of mathematical object} |
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% |
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%\begin{itemize} |
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%\item primitive data |
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%\begin{eqnarray*} |
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%\mbox{Integer32} &:& ({\tt CMO\_INT32}, {\sl int32}\ \mbox{n}) \\ |
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%\mbox{Cstring}&:& ({\tt CMO\_STRING},{\sl int32}\, \mbox{ n}, {\sl string}\, \mbox{s}) \\ |
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%\mbox{List} &:& ({\tt CMO\_LIST}, {\sl int32}\, len, ob[0], \ldots,ob[m-1]) |
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%\end{eqnarray*} |
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% |
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%\item numbers and polynomials |
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%\begin{eqnarray*} |
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%\mbox{ZZ} &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots |
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%{\sl byte}\, \mbox{a[$|$f$|$]} ) \\ |
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%\mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, n, \mbox{e[1]}, \ldots, \mbox{e[n]}, \mbox{Coef}) \\ |
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%\mbox{Coef}&:& \mbox{ZZ} | \mbox{Integer32} \\ |
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%\mbox{Dpolynomial}&:& ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},\\ |
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% & & m, \mbox{DringDefinition}, \mbox{Monomial32}, \ldots)\\ |
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%\mbox{DringDefinition} |
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% &:& \mbox{DMS of N variables} \\ |
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% & & ({\tt CMO\_RING\_BY\_NAME}, name) \\ |
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% & & ({\tt CMO\_DMS\_GENERIC}) \\ |
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%\end{eqnarray*} |
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%\end{itemize} |
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%\end{slide} |
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% |
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%\begin{slide}{} |
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%\fbox{\fbc Stack based communication} |
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% |
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%\begin{itemize} |
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%\item Data arrived a client |
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% |
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%Pushed to the stack |
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% |
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%\item Result |
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% |
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%Pushd to the stack |
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% |
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%Written to the stream when requested by a command |
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% |
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%\item The reason why we use the stack |
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% |
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%\begin{itemize} |
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%\item Stack = I/O buffer for (possibly large) objects |
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% |
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%Multiple requests can be sent before their execution |
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% |
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%A server does not get stuck in sending results |
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%\end{itemize} |
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%\end{itemize} |
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%\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc OpenXM (Open message eXchange protocol for Mathematics) } |
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\begin{itemize} |
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\item {\itc An environment for parallel distributed computation} |
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Both for interactive, non-interactive environment |
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|
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\item {\itc OpenXM RFC-100 = Client-server architecture} |
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Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server |
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OX (OpenXM) message : command and data |
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|
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\item {\itc Data} |
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Encoding : CMO (Common Mathematical Object format) |
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Serialized representation of mathematical object |
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|
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--- Main idea was borrowed from OpenMath |
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({\urlc \tt http://www.openmath.org}) |
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|
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\item {\itc Command} |
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|
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stack machine command --- server is a stackmachine |
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|
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+ server's own command sequences --- hybrid server |
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\end{itemize} |
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\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc Example of distributed computation --- $F_4$ vs. $Buchberger$ } |
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|
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\begin{verbatim} |
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/* competitive Gbase computation over GF(M) */ |
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/* Cf. A.28 in SINGULAR Manual */ |
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/* Process list is specified as buch_vs_f4_mod(...|proc=P) */ |
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def buch_vs_f4_mod(G,V,M,O) |
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{ |
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P = getopt(proc); |
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if ( type(P) == -1 ) return dp_f4_mod_main(G,V,M,O); |
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P0 = P[0]; P1 = P[1]; P = [P0,P1]; |
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map(ox_reset,P); /* resets the both servers */ |
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ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O); /* for F4 */ |
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ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O); /* for Buchberger */ |
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map(ox_push_cmd,P,262); /* 262 = OX_popCMO */ |
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F = ox_select(P); /* waits a server to return the result */ |
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R = ox_get(F[0]); /* gets the result from the winner */ |
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if ( F[0] == P0 ) { Win = "F4"; Lose = P1;} |
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else { Win = "Buchberger"; Lose = P0; } |
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ox_reset(Lose); /* simply resets the loser */ |
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return [Win,R]; |
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} |
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\end{verbatim} |
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\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc Real speedup by parallelism --- polynomial multiplication} |
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|
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{\itc Product of dense univariate polynomials with 3000bit coefficients} |
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|
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{\itc Algorithm : FFT+Chinese remainder (by Shoup)} |
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|
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\epsfxsize=20cm |
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\epsffile{3k.ps} |
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|
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{\itc Communication cost} |
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|
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\begin{itemize} |
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\item $O(n{\color{red}\log L})$ with server-server communication (OX-RFC102) |
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\item $O(n{\color{red}L})$ without server-server communication (OX-RFC100) |
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\end{itemize} |
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($L$: number of processes, $n$: degree) |
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|
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\end{slide} |
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|
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\begin{slide}{} |
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\fbox{\fbc References} |
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|
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[BERN97] L. Bernardin, On square-free factorization of |
| |
multivariate polynomials over a finite field, Theoretical |
| |
Computer Science 187 (1997), 105-116. |
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|
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[FAUG99] J.C. Faug\`ere, |
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A new efficient algorithm for computing Groebner bases ($F_4$), |
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Journal of Pure and Applied Algebra (139) 1-3 (1999), 61-88. |
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|
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[GRAY98] S. Gray et al, |
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Design and Implementation of MP, A Protocol for Efficient Exchange of |
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Mathematical Expression, |
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J. Symb. Comp. {\bf 25} (1998), 213-238. |
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|
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[HOEI00] M. van Hoeij, Factoring polynomials and the knapsack problem, |
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to appear in Journal of Number Theory (2000). |
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|
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[LIAO01] W. Liao et al, |
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OMEI: An Open Mathematical Engine Interface, |
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Proc. ASCM2001 (2001), 82-91. |
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[NORO97] M. Noro, J. McKay, |
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\end{slide} |
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\end{document} |
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