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version 1.7, 2001/10/10 06:32:10 version 1.8, 2001/10/11 01:34:42
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 % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.6 2001/10/09 11:44:43 noro Exp $  % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.7 2001/10/10 06:32:10 noro Exp $
 \setlength{\parskip}{10pt}  \setlength{\parskip}{10pt}
   
 \begin{slide}{}  \begin{slide}{}
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 \end{center}  \end{center}
 \end{slide}  \end{slide}
   
   %\begin{slide}{}
   %\fbox{Integration of mathematical software systems}
   %
   %\begin{itemize}
   %\item Data integration
   %
   %\begin{itemize}
   %\item OpenMath ({\tt http://www.openmath.org}) , MP [GRAY98]
   %\end{itemize}
   %
   %Standards for representing mathematical objects
   %
   %\item Control integration
   %
   %\begin{itemize}
   %\item MCP [WANG99], OMEI [LIAO01]
   %\end{itemize}
   %
   %Protocols for remote subroutine calls or session management
   %
   %\item Combination of two integrations
   %
   %\begin{itemize}
   %\item MathLink, OpenMath+MCP, MP+MCP
   %
   %and OpenXM ({\tt http://www.openxm.org})
   %\end{itemize}
   %
   %Both are necessary for practical implementation
   %
   %\end{itemize}
   %\end{slide}
 \begin{slide}{}  \begin{slide}{}
 \fbox{Integration of mathematical software systems}  
   
 \begin{itemize}  
 \item Data integration  
   
 \begin{itemize}  
 \item OpenMath ({\tt http://www.openmath.org}) , MP [GRAY98]  
 \end{itemize}  
   
 Standards for representing mathematical objects  
   
 \item Control integration  
   
 \begin{itemize}  
 \item MCP [WANG99], OMEI [LIAO01]  
 \end{itemize}  
   
 Protocols for remote subroutine calls or session management  
   
 \item Combination of two integrations  
   
 \begin{itemize}  
 \item MathLink, OpenMath+MCP, MP+MCP  
   
 and OpenXM ({\tt http://www.openxm.org})  
 \end{itemize}  
   
 Both are necessary for practical implementation  
   
 \end{itemize}  
 \end{slide}  
 \begin{slide}{}  
 \fbox{OpenXM (Open message eXchange protocol for Mathematics) }  
   
 \begin{itemize}  
 \item An environment for parallel distributed computation  
   
 Both for interactive, non-interactive environment  
   
 \item Client-server architecture  
   
 Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server  
   
 OX (OpenXM) message : command and data  
   
 \item Data  
   
 Encoding : CMO (Common Mathematical Object format)  
   
 Serialized representation of mathematical object  
   
 --- Main idea was borrowed from OpenMath  
 \item Command  
   
 stack machine command --- server is a stackmachine  
   
 + server's own command sequences --- hybrid server  
 \end{itemize}  
 \end{slide}  
   
 \begin{slide}{}  
 \fbox{A computer algebra system Risa/Asir}  \fbox{A computer algebra system Risa/Asir}
   
 ({\tt http://www.math.kobe-u.ac.jp/Asir/asir.html})  ({\tt http://www.math.kobe-u.ac.jp/Asir/asir.html})
   
 \begin{itemize}  \begin{itemize}
 \item Traditional style software for polynomial computation  \item Software mainly for polynomial computation
   
 No domain specification, automatic expansion  
   
 \item User language with C-like syntax  \item User language with C-like syntax
   
 C language without type declaration, with list processing  C language without type declaration, with list processing
Line 88  C language without type declaration, with list process
Line 57  C language without type declaration, with list process
   
 Whole source tree is available via CVS  Whole source tree is available via CVS
   
   The latest version : see {\tt http://www.openxm.org}
   
 \item OpenXM interface  \item OpenXM interface
   
 \begin{itemize}  \begin{itemize}
   \item OpenXM
   
   An infrastructure for exchanging mathematical data
 \item Risa/Asir is a main client in OpenXM package.  \item Risa/Asir is a main client in OpenXM package.
 \item An OpenXM server {\tt ox\_asir}  \item An OpenXM server {\tt ox\_asir}
 \item A library with OpenXM library interface {\tt libasir.a}  \item A library with OpenXM library interface {\tt libasir.a}
Line 102  Whole source tree is available via CVS
Line 76  Whole source tree is available via CVS
 \fbox{Goal of developing Risa/Asir}  \fbox{Goal of developing Risa/Asir}
   
 \begin{itemize}  \begin{itemize}
 \item Efficient implementation in specific area  \item Testing new algorithms
   
 \begin{itemize}  \begin{itemize}
 \item Polynomial factorization  \item Development started in Fujitsu labs
   
 \item Groebner basis related computation  Polynomial factorization, Groebner basis related computation,
   cryptosystems , quantifier elimination , $\ldots$
 Main target : coefficient swells in characteristic 0 cases  
   
 Main tool : modular method  
 \end{itemize}  \end{itemize}
   
 \item Front-end or server of a general purpose math software  \item To be a general purpose, open system
   
 We do not persist in self-containedness  Since 1997, we have been developing OpenXM package
   containing various servers and clients
   
 \begin{itemize}  Risa/Asir is a component of OpenXM
   
 \item contains PARI library ({\tt http://www.parigp-home.de}) from the very beginning  \item Environment for parallel and distributed computation
   
 \item also acts as a main client of OpenXM package  
   
 One can use various OpenXM servers  
   
 \end{itemize}  \end{itemize}
   
 \end{itemize}  
 \end{slide}  \end{slide}
   
 \begin{slide}{}  %\begin{slide}{}
 \fbox{Capability for polynomial computation}  %\fbox{Capability for polynomial computation}
   %
   %\begin{itemize}
   %\item Fundamental polynomial arithmetics
   %
   %recursive representation and distributed representation
   %
   %\item Polynomial factorization
   %
   %\begin{itemize}
   %\item Univariate : over {\bf Q}, algebraic number fields and finite fields
   %
   %\item Multivariate : over {\bf Q}
   %\end{itemize}
   %
   %\item Groebner basis computation
   %
   %\begin{itemize}
   %\item Buchberger and $F_4$ [FAUG99] algorithm
   %
   %\item Change of ordering/RUR [ROUI96] of 0-dimensional ideals
   %
   %\item Primary ideal decomposition
   %
   %\item Computation of $b$-function (in Weyl Algebra)
   %\end{itemize}
   %\end{itemize}
   %\end{slide}
   
 \begin{itemize}  
 \item Fundamental polynomial arithmetics  
   
 recursive representation and distributed representation  
   
 \item Polynomial factorization  
   
 \begin{itemize}  
 \item Univariate : over {\bf Q}, algebraic number fields and finite fields  
   
 \item Multivariate : over {\bf Q}  
 \end{itemize}  
   
 \item Groebner basis computation  
   
 \begin{itemize}  
 \item Buchberger and $F_4$ [FAUG99] algorithm  
   
 \item Change of ordering/RUR [ROUI96] of 0-dimensional ideals  
   
 \item Primary ideal decomposition  
   
 \item Computation of $b$-function (in Weyl Algebra)  
 \end{itemize}  
 \end{itemize}  
 \end{slide}  
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{History of development : Polynomial factorization}  \fbox{History of development : Polynomial factorization}
   
Line 185  Intensive use of successive extension, non-squarefree 
Line 151  Intensive use of successive extension, non-squarefree 
   
 Univariate factorization over large finite fields  Univariate factorization over large finite fields
   
   Motivated by a reseach project in Fujitsu on cryptography
   
 \item 2000-current  \item 2000-current
   
 Multivariate factorization over small finite fields (in progress)  Multivariate factorization over small finite fields (in progress)
Line 205  Trace lifting with homogenization
Line 173  Trace lifting with homogenization
   
 Omitting GB check by compatible prime [NOYO99]  Omitting GB check by compatible prime [NOYO99]
   
 Modular change of ordering/RUR [NOYO99]  Modular change of ordering/RUR[ROUI96] [NOYO99]
   
 Primary ideal decomposition [SHYO96]  Primary ideal decomposition [SHYO96]
   
Line 216  Solved {\it McKay} system for the first time
Line 184  Solved {\it McKay} system for the first time
   
 \item 1998-2000  \item 1998-2000
   
 Test implementation of $F_4$  Test implementation of $F_4$ [FAUG99]
   
 \item 2000-current  \item 2000-current
   
 Buchberger algorithm in Weyl algebra [TAKA90]  Buchberger algorithm in Weyl algebra
   
 Efficient $b$-function computation by a modular method  Efficient $b$-function computation[OAKU97] by a modular method
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Performance --- Factorizer}  
   
 \begin{itemize}  
 \item 4 years ago  
   
 Over {\bf Q} : fine compared with existing software  
 like REDUCE, Mathematica, maple  
   
 Univariate, over algebraic number fields :  
 fine because of some tricks for polynomials  
 derived from norms.  
   
 \item Current  
   
 Multivariate : moderate  
   
 Univariate : completely obsoleted by M. van Hoeij's new algorithm  
 [HOEI00]  
 \end{itemize}  
   
 \end{slide}  
   
 \begin{slide}{}  
 \fbox{Timing data --- Factorization}  \fbox{Timing data --- Factorization}
   
 \underline{Univariate; over {\bf Q}}  \underline{Univariate; over {\bf Q}}
   
 $N_i$ : a norm of a poly, $\deg(N_i) = i$  $N_i$ : a norm of a polynomial, $\deg(N_i) = i$
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|} \hline
                 & $N_{105}$ & $N_{120}$ & $N_{168}$ & $N_{210}$ \\ \hline                  & $N_{105}$ & $N_{120}$ & $N_{168}$ & $N_{210}$ \\ \hline
 Asir    & 0.86  & 59 & 840 & hard \\ \hline  Asir    & 0.86  & 59 & 840 & hard \\ \hline
 Asir NormFactor & 1.6   & 2.2& 6.1& hard \\ \hline  Asir NormFactor & 1.6   & 2.2& 6.1& hard \\ \hline
 Singular& hard? & hard?& hard? & hard? \\ \hline  %Singular& hard?        & hard?& hard? & hard? \\ \hline
 CoCoA 4 & 0.2   & 7.1   & 16 & 0.5 \\ \hline\hline  CoCoA 4 & 0.2   & 7.1   & 16 & 0.5 \\ \hline\hline
 NTL-5.2 & 0.16  & 0.9   & 1.4 & 0.4 \\ \hline  NTL-5.2 & 0.16  & 0.9   & 1.4 & 0.4 \\ \hline
 \end{tabular}  \end{tabular}
Line 273  $W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\t
Line 218  $W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\t
 \begin{tabular}{|c||c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|} \hline
         & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline          & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline
 Asir    & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline  Asir    & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline
 Singular& $>$15min      & ---   & ---& ---& ---\\ \hline  %Singular& $>$15min     & ---   & ---& ---& ---\\ \hline
 CoCoA 4 & 5.2 & $>$15min        & $>$15min & $>$15min & 117 \\ \hline\hline  CoCoA 4 & 5.2 & $>$15min        & $>$15min & $>$15min & 117 \\ \hline\hline
 Mathematica& 0.2        & 16    & 23 & 36 & 1.1 \\ \hline  Mathematica 4& 0.2      & 16    & 23 & 36 & 1.1 \\ \hline
   Maple 7& 0.5    & 18    & 48  &  & 1.3 \\ \hline
 \end{tabular}  \end{tabular}
 \end{center}  \end{center}
   
 --- : not tested  %--- : not tested
 \end{slide}  \end{slide}
 \begin{slide}{}  
 \fbox{Performance --- Groebner basis related computation}  
   
 \begin{itemize}  
 \item 7 years ago  
   
 Trace lifting : rather fine but coefficient swells often occur  
   
 Homogenization+trace lifting : robust and fast in the above cases  
   
 \item 4 years ago  
   
 Modular RUR was comparable with Rouillier's implementation.  
   
 DRL basis of {\it McKay}:  
   
 5 days on Risa/Asir, 53 seconds on Faug\`ere FGb  
 \item Current  
   
 $F_4$ in FGb : much more efficient than $F_4$ in Risa/Asir  
   
 Buchberger in Singular ({\tt http://www.singular.uni-kl.de})  
 : faster than Risa/Asir  
   
   
 $\Leftarrow$ efficient monomial and polynomial computation  
   
 \end{itemize}  
 \end{slide}  
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Timing data --- DRL Groebner basis computation}  \fbox{Timing data --- DRL Groebner basis computation}
   
Line 320  $\Leftarrow$ efficient monomial and polynomial computa
Line 237  $\Leftarrow$ efficient monomial and polynomial computa
                 & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline                  & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline
 Asir $Buchberger$       & 31 & 1687  & 2.6  & 27 & 294  & 4309 & --- \\ \hline  Asir $Buchberger$       & 31 & 1687  & 2.6  & 27 & 294  & 4309 & --- \\ \hline
 Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline  Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline
 CoCoA 4 & 241 & & 3.8 & 35 & 402 & & --- \\ \hline\hline  CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021  & --- \\ \hline\hline
 Asir $F_4$      & 5.3 & 129 & 0.5  & 4.5 & 31  & 273 & 2641 \\ \hline  Asir $F_4$      & 5.3 & 129 & 0.5  & 4.5 & 31  & 273 & 2641 \\ \hline
 FGb(estimated)  & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline  FGb(estimated)  & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline
 \end{tabular}  \end{tabular}
Line 340  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
Line 257  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
 \end{center}  \end{center}
 --- : not tested  --- : not tested
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{How do we proceed?}  \fbox{Summary of performance}
   
 \underline{Total performance : not excellent, but not so bad}  \begin{itemize}
   \item Factorizer
   
 \begin{itemize}  \begin{itemize}
 \item Trying to improve our implementation  \item Multivariate : reasonable performance
   
 This is very important as a motivation of further development  \item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00]
   \end{itemize}
   
   \item Groebner basis computation
   
 \begin{itemize}  \begin{itemize}
   \item Buchberger
   
 \item Computation of $b$-function  Singular shows nice perfomance
   
 fast but not satisfactory  Trace lifting is efficient in some cases over {\bf Q}
   
 $\Rightarrow$ Groebner basis computation in Weyl  \item $F_4$
 algebra should be improved  
   FGb is much faster than Risa/Asir
   
   But we observe that {\it McKay} is computed efficiently by $F_4$
 \end{itemize}  \end{itemize}
   \end{itemize}
   
 \item Developing new OpenXM servers  \end{slide}
   
 {ox\_NTL} for univariate factorization,  \begin{slide}{}
   \fbox{Summary}
   
 {ox\_???} for Groebner basis computation, etc.  \begin{itemize}
   \item Total performance is not excellent, but not so bad
   
 $\Rightarrow$ Risa/Asir can be a front-end of efficient servers  \item A completely open system
   
   The whole source is available
   
   \item Interface compliant to OpenXM RFC-100
   
   The interface is fully documented
 \end{itemize}  \end{itemize}
   
 \begin{center}  
 \underline{In both cases, OpenXM interface is important}  
 \end{center}  
 \end{slide}  \end{slide}
   
   
Line 430  $\Rightarrow$ Risa/Asir can be a front-end of efficien
Line 361  $\Rightarrow$ Risa/Asir can be a front-end of efficien
 %\end{slide}  %\end{slide}
   
 \begin{slide}{}  \begin{slide}{}
   \fbox{OpenXM (Open message eXchange protocol for Mathematics) }
   
   \begin{itemize}
   \item An environment for parallel distributed computation
   
   Both for interactive, non-interactive environment
   
   \item OpenXM RFC-100 = Client-server architecture
   
   Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server
   
   OX (OpenXM) message : command and data
   
   \item Data
   
   Encoding : CMO (Common Mathematical Object format)
   
   Serialized representation of mathematical object
   
   --- Main idea was borrowed from OpenMath
   
   ({\tt http://www.openmath.org})
   
   \item Command
   
   stack machine command --- server is a stackmachine
   
   + server's own command sequences --- hybrid server
   \end{itemize}
   \end{slide}
   
   \begin{slide}{}
 \fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ }  \fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ }
   
 \begin{verbatim}  \begin{verbatim}
Line 470  Design and Implementation of MP, A Protocol for Effici
Line 433  Design and Implementation of MP, A Protocol for Effici
 Mathematical Expression,  Mathematical Expression,
 J. Symb. Comp. {\bf 25} (1998), 213-238.  J. Symb. Comp. {\bf 25} (1998), 213-238.
   
 [HOEI00] M. van Heoij, Factoring polynomials and the knapsack problem,  [HOEI00] M. van Hoeij, Factoring polynomials and the knapsack problem,
 to appear in Journal of Number Theory (2000).  to appear in Journal of Number Theory (2000).
   
 [LIAO01] W. Liao et al,  [LIAO01] W. Liao et al,
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