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version 1.2, 2001/10/04 04:12:29 version 1.10, 2001/10/11 09:44:56
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 % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.1 2001/10/03 08:32:58 noro Exp $  % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.9 2001/10/11 08:43:08 noro Exp $
 \setlength{\parskip}{10pt}  \setlength{\parskip}{10pt}
   
 \begin{slide}{}  \begin{slide}{}
 \begin{center}  \begin{center}
 \fbox{\large Part I : Overview and history of Risa/Asir}  \fbox{\large Part I : OpenXM and Risa/Asir --- overview and history}
 \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{A computer algebra system Risa/Asir}  \fbox{A computer algebra system Risa/Asir}
   
 \begin{itemize}  ({\tt http://www.math.kobe-u.ac.jp/Asir/asir.html})
 \item Old style software for polynomial computation  
   
 \begin{itemize}  \begin{itemize}
 \item Domain specification is not necessary prior to computation  \item Software mainly for polynomial computation
 \item automatic conversion of inputs into internal canonical forms  
 \end{itemize}  
   
 \item User language with C-like syntax  \item User language with C-like syntax
   
 \begin{itemize}  C language without type declaration, with list processing
 \item No type declaration of variables  
 \item Builtin debugger for user programs  
 \end{itemize}  
   
   \item Builtin {\tt gdb}-like debugger for user programs
   
 \item Open source  \item Open source
   
 \begin{itemize}  Whole source tree is available via CVS
 \item Whole source tree is available via CVS  
 \end{itemize}  
   
 \item OpenXM ((Open message eXchange protocol for Mathematics) interface  The latest version : see {\tt http://www.openxm.org}
   
 \begin{itemize}  \item OpenXM interface
 \item As a client : can call procedures on other OpenXM servers  
 \item As a server : offers all its functionalities to OpenXM clients  
 \item As a library : OpenXM functionality is available via subroutine calls  
 \end{itemize}  
 \end{itemize}  
 \end{slide}  
   
 \begin{slide}{}  
 \fbox{Major functionalities}  
   
 \begin{itemize}  \begin{itemize}
 \item Fundamental polynomial arithmetics  \item OpenXM
   
 \begin{itemize}  An infrastructure for exchanging mathematical data
 \item Internal form of a polynomial : recursive representaion or distributed  \item Risa/Asir is a main client in OpenXM package.
 representation  \item An OpenXM server {\tt ox\_asir}
   \item A library with OpenXM library interface {\tt libasir.a}
 \end{itemize}  \end{itemize}
   
 \item Polynomial factorization  
   
 \begin{itemize}  
 \item Univariate factorization over the rationals, algebraic number fields and various finite fields  
   
 \item Multivariate factorization over the rationals  
 \end{itemize}  \end{itemize}
   
 \item Groebner basis computation  
   
 \begin{itemize}  
 \item Buchberger and $F_4$ [Faug\'ere] algorithm  
   
 \item Change of ordering/RUR [Rouillier] of 0-dimensional ideals  
   
 \item Primary ideal decomposition  
   
 \item Computation of $b$-function  
 \end{itemize}  
   
 \item PARI [PARI] library interface  
   
 \item Paralell distributed computation under OpenXM  
 \end{itemize}  
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{History of development : ---1994}  \fbox{Goal of developing Risa/Asir}
   
 \begin{itemize}  \begin{itemize}
 \item --1989  \item Testing new algorithms
   
 Several subroutines were developed for a Prolog program.  
   
 \item 1989--1992  
   
 \begin{itemize}  \begin{itemize}
 \item Reconfigured as Risa/Asir with a parser and Boehm's conservative GC [Boehm]  \item Development started in Fujitsu labs
   
 \item Developed univariate and multivariate factorizers over the rationals.  Polynomial factorization, Groebner basis related computation,
   cryptosystems , quantifier elimination , $\ldots$
 \end{itemize}  \end{itemize}
   
 \item 1992--1994  \item To be a general purpose, open system
   
 \begin{itemize}  Since 1997, we have been developing OpenXM package
 \item Started implementation of Buchberger algorithm  containing various servers and clients
   
 Written in user language $\Rightarrow$ rewritten in C (by Murao)  Risa/Asir is a component of OpenXM
   
 $\Rightarrow$ trace lifting [Traverso]  \item Environment for parallel and distributed computation
   
 \item Univariate factorization over algebraic number fields  
   
 Intensive use of successive extension, non-squarefree norms  
 \end{itemize}  \end{itemize}
 \end{itemize}  
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  %\begin{slide}{}
 \fbox{History of development : 1994-1996}  %\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 Free distribution of binary versions from Fujitsu  
   
 \item Primary ideal decomposition  
   
 \begin{itemize}  
 \item Shimoyama-Yokoyama algorithm [SY]  
 \end{itemize}  
   
 \item Improvement of Buchberger algorithm  
   
 \begin{itemize}  
 \item Trace lifting+homogenization  
   
 \item Omitting check by compatible prime  
   
 \item Modular change of ordering, Modular RUR  
   
 These are joint works with Yokoyama [NY]  
 \end{itemize}  
 \end{itemize}  
   
 \end{slide}  
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{History of development : 1996-1998}  \fbox{History of development : Polynomial factorization}
   
 \begin{itemize}  \begin{itemize}
 \item Distributed compuatation  \item 1989
   
 \begin{itemize}  Start of Risa/Asir with Boehm's conservative GC
 \item A prototype of OpenXM  
 \end{itemize}  
   
 \item Improvement of Buchberger algorithm  ({\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc})
   
 \begin{itemize}  \item 1989-1992
 \item Content reduction during nomal form computation  
   
 \item Its parallelization by the above facility  Univariate and multivariate factorizers over {\bf Q}
   
 \item Computation of odd order replicable functions [Noro]  \item 1992-1994
   
 Risa/Asir : it took 5days to compute a DRL basis ({\it McKay})  Univariate factorization over algebraic number fields
   
 Faug\`ere FGb : computation of the DRL basis 53sec  Intensive use of successive extension, non-squarefree norms
 \end{itemize}  
   
   \item 1996-1998
   
 \item Univariate factorization over large finite fields  Univariate factorization over large finite fields
   
 \begin{itemize}  Motivated by a reseach project in Fujitsu on cryptography
 \item To implement Schoof-Elkies-Atkin algorithm  
   
 Counting rational points on elliptic curves  \item 2000-current
   
 --- not free But related functions are freely available  Multivariate factorization over small finite fields (in progress)
 \end{itemize}  \end{itemize}
 \end{itemize}  
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{History of development : 1998-2000}  \fbox{History of development : Groebner basis}
 \begin{itemize}  
 \item OpenXM  
   
 \begin{itemize}  \begin{itemize}
 \item OpenXM specification was written by Noro and Takayama  \item 1992-1994
   
 Borrowed idea on encoding, phrase book from OpenMath [OpenMath]  User language $\Rightarrow$ C version; trace lifting [TRAV88]
   
 \item Functions for distributed computation were rewritten  \item 1994-1996
 \end{itemize}  
   
 \item Risa/Asir on Windows  Trace lifting with homogenization
   
 \begin{itemize}  Omitting GB check by compatible prime [NOYO99]
 \item Requirement from a company for which Noro worked  
   
 Written in Visual C++  Modular change of ordering/RUR[ROUI96] [NOYO99]
 \end{itemize}  
   
 \item Test implementation of $F_4$  Primary ideal decomposition [SHYO96]
   
 \begin{itemize}  \item 1996-1998
 \item Implemented according to [Faug\`ere]  
   
 \item Over $GF(p)$ : pretty good  Efficient content reduction during NF computation [NORO97]
   Solved {\it McKay} system for the first time
   
 \item Over the rationals : not so good except for {\it McKay}  \item 1998-2000
 \end{itemize}  
 \end{itemize}  
 \end{slide}  
   
 \begin{slide}{}  Test implementation of $F_4$ [FAUG99]
 \fbox{History of development : 2000-current}  
 \begin{itemize}  
 \item The source code is freely available  
   
 \begin{itemize}  \item 2000-current
 \item Noro moved from Fujitsu to Kobe university  
   
 Started Kobe branch [Risa/Asir]  Buchberger algorithm in Weyl algebra
 \end{itemize}  
   
 \item OpenXM [OpenXM]  Efficient $b$-function computation[OAKU97] by a modular method
   
 \begin{itemize}  
 \item Revising the specification : OX-RFC100, 101, (102)  
   
 \item OX-RFC102 : communications between servers via MPI  
 \end{itemize}  \end{itemize}
   
 \item Rings of differential operators  
   
 \begin{itemize}  
 \item Buchberger algorithm [Takayama]  
   
 \item $b$-function computation [OT]  
   
 Minimal polynomial computation by modular method  
 \end{itemize}  
 \end{itemize}  
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Status of each component --- Factorizer}  \fbox{Timing data --- Factorization}
   
 \begin{itemize}  \underline{Univariate; over {\bf Q}}
 \item 10 years ago  
   
 its performace was fine compared with existing software  $N_i$ : a norm of a polynomial, $\deg(N_i) = i$
 like REDUCE, Mathematica.  \begin{center}
   \begin{tabular}{|c||c|c|c|c|} \hline
                   & $N_{105}$ & $N_{120}$ & $N_{168}$ & $N_{210}$ \\ \hline
   Asir    & 0.86  & 59 & 840 & hard \\ \hline
   Asir NormFactor & 1.6   & 2.2& 6.1& hard \\ \hline
   %Singular& hard?        & hard?& hard? & hard? \\ \hline
   CoCoA 4 & 0.2   & 7.1   & 16 & 0.5 \\ \hline\hline
   NTL-5.2 & 0.16  & 0.9   & 1.4 & 0.4 \\ \hline
   \end{tabular}
   \end{center}
   
 \item 4 years ago  \underline{Multivariate; over {\bf Q}}
   
 Univarate factorization over algebraic number fields was  $W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\tt asir2000/lib/fctrdata}
 still fine because of some tricks on factoring polynomials  \begin{center}
 derived from norms.  \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
   Asir    & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline
   %Singular& $>$15min     & ---   & ---& ---& ---\\ \hline
   CoCoA 4 & 5.2 & $>$15min        & $>$15min & $>$15min & 117 \\ \hline\hline
   Mathematica 4& 0.2      & 16    & 23 & 36 & 1.1 \\ \hline
   Maple 7& 0.5    & 18    & 967  & 48 & 1.3 \\ \hline
   \end{tabular}
   \end{center}
   
 \item Current  %--- : not tested
   
 Multivariate : not so bad  
   
 Univariate : completely obsolete by M. van Hoeij's new algorithm  
 [Hoeij]  
 \end{itemize}  
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Status of each component --- Groebner basis related functions}  \fbox{Timing data --- DRL Groebner basis computation}
   
 \begin{itemize}  \underline{Over $GF(32003)$}
 \item 8 years ago  \begin{center}
   \begin{tabular}{|c||c|c|c|c|c|c|c|} \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
   Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \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
   FGb(estimated)  & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline
   \end{tabular}
   \end{center}
   
 The performace was poor with only the sugar strategy.  \underline{Over {\bf Q}}
   
 \item 7 years ago  \begin{center}
   \begin{tabular}{|c||c|c|c|c|c|} \hline
 Rather fine with trace lifting but Faug\`ere's (old)Gb was more                  & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline
 efficient.  Asir $Buchberger$       & 389 & 594 & 29 & 299 & 34950 \\ \hline
   Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline
 Homogenization+trace lifting made it possible to compute  CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline
 wider range of Groebner bases.  Asir $F_4$      &  989 & 456 & 90 & 991 & 4939 \\ \hline
   FGb(estimated)  & 8 &11 & 0.6 & 5 & 10 \\ \hline
 \item 4 years ago  \end{tabular}
   \end{center}
 Modular RUR was comparable with Rouillier's implementation.  --- : not tested
   
 \item Current  
   
 FGb seems much more efficient than our $F_4$ implementation.  
   
 Singular's Groebner basis computation is also several times  
 faster than Risa/Asir, because Singular seems to have efficient  
 monomial and polynomial representation.  
   
 \end{itemize}  
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{OpenXM}  \fbox{Summary of performance}
   
 \begin{itemize}  \begin{itemize}
 \item An environment for parallel distributed computation  \item Factorizer
   
 Both for interactive, non-interactive environment  
   
 \item Message passing  
   
 OX (OpenXM) message : command and data  
   
 \item Hybrid command execution  
   
 \begin{itemize}  \begin{itemize}
 \item Stack machine command  \item Multivariate : reasonable performance
   
 push, pop, function execution, $\ldots$  \item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00]
   
 \item accepts its own command sequences  
   
 {\tt execute\_string} --- easy to use  
 \end{itemize}  \end{itemize}
   
 \item Data is represented as CMO  \item Groebner basis computation
   
 CMO (Common Mathematical Object format)  
   
 --- Serialized representation of mathematical object  
   
 {\sl Integer32}, {\sl Cstring}, {\sl List}, {\sl ZZ}, $\ldots$  
 \end{itemize}  
 \end{slide}  
   
   
 \begin{slide}{}  
 \fbox{OpenXM and OpenMath}  
   
 \begin{itemize}  \begin{itemize}
 \item OpenMath  \item Buchberger
   
 \begin{itemize}  Singular shows nice perfomance
 \item A standard for representing mathematical objects  
   
 \item CD (Content Dictionary) : assigns semantics to symbols  Trace lifting is efficient in some cases over {\bf Q}
   
 \item Phrasebook : convesion between internal and OpenMath objects.  \item $F_4$
   
 \item Encoding : format for actual data exchange  FGb is much faster than Risa/Asir
 \end{itemize}  
   
 \item OpenXM  But we observe that {\it McKay} is computed efficiently by $F_4$
   
 \begin{itemize}  
 \item Specification for encoding and exchanging messages  
   
 \item It also specifies behavior of servers and session management  
 \end{itemize}  \end{itemize}
   
 \end{itemize}  \end{itemize}
   
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{OpenXM server interface in Risa/Asir}  \fbox{What is the merit to use Risa/Asir?}
   
 \begin{itemize}  \begin{itemize}
 \item TCP/IP stream  \item Total performance is not excellent, but not bad
   
 \begin{itemize}  \item A completely open system
 \item Launcher  
   
 A client executes a launcher on a host.  The whole source is available
   
 The launcher launches a server on the same host.  \item Interface compliant to OpenXM RFC-100
   
 \item Server  The interface is fully documented
   
 A server reads from the descriptor 3, write to the descriptor 4.  \item It serves as a test bench to try new ideas
   
   Interactive debugger is very useful
 \end{itemize}  \end{itemize}
   
 \item Subroutine call  
   
 Risa/Asir subroutine library provides interfaces corresponding to  
 pushing and popping data and executing stack commands.  
 \end{itemize}  
 \end{slide}  \end{slide}
   
 \begin{slide}{}  
 \fbox{OpenXM client interface in Risa/Asir}  
   
 \begin{itemize}  
 \item Primitive interface functions  
   
 Pushing and popping data, sending commands etc.  
   
 \item Convenient functions  
   
 Launching servers, calling remote functions,  
  interrupting remote executions etc.  
   
 \item Parallel distributed computation is easy  
   
 Simple parallelization is practically important  
   
 Competitive computation is easily realized  
 \end{itemize}  
 \end{slide}  
   
   
 %\begin{slide}{}  %\begin{slide}{}
 %\fbox{CMO = Serialized representation of mathematical object}  %\fbox{CMO = Serialized representation of mathematical object}
 %  %
Line 453  Competitive computation is easily realized 
Line 357  Competitive computation is easily realized 
 %\begin{itemize}  %\begin{itemize}
 %\item Stack = I/O buffer for (possibly large) objects  %\item Stack = I/O buffer for (possibly large) objects
 %  %
 %Multiple requests can be sent before their exection  %Multiple requests can be sent before their execution
 %  %
 %A server does not get stuck in sending results  %A server does not get stuck in sending results
 %\end{itemize}  %\end{itemize}
Line 461  Competitive computation is easily realized 
Line 365  Competitive computation is easily realized 
 %\end{slide}  %\end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Executing functions on a server (I) --- {\tt SM\_executeFunction}}  \fbox{OpenXM (Open message eXchange protocol for Mathematics) }
   
 \begin{enumerate}  \begin{itemize}
 \item (C $\rightarrow$ S) Arguments are sent in binary encoded form.  \item An environment for parallel distributed computation
 \item (C $\rightarrow$ S) The number of aruments is sent as {\sl Integer32}.  
 \item (C $\rightarrow$ S) A function name is sent as {\sl Cstring}.  
 \item (C $\rightarrow$ S) A command {\tt SM\_executeFunction} is sent.  
 \item The result is pushed to the stack.  
 \item (C $\rightarrow$ S) A command {\tt SM\_popCMO} is sent.  
 \item (S $\rightarrow$ C) The result is sent in binary encoded form.  
 \end{enumerate}  
   
 $\Rightarrow$ Communication is fast, but functions for binary data  Both for interactive, non-interactive environment
 conversion are necessary.  
 \end{slide}  
   
 \begin{slide}{}  \item OpenXM RFC-100 = Client-server architecture
 \fbox{Executing functions on a server (II) --- {\tt SM\_executeString}}  
   
 \begin{enumerate}  Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server
 \item (C $\rightarrow$ S) A character string represeting a request in a server's  
 user language is sent as {\sl Cstring}.  
 \item (C $\rightarrow$ S) A command {\tt SM\_executeString} is sent.  
 \item The result is pushed to the stack.  
 \item (C $\rightarrow$ S) A command {\tt SM\_popString} is sent.  
 \item (S $\rightarrow$ C) The result is sent in readable form.  
 \end{enumerate}  
   
 $\Rightarrow$ Communication may be slow, but the client parser may be  OX (OpenXM) message : command and data
 enough to read the result.  
   \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}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
Line 521  def grvsf4(G,V,M,O)
Line 424  def grvsf4(G,V,M,O)
 \begin{slide}{}  \begin{slide}{}
 \fbox{References}  \fbox{References}
   
 [Bernardin] L. Bernardin, On square-free factorization of  [BERN97] L. Bernardin, On square-free factorization of
 multivariate polynomials over a finite field, Theoretical  multivariate polynomials over a finite field, Theoretical
 Computer Science 187 (1997), 105-116.  Computer Science 187 (1997), 105-116.
   
 [Boehm] {\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc}  [FAUG99] J.C. Faug\`ere,
   
 [Faug\`ere] J.C. Faug\`ere,  
 A new efficient algorithm for computing Groebner bases  ($F_4$),  A new efficient algorithm for computing Groebner bases  ($F_4$),
 Journal of Pure and Applied Algebra (139) 1-3 (1999), 61-88.  Journal of Pure and Applied Algebra (139) 1-3 (1999), 61-88.
   
 [Hoeij] M. van Heoij, Factoring polynomials and the knapsack problem,  [GRAY98] S. Gray et al,
   Design and Implementation of MP, A Protocol for Efficient Exchange of
   Mathematical Expression,
   J. Symb. Comp. {\bf 25} (1998), 213-238.
   
   [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).
   
 [SY] T. Shimoyama, K. Yokoyama, Localization and Primary Decomposition of Polynomial Ideals.  J. Symb. Comp. {\bf 22} (1996), 247-277.  [LIAO01] W. Liao et al,
   OMEI: An Open Mathematical Engine Interface,
   Proc. ASCM2001 (2001), 82-91.
   [NORO97] M. Noro, J. McKay,
   Computation of replicable functions on Risa/Asir.
   Proc. PASCO'97, ACM Press (1997), 130-138.
   \end{slide}
   
 [NY] M. Noro, K. Yokoyama,  \begin{slide}{}
   
   [NOYO99] M. Noro, K. Yokoyama,
 A Modular Method to Compute the Rational Univariate  A Modular Method to Compute the Rational Univariate
 Representation of Zero-Dimensional Ideals.  Representation of Zero-Dimensional Ideals.
 J. Symb. Comp. {\bf 28}/1 (1999), 243-263.  J. Symb. Comp. {\bf 28}/1 (1999), 243-263.
   
 [OpenMath] {\tt http://www.openmath.org}  [OAKU97] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic
   local cohomology groups of $D$-modules.
   Advances in Applied Mathematics, 19 (1997), 61-105.
   
 [OpenXM] {\tt http://www.openxm.org}  [ROUI96] F. Rouillier,
   
 [PARI] {\tt http://www.parigp-home.de}  
   
 [Risa/Asir] {\tt http://www.math.kobe-u.ac.jp/Asir/asir.html}  
   
 [Rouillier] F. Rouillier,  
 R\'esolution des syst\`emes z\'ero-dimensionnels.  R\'esolution des syst\`emes z\'ero-dimensionnels.
 Doctoral Thesis(1996), University of Rennes I, France.  Doctoral Thesis(1996), University of Rennes I, France.
   
 [Traverso] C. Traverso, \gr trace algorithms. Proc. ISSAC '88 (LNCS 358), 125-138.  [SHYO96] T. Shimoyama, K. Yokoyama, Localization and Primary Decomposition of Polynomial Ideals.  J. Symb. Comp. {\bf 22} (1996), 247-277.
   
   [TRAV88] C. Traverso, \gr trace algorithms. Proc. ISSAC '88 (LNCS 358), 125-138.
   
   [WANG99] P. S. Wang,
   Design and Protocol for Internet Accessible Mathematical Computation,
   Proc. ISSAC '99 (1999), 291-298.
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
Line 606  Berlekamp-Zassenhaus
Line 521  Berlekamp-Zassenhaus
   
 Trager's algorithm + some improvement  Trager's algorithm + some improvement
   
 \item Over finite fieds  \item Over finite fields
   
 DDF + Cantor-Zassenhaus; FFT for large finite fields  DDF + Cantor-Zassenhaus; FFT for large finite fields
 \end{itemize}  \end{itemize}
Line 618  DDF + Cantor-Zassenhaus; FFT for large finite fields
Line 533  DDF + Cantor-Zassenhaus; FFT for large finite fields
   
 Classical EZ algorithm  Classical EZ algorithm
   
 \item Over small finite fieds  \item Over small finite fields
   
 Modified Bernardin's square free algorithm [Bernardin],  Modified Bernardin's square free algorithm [BERN97],
   
 possibly Hensel lifting over extension fields  possibly Hensel lifting over extension fields
 \end{itemize}  \end{itemize}
Line 645  Guess of a groebner basis by detecting zero reduction 
Line 560  Guess of a groebner basis by detecting zero reduction 
 Homogenization+guess+dehomogenization+check  Homogenization+guess+dehomogenization+check
 \end{itemize}  \end{itemize}
   
 \item Rings of differential operators  \item Weyl Algebra
   
 \begin{itemize}  \begin{itemize}
 \item Groebner basis of a left ideal  \item Groebner basis of a left ideal
Line 663  Key : an efficient implementation of Leibniz rule
Line 578  Key : an efficient implementation of Leibniz rule
 \begin{itemize}  \begin{itemize}
 \item More efficient than our Buchberger algorithm implementation  \item More efficient than our Buchberger algorithm implementation
   
 but less efficient than FGb by Faugere  but less efficient than FGb by Faug\`ere
 \end{itemize}  \end{itemize}
   
 \item Over the rationals  \item Over the rationals
Line 730  An ideal whose radical is prime
Line 645  An ideal whose radical is prime
 \begin{slide}{}  \begin{slide}{}
 \fbox{Computation of $b$-function}  \fbox{Computation of $b$-function}
   
 $D$ : the ring of differential operators  $D=K\langle x,\partial \rangle$ : Weyl algebra
   
 $b(s)$ : a polynomial of the smallest degree s.t.  $b(s)$ : a polynomial of the smallest degree s.t.
 there exists $P(s) \in D[s]$, $P(s)f^{s+1}=b(s)f^s$  there exists $P(s) \in D[s]$, $P(s)f^{s+1}=b(s)f^s$
Line 779  The knapsack factorization is available via {\tt pari(
Line 694  The knapsack factorization is available via {\tt pari(
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
   \begin{slide}{}
   \fbox{OpenXM server interface in Risa/Asir}
   
   \begin{itemize}
   \item TCP/IP stream
   
   \begin{itemize}
   \item Launcher
   
   A client executes a launcher on a host.
   
   The launcher launches a server on the same host.
   
   \item Server
   
   Reads from the descriptor 3
   
   Writes to the descriptor 4
   
   \end{itemize}
   
   \item Subroutine call
   
   In Risa/Asir subroutine library {\tt libasir.a}:
   
   OpenXM functionalities are implemented as function calls
   
   pushing and popping data, executing stack commands etc.
   \end{itemize}
   \end{slide}
   
   \begin{slide}{}
   \fbox{OpenXM client interface in Risa/Asir}
   
   \begin{itemize}
   \item Primitive interface functions
   
   Pushing and popping data, sending commands etc.
   
   \item Convenient functions
   
   Launching servers,
   
   Calling remote functions,
   
   Resetting remote executions etc.
   
   \item Parallel distributed computation
   
   Simple parallelization is practically important
   
   Competitive computation is easily realized ($\Rightarrow$ demo)
   \end{itemize}
   \end{slide}
   
   \begin{slide}{}
   \fbox{Executing functions on a server (I) --- {\tt SM\_executeFunction}}
   
   \begin{enumerate}
   \item (C $\rightarrow$ S) Arguments are sent in binary encoded form.
   \item (C $\rightarrow$ S) The number of arguments is sent as {\sl Integer32}.
   \item (C $\rightarrow$ S) A function name is sent as {\sl Cstring}.
   \item (C $\rightarrow$ S) A command {\tt SM\_executeFunction} is sent.
   \item The result is pushed to the stack.
   \item (C $\rightarrow$ S) A command {\tt SM\_popCMO} is sent.
   \item (S $\rightarrow$ C) The result is sent in binary encoded form.
   \end{enumerate}
   
   $\Rightarrow$ Communication is fast, but functions for binary data
   conversion are necessary.
   \end{slide}
   
   \begin{slide}{}
   \fbox{Executing functions on a server (II) --- {\tt SM\_executeString}}
   
   \begin{enumerate}
   \item (C $\rightarrow$ S) A character string representing a request in a server's
   user language is sent as {\sl Cstring}.
   \item (C $\rightarrow$ S) A command {\tt SM\_executeString} is sent.
   \item The result is pushed to the stack.
   \item (C $\rightarrow$ S) A command {\tt SM\_popString} is sent.
   \item (S $\rightarrow$ C) The result is sent in readable form.
   \end{enumerate}
   
   $\Rightarrow$ Communication may be slow, but the client parser may be
   enough to read the result.
   \end{slide}
   
   %\begin{slide}{}
   %\fbox{History of development : ---1994}
   %
   %\begin{itemize}
   %\item --1989
   %
   %Several subroutines were developed for a Prolog program.
   %
   %\item 1989--1992
   %
   %\begin{itemize}
   %\item Reconfigured as Risa/Asir with a parser and Boehm's conservative GC
   %
   %\item Developed univariate and multivariate factorizers over the rationals.
   %\end{itemize}
   %
   %\item 1992--1994
   %
   %\begin{itemize}
   %\item Started implementation of Buchberger algorithm
   %
   %Written in user language $\Rightarrow$ rewritten in C (by Murao)
   %
   %$\Rightarrow$ trace lifting [TRAV88]
   %
   %\item Univariate factorization over algebraic number fields
   %
   %Intensive use of successive extension, non-squarefree norms
   %\end{itemize}
   %\end{itemize}
   %
   %\end{slide}
   %
   %\begin{slide}{}
   %\fbox{History of development : 1994-1996}
   %
   %\begin{itemize}
   %\item Free distribution of binary versions from Fujitsu
   %
   %\item Primary ideal decomposition
   %
   %\begin{itemize}
   %\item Shimoyama-Yokoyama algorithm [SHYO96]
   %\end{itemize}
   %
   %\item Improvement of Buchberger algorithm
   %
   %\begin{itemize}
   %\item Trace lifting+homogenization
   %
   %\item Omitting check by compatible prime
   %
   %\item Modular change of ordering, Modular RUR
   %
   %These are joint works with Yokoyama [NOYO99]
   %\end{itemize}
   %\end{itemize}
   %
   %\end{slide}
   %
   %\begin{slide}{}
   %\fbox{History of development : 1996-1998}
   %
   %\begin{itemize}
   %\item Distributed computation
   %
   %\begin{itemize}
   %\item A prototype of OpenXM
   %\end{itemize}
   %
   %\item Improvement of Buchberger algorithm
   %
   %\begin{itemize}
   %\item Content reduction during normal form computation
   %
   %\item Its parallelization by the above facility
   %
   %\item Computation of odd order replicable functions [NORO97]
   %
   %Risa/Asir : it took 5days to compute a DRL basis ({\it McKay})
   %
   %Faug\`ere FGb : computation of the DRL basis 53sec
   %\end{itemize}
   %
   %
   %\item Univariate factorization over large finite fields
   %
   %\begin{itemize}
   %\item To implement Schoof-Elkies-Atkin algorithm
   %
   %Counting rational points on elliptic curves
   %
   %--- not free But related functions are freely available
   %\end{itemize}
   %\end{itemize}
   %
   %\end{slide}
   %
   %\begin{slide}{}
   %\fbox{History of development : 1998-2000}
   %\begin{itemize}
   %\item OpenXM
   %
   %\begin{itemize}
   %\item OpenXM specification was written by Noro and Takayama
   %
   %Borrowed idea on encoding, phrase book from OpenMath
   %
   %\item Functions for distributed computation were rewritten
   %\end{itemize}
   %
   %\item Risa/Asir on Windows
   %
   %\begin{itemize}
   %\item Requirement from a company for which Noro worked
   %
   %Written in Visual C++
   %\end{itemize}
   %
   %\item Test implementation of $F_4$
   %
   %\begin{itemize}
   %\item Implemented according to [FAUG99]
   %
   %\item Over $GF(p)$ : pretty good
   %
   %\item Over the rationals : not so good except for {\it McKay}
   %\end{itemize}
   %\end{itemize}
   %\end{slide}
   %
   %\begin{slide}{}
   %\fbox{History of development : 2000-current}
   %\begin{itemize}
   %\item The source code is freely available
   %
   %\begin{itemize}
   %\item Noro moved from Fujitsu to Kobe university
   %
   %Started Kobe branch
   %\end{itemize}
   %
   %\item OpenXM
   %
   %\begin{itemize}
   %\item Revising the specification : OX-RFC100, 101, (102)
   %
   %\item OX-RFC102 : communications between servers via MPI
   %\end{itemize}
   %
   %\item Weyl algebra
   %
   %\begin{itemize}
   %\item Buchberger algorithm [TAKA90]
   %
   %\item $b$-function computation [OAKU97]
   %
   %Minimal polynomial computation by modular method
   %\end{itemize}
   %\end{itemize}
   %
   %\end{slide}
 \begin{slide}{}  \begin{slide}{}
 \end{slide}  \end{slide}
   
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