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RCS file: /home/cvs/OpenXM/doc/Papers/Attic/dagb-noro.tex,v
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retrieving revision 1.7
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--- OpenXM/doc/Papers/Attic/dagb-noro.tex	2001/10/03 08:32:58	1.1
+++ OpenXM/doc/Papers/Attic/dagb-noro.tex	2001/10/10 06:32:10	1.7
@@ -1,291 +1,517 @@
-% $OpenXM$
+% $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.6 2001/10/09 11:44:43 noro Exp $
 \setlength{\parskip}{10pt}
 
 \begin{slide}{}
-\fbox{A computer algebra system Risa/Asir}
+\begin{center}
+\fbox{\large Part I : OpenXM and Risa/Asir --- overview and history}
+\end{center}
+\end{slide}
 
+\begin{slide}{}
+\fbox{Integration of mathematical software systems}
+
 \begin{itemize}
-\item Old style software for polynomial computation
+\item Data integration
 
 \begin{itemize}
-\item Domain specification is not necessary prior to computation
-\item automatic conversion of inputs into internal canonical forms
+\item OpenMath ({\tt http://www.openmath.org}) , MP [GRAY98]
 \end{itemize}
 
-\item User language with C-like syntax
+Standards for representing mathematical objects
 
+\item Control integration
+
 \begin{itemize}
-\item No type declaration of variables
-\item Builtin debugger for user programs
+\item MCP [WANG99], OMEI [LIAO01]
 \end{itemize}
 
-\item Open source
+Protocols for remote subroutine calls or session management
 
+\item Combination of two integrations
+
 \begin{itemize}
-\item Whole source tree is available via CVS
+\item MathLink, OpenMath+MCP, MP+MCP
+
+and OpenXM ({\tt http://www.openxm.org})
 \end{itemize}
 
-\item OpenXM interface
+Both are necessary for practical implementation
 
-\begin{itemize}
-\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}
+\fbox{OpenXM (Open message eXchange protocol for Mathematics) }
 
 \begin{itemize}
-\item Fundamental polynomial arithmetics
+\item An environment for parallel distributed computation
 
-\begin{itemize}
-\item Internal form of a polynomial : recursive representaion or distributed
-representation
-\end{itemize}
+Both for interactive, non-interactive environment
 
-\item Polynomial factorization
+\item Client-server architecture
 
-\begin{itemize}
-\item Univariate factorization over the rationals, algebraic number fields and various finite fields
+Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server
 
-\item Multivariate factorization over the rationals
+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}
 
-\item Groebner basis computation
+\begin{slide}{}
+\fbox{A computer algebra system Risa/Asir}
 
+({\tt http://www.math.kobe-u.ac.jp/Asir/asir.html})
+
 \begin{itemize}
-\item Buchberger and $F_4$ algorithm
+\item Traditional style software for polynomial computation
 
-\item Change of ordering/RUR of 0-dimensional ideals
+No domain specification, automatic expansion
 
-\item Primary ideal decomposition
+\item User language with C-like syntax
 
-\item Computation of $b$-function
-\end{itemize}
+C language without type declaration, with list processing
 
-\item PARI library interface 
+\item Builtin {\tt gdb}-like debugger for user programs
 
-\item Paralell distributed computation under OpenXM 
+\item Open source
+
+Whole source tree is available via CVS
+
+\item OpenXM interface
+
+\begin{itemize}
+\item Risa/Asir is a main client in OpenXM package.
+\item An OpenXM server {\tt ox\_asir}
+\item A library with OpenXM library interface {\tt libasir.a}
 \end{itemize}
+\end{itemize}
 \end{slide}
 
 \begin{slide}{}
-\fbox{History of development : ---1994}
+\fbox{Goal of developing Risa/Asir}
 
 \begin{itemize}
-\item --1989
+\item Efficient implementation in specific area
 
-Several subroutines were developed for a Prolog program.
+\begin{itemize}
+\item Polynomial factorization
 
-\item 1989--1992
+\item Groebner basis related computation
 
-\begin{itemize}
-\item Reconfigured as Risa/Asir with the parser and Boehm's conservative GC.
+Main target : coefficient swells in characteristic 0 cases
 
-\item Developed univariate and multivariate factorizers over the rationals.
+Main tool : modular method
 \end{itemize}
 
-\item 1992--1994
+\item Front-end or server of a general purpose math software
 
+We do not persist in self-containedness
+
 \begin{itemize}
-\item Started implementation of Groebner basis computation
 
-User language $\Rightarrow$ rewritten in C (by Murao) $\Rightarrow$
-trace lifting
+\item contains PARI library ({\tt http://www.parigp-home.de}) from the very beginning
 
-\item Univariate factorization over algebraic number fields
+\item also acts as a main client of OpenXM package
 
-Intensive use of successive extension, non-squarefree norms
+One can use various OpenXM servers
+
 \end{itemize}
-\end{itemize}
 
+\end{itemize}
 \end{slide}
 
 \begin{slide}{}
-\fbox{History of development : 1994-1996}
+\fbox{Capability for polynomial computation}
 
 \begin{itemize}
-\item Free distribution of binary versions
+\item Fundamental polynomial arithmetics
 
-\item Primary ideal decomposition
+recursive representation and distributed representation
 
+\item Polynomial factorization
+
 \begin{itemize}
-\item Shimoyama-Yokoyama algorithm
+\item Univariate : over {\bf Q}, algebraic number fields and finite fields
+
+\item Multivariate : over {\bf Q}
 \end{itemize}
 
-\item Improvement of Buchberger algorithm
+\item Groebner basis computation
 
 \begin{itemize}
-\item Trace lifting+homogenization
+\item Buchberger and $F_4$ [FAUG99] algorithm
 
-\item Omitting check by compatible prime
+\item Change of ordering/RUR [ROUI96] of 0-dimensional ideals
 
-\item Modular change of ordering, Modular RUR
+\item Primary ideal decomposition
 
-\item Noro met Faug\`ere at RISC-Linz and he mentioned $F_4$.
+\item Computation of $b$-function (in Weyl Algebra)
 \end{itemize}
 \end{itemize}
-
 \end{slide}
 
 \begin{slide}{}
-\fbox{History of development : 1996-1998}
+\fbox{History of development : Polynomial factorization}
 
 \begin{itemize}
-\item Distributed compuatation
+\item 1989
 
-\begin{itemize}
-\item A prototype of OpenXM
-\end{itemize}
+Start of Risa/Asir with Boehm's conservative GC 
 
-\item Improvement of Buchberger algorithm
+({\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc})
 
-\begin{itemize}
-\item Content reduction during nomal form computation
+\item 1989-1992
 
-\item Its parallelization by the above facility
+Univariate and multivariate factorizers over {\bf Q}
 
-\item Application : computation of odd order replicable functions
+\item 1992-1994
 
-Risa/Asir : it took 5days to compute a DRL basis ({\it McKay})
+Univariate factorization over algebraic number fields
 
-From Faug\`ere : computation of the DRL basis 53sec
-\end{itemize}
+Intensive use of successive extension, non-squarefree norms
 
+\item 1996-1998
 
-\item Univariate factorization over large finite fields
+Univariate factorization over large finite fields
 
-\begin{itemize}
-\item To implement Schoof-Elkies-Atkin algorithm 
+\item 2000-current 
 
-Counting rational points on elliptic curves --- not free
-
-But related functions are freely available
+Multivariate factorization over small finite fields (in progress)
 \end{itemize}
-\end{itemize}
-
 \end{slide}
 
 \begin{slide}{}
-\fbox{History of development : 1998-2000}
-\begin{itemize}
-\item OpenXM
+\fbox{History of development : Groebner basis}
 
 \begin{itemize}
-\item OpenXM specification was written by Noro and Takayama
+\item 1992-1994
 
-\item Functions for distributed computation were rewritten
-\end{itemize}
+User language $\Rightarrow$ C version; trace lifting [TRAV88]
 
-\item Risa/Asir on Windows
+\item 1994-1996
 
-\begin{itemize}
-\item Requirement from a company for which Noro worked
+Trace lifting with homogenization
 
-Written in Visual C++
-\end{itemize}
+Omitting GB check by compatible prime [NOYO99]
 
-\item Test implementation of $F_4$
+Modular change of ordering/RUR [NOYO99]
 
-\begin{itemize}
-\item Over $GF(p)$ : pretty good
+Primary ideal decomposition [SHYO96]
 
-\item Over the rationals : not so good except for {\it McKay}
+\item 1996-1998
+
+Efficient content reduction during NF computation [NORO97]
+Solved {\it McKay} system for the first time
+
+\item 1998-2000
+
+Test implementation of $F_4$
+
+\item 2000-current
+
+Buchberger algorithm in Weyl algebra [TAKA90]
+
+Efficient $b$-function computation by a modular method
 \end{itemize}
-\end{itemize}
 \end{slide}
 
 \begin{slide}{}
-\fbox{History of development : 2000-current}
-\begin{itemize}
-\item The source code is freely available
+\fbox{Performance --- Factorizer}
 
 \begin{itemize}
-\item Noro moved from Fujitsu to Kobe university.
+\item 4 years ago
 
-\item Fujitsu kindly permitted to make Risa/Asir open source.
-\end{itemize}
+Over {\bf Q} : fine compared with existing software
+like REDUCE, Mathematica, maple
 
-\item OpenXM
+Univariate, over algebraic number fields :
+fine because of some tricks for polynomials
+derived from norms.
 
-\begin{itemize}
-\item Revising the specification : OX-RFC100, 101, (102)
+\item Current
 
-\item OX-RFC102 : ommunications between servers via MPI
+Multivariate : moderate
+
+Univariate : completely obsoleted by M. van Hoeij's new algorithm
+[HOEI00]
 \end{itemize}
 
-\item Rings of differential operators
+\end{slide}
 
-\begin{itemize}
-\item Buchberger algorithm
+\begin{slide}{}
+\fbox{Timing data --- Factorization}
 
-\item $b$-function computation
+\underline{Univariate; over {\bf Q}}
 
-Minimal polynomial computation by modular method
-\end{itemize}
-\end{itemize}
+$N_i$ : a norm of a poly, $\deg(N_i) = i$
+\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}
 
-\end{slide}
+\underline{Multivariate; over {\bf Q}}
 
+$W_{i,j,k} = Wang[i]\cdot Wang[j]\cdot Wang[k]$ in {\tt asir2000/lib/fctrdata}
+\begin{center}
+\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& 0.2 	& 16 	& 23 & 36 & 1.1 \\ \hline
+\end{tabular}
+\end{center}
+
+--- : not tested
+\end{slide}
 \begin{slide}{}
-\fbox{Status of each component --- Factorizer}
+\fbox{Performance --- Groebner basis related computation}
 
 \begin{itemize}
-\item 10 years ago
+\item 7 years ago
 
-its performace was fine compared with existing software
-like REDUCE, Maple, Mathematica.
+Trace lifting : rather fine but coefficient swells often occur
 
+Homogenization+trace lifting : robust and fast in the above cases
+
 \item 4 years ago
 
-Univarate factorization over algebraic number fields was
-still fine because of some tricks on factoring polynomials
-derived from norms.
+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
 
-Multivariate : not so bad
+$F_4$ in FGb : much more efficient than $F_4$ in Risa/Asir
 
-Univariate : completely obsolete by M. van Hoeij's new algorithm
-\end{itemize}
+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}{}
-\fbox{Status of each component --- Groebner basis related functions}
+\fbox{Timing data --- DRL Groebner basis computation}
 
+\underline{Over $GF(32003)$}
+\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 & & 3.8 & 35 & 402 & & --- \\ \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}
+
+\underline{Over {\bf Q}}
+
+\begin{center}
+\begin{tabular}{|c||c|c|c|c|c|} \hline
+		& $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline
+Asir $Buchberger$ 	& 389 & 594 & 29 & 299 & 34950 \\ \hline
+Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline
+CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline
+Asir $F_4$ 	&  989 & 456 & 90 & 991 & 4939 \\ \hline
+FGb(estimated)	& 8 &11 & 0.6 & 5 & 10 \\ \hline
+\end{tabular}
+\end{center}
+--- : not tested
+\end{slide}
+\begin{slide}{}
+\fbox{How do we proceed?}
+
+\underline{Total performance : not excellent, but not so bad}
+
 \begin{itemize}
-\item 8 years ago
+\item Trying to improve our implementation
 
-The performace was poor with only the sugar strategy.
+This is very important as a motivation of further development
 
-\item 7 years ago
+\begin{itemize}
 
-Rather fine with trace lifting but Faug\`ere's (old)Gb was more
-efficient.  
+\item Computation of $b$-function
 
-Homogenization+trace lifting made it possible to compute
-wider range of Groebner bases.
+fast but not satisfactory
 
-\item 4 years ago
+$\Rightarrow$ Groebner basis computation in Weyl
+algebra should be improved
+\end{itemize}
 
-Modular RUR was comparable with Rouillier's implementation.
+\item Developing new OpenXM servers
 
-\item Current
+{ox\_NTL} for univariate factorization, 
 
-FGb seems much more efficient than our $F_4$ implementation.
+{ox\_???} for Groebner basis computation, etc.
 
-Singular's Groebner basis computation is also several times
-faster than Risa/Asir, because Singular seems to have efficient
-monomial and polynomial representation.
+$\Rightarrow$ Risa/Asir can be a front-end of efficient servers
 
 \end{itemize}
+
+\begin{center}
+\underline{In both cases, OpenXM interface is important}
+\end{center}
 \end{slide}
 
+
+%\begin{slide}{}
+%\fbox{CMO = Serialized representation of mathematical object}
+%
+%\begin{itemize}
+%\item primitive data
+%\begin{eqnarray*}
+%\mbox{Integer32} &:& ({\tt CMO\_INT32}, {\sl int32}\ \mbox{n}) \\
+%\mbox{Cstring}&:& ({\tt CMO\_STRING},{\sl int32}\,  \mbox{ n}, {\sl string}\, \mbox{s}) \\
+%\mbox{List} &:& ({\tt CMO\_LIST}, {\sl int32}\, len, ob[0], \ldots,ob[m-1])
+%\end{eqnarray*}
+%
+%\item numbers and polynomials
+%\begin{eqnarray*}
+%\mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots
+%{\sl byte}\, \mbox{a[$|$f$|$]} ) \\
+%\mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, n, \mbox{e[1]}, \ldots, \mbox{e[n]}, \mbox{Coef}) \\
+%\mbox{Coef}&:& \mbox{ZZ} | \mbox{Integer32} \\
+%\mbox{Dpolynomial}&:& ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},\\
+%                  & & m, \mbox{DringDefinition}, \mbox{Monomial32}, \ldots)\\
+%\mbox{DringDefinition}
+%                  &:& \mbox{DMS of N variables} \\
+%                  & & ({\tt CMO\_RING\_BY\_NAME}, name) \\
+%                  & & ({\tt CMO\_DMS\_GENERIC}) \\
+%\end{eqnarray*}
+%\end{itemize}
+%\end{slide}
+%
+%\begin{slide}{}
+%\fbox{Stack based communication}
+%
+%\begin{itemize}
+%\item Data arrived a client
+%
+%Pushed to the stack
+%
+%\item Result
+%
+%Pushd to the stack
+%
+%Written to the stream when requested by a command
+%
+%\item The reason why we use the stack
+%
+%\begin{itemize}
+%\item Stack = I/O buffer for (possibly large) objects
+%
+%Multiple requests can be sent before their execution
+%
+%A server does not get stuck in sending results
+%\end{itemize}
+%\end{itemize}
+%\end{slide}
+
 \begin{slide}{}
+\fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ }
+
+\begin{verbatim}
+/* competitive Gbase computation over GF(M) */
+/* Cf. A.28 in SINGULAR Manual */
+/* Process list is specified as an option : grvsf4(...|proc=P) */
+def grvsf4(G,V,M,O)
+{
+  P = getopt(proc);
+  if ( type(P) == -1 ) return dp_f4_mod_main(G,V,M,O);
+  P0 = P[0]; P1 = P[1]; P = [P0,P1];
+  map(ox_reset,P);
+  ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O);
+  ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O);
+  map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
+  F = ox_select(P); R = ox_get(F[0]);
+  if ( F[0] == P0 ) { Win = "F4"; Lose = P1;}
+  else { Win = "Buchberger"; Lose = P0; }
+  ox_reset(Lose); /* simply resets the loser */
+  return [Win,R];
+}
+\end{verbatim}
+\end{slide}
+
+\begin{slide}{}
+\fbox{References}
+
+[BERN97] L. Bernardin, On square-free factorization of 
+multivariate polynomials over a finite field, Theoretical
+Computer Science 187 (1997), 105-116.
+
+[FAUG99] J.C. Faug\`ere,
+A new efficient algorithm for computing Groebner bases  ($F_4$),
+Journal of Pure and Applied Algebra (139) 1-3 (1999), 61-88.
+
+[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 Heoij, Factoring polynomials and the knapsack problem,
+to appear in Journal of Number Theory (2000).
+
+[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}
+
+\begin{slide}{}
+
+[NOYO99] M. Noro, K. Yokoyama, 
+A Modular Method to Compute the Rational Univariate
+Representation of Zero-Dimensional Ideals.
+J. Symb. Comp. {\bf 28}/1 (1999), 243-263.
+
+[OAKU97] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic
+local cohomology groups of $D$-modules.
+Advances in Applied Mathematics, 19 (1997), 61-105.
+
+[ROUI96] F. Rouillier,
+R\'esolution des syst\`emes z\'ero-dimensionnels. 
+Doctoral Thesis(1996), University of Rennes I, France.
+
+[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}
+
+\begin{slide}{}
+\begin{center}
+\fbox{\large Part II : Algorithms and implementations in Risa/Asir}
+\end{center}
+\end{slide}
+
+\begin{slide}{}
 \fbox{Ground fields}
 
 \begin{itemize}
@@ -328,7 +554,7 @@ Berlekamp-Zassenhaus
 
 Trager's algorithm + some improvement
 
-\item Over finite fieds
+\item Over finite fields
 
 DDF + Cantor-Zassenhaus; FFT for large finite fields
 \end{itemize}
@@ -340,9 +566,11 @@ DDF + Cantor-Zassenhaus; FFT for large finite fields
 
 Classical EZ algorithm
 
-\item Over finite fieds
+\item Over small finite fields
 
-Modified Bernardin square free, bivariate Hensel
+Modified Bernardin's square free algorithm [BERN97],
+
+possibly Hensel lifting over extension fields
 \end{itemize}
 
 \end{itemize}
@@ -365,12 +593,12 @@ Guess of a groebner basis by detecting zero reduction 
 Homogenization+guess+dehomogenization+check
 \end{itemize}
 
-\item Rings of differential operators
+\item Weyl Algebra
 
 \begin{itemize}
 \item Groebner basis of a left ideal
 
-An efficient implementation of Leibniz rule
+Key : an efficient implementation of Leibniz rule
 \end{itemize}
 
 \end{itemize}
@@ -381,9 +609,9 @@ An efficient implementation of Leibniz rule
 \begin{itemize}
 \item Over small finite fields ($GF(p)$, $p < 2^{30}$)
 \begin{itemize}
-\item More efficient than Buchberger algorithm
+\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}
 
 \item Over the rationals
@@ -391,16 +619,18 @@ but less efficient than FGb by Faugere
 \begin{itemize}
 \item Very naive implementation
 
+Modular computation + CRT + Checking the result at each degree
+
 \item Less efficient than Buchberger algorithm
 
-except for one example
+except for one example (={\it McKay})
 \end{itemize}
 
 \end{itemize}
 \end{slide}
 
 \begin{slide}{}
-\fbox{Change of ordering for zero-dimimensional ideals}
+\fbox{Change of ordering for zero-dimensional ideals}
 
 \begin{itemize}
 \item Any ordering to lex ordering
@@ -448,7 +678,7 @@ An ideal whose radical is prime
 \begin{slide}{}
 \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.
 there exists $P(s) \in D[s]$, $P(s)f^{s+1}=b(s)f^s$
@@ -494,42 +724,10 @@ evaluated by {\tt eval()}
 
 The knapsack factorization is available via {\tt pari(factor,{\it poly})}
 \end{itemize}
-
-
 \end{itemize}
 \end{slide}
 
 \begin{slide}{}
-\fbox{OpenXM}
-
-\begin{itemize}
-\item An environment for parallel distributed computation
-
-Both for interactive, non-interactive environment
-
-\item Message passing
-
-OX (OpenXM) message : command and data
-
-\item Hybrid command execution
-
-\begin{itemize}
-\item Stack machine command
-
-push, pop, function execution, $\ldots$
-
-\item accepts its own command sequences
-
-{\tt execute\_string} --- easy to use
-\end{itemize}
-
-\item Data is represented as CMO
-
-CMO --- Common Mathematical Object format
-\end{itemize}
-\end{slide}
-
-\begin{slide}{}
 \fbox{OpenXM server interface in Risa/Asir}
 
 \begin{itemize}
@@ -544,14 +742,19 @@ The launcher launches a server on the same host.
 
 \item Server
 
-A server reads from the descriptor 3, write to the descriptor 4.
+Reads from the descriptor 3
 
+Writes to the descriptor 4
+
 \end{itemize}
 
 \item Subroutine call
 
-Risa/Asir subroutine library provides interfaces corresponding to
-pushing and popping data and executing stack commands.
+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}
 
@@ -565,77 +768,26 @@ Pushing and popping data, sending commands etc.
 
 \item Convenient functions
 
-Launching servers, calling remote functions,
- interrupting remote executions etc.
+Launching servers,
 
-\item Parallel distributed computation is easy
+Calling remote functions,
 
-Simple parallelization is practically important
+Resetting remote executions etc.
 
-Competitive computation is easily realized 
-\end{itemize}
-\end{slide}
+\item Parallel distributed computation
 
+Simple parallelization is practically important
 
-\begin{slide}{}
-\fbox{CMO = Serialized representation of mathematical object}
-
-\begin{itemize}
-\item primitive data
-\begin{eqnarray*}
-\mbox{Integer32} &:& ({\tt CMO\_INT32}, {\sl int32}\ \mbox{n}) \\
-\mbox{Cstring}&:& ({\tt CMO\_STRING},{\sl int32}\,  \mbox{ n}, {\sl string}\, \mbox{s}) \\
-\mbox{List} &:& ({\tt CMO\_LIST}, {\sl int32}\, len, ob[0], \ldots,ob[m-1])
-\end{eqnarray*}
-
-\item numbers and polynomials
-\begin{eqnarray*}
-\mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots
-{\sl byte}\, \mbox{a[$|$f$|$]} ) \\
-\mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, n, \mbox{e[1]}, \ldots, \mbox{e[n]}, \mbox{Coef}) \\
-\mbox{Coef}&:& \mbox{ZZ} | \mbox{Integer32} \\
-\mbox{Dpolynomial}&:& ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},\\
-                  & & m, \mbox{DringDefinition}, \mbox{Monomial32}, \ldots)\\
-\mbox{DringDefinition}
-                 &:& \mbox{DMS of N variables} \\
-		 & & ({\tt CMO\_RING\_BY\_NAME}, name) \\
-                 & & ({\tt CMO\_DMS\_GENERIC}) \\
-\end{eqnarray*}
+Competitive computation is easily realized ($\Rightarrow$ demo)
 \end{itemize}
 \end{slide}
 
 \begin{slide}{}
-\fbox{Stack based communication}
-
-\begin{itemize}
-\item Data arrived a client
-
-Pushed to the stack
-
-\item Result
-
-Pushd to the stack
-
-Written to the stream when requested by a command
-
-\item The reason why we use the stack
-
-\begin{itemize}
-\item Stack = I/O buffer for (possibly large) objects
-
-Multiple requests can be sent before their exection
-
-A server does not get stuck in sending results
-\end{itemize}
-\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 aruments is sent as {\sl Integer32}.
+\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.
@@ -651,7 +803,7 @@ conversion are necessary.
 \fbox{Executing functions on a server (II) --- {\tt SM\_executeString}}
 
 \begin{enumerate}
-\item (C $\rightarrow$ S) A character string represeting a request in a server's
+\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.
@@ -663,35 +815,168 @@ $\Rightarrow$ Communication may be slow, but the clien
 enough to read the result.
 \end{slide}
 
-\begin{slide}{}
-\fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ }
-
-\begin{verbatim}
-/* competitive Gbase computation over GF(M) */
-/* Cf. A.28 in SINGULAR Manual */
-/* Process list is specified as an option : grvsf4(...|proc=P) */
-def grvsf4(G,V,M,O)
-{
-  P = getopt(proc);
-  if ( type(P) == -1 ) return dp_f4_mod_main(G,V,M,O);
-  P0 = P[0]; P1 = P[1]; P = [P0,P1];
-  map(ox_reset,P);
-  ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O);
-  ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O);
-  map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
-  F = ox_select(P); R = ox_get(F[0]);
-  if ( F[0] == P0 ) { Win = "F4"; Lose = P1;}
-  else { Win = "Buchberger"; Lose = P0; }
-  ox_reset(Lose); /* simply resets the loser */
-  return [Win,R];
-}
-\end{verbatim}
-
-\end{slide}
-
-\begin{slide}{}
-\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}{}
 \end{slide}