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 % $OpenXM$  % $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.6 2001/10/09 11:44:43 noro Exp $
 \setlength{\parskip}{10pt}  \setlength{\parskip}{10pt}
   
 \begin{slide}{}  \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}  \begin{itemize}
 \item Old style software for polynomial computation  \item Data integration
   
 \begin{itemize}  \begin{itemize}
 \item Domain specification is not necessary prior to computation  \item OpenMath ({\tt http://www.openmath.org}) , MP [GRAY98]
 \item automatic conversion of inputs into internal canonical forms  
 \end{itemize}  \end{itemize}
   
 \item User language with C-like syntax  Standards for representing mathematical objects
   
   \item Control integration
   
 \begin{itemize}  \begin{itemize}
 \item No type declaration of variables  \item MCP [WANG99], OMEI [LIAO01]
 \item Builtin debugger for user programs  
 \end{itemize}  \end{itemize}
   
 \item Open source  Protocols for remote subroutine calls or session management
   
   \item Combination of two integrations
   
 \begin{itemize}  \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}  \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{itemize}  
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Major functionalities}  \fbox{OpenXM (Open message eXchange protocol for Mathematics) }
   
 \begin{itemize}  \begin{itemize}
 \item Fundamental polynomial arithmetics  \item An environment for parallel distributed computation
   
 \begin{itemize}  Both for interactive, non-interactive environment
 \item Internal form of a polynomial : recursive representaion or distributed  
 representation  
 \end{itemize}  
   
 \item Polynomial factorization  \item Client-server architecture
   
 \begin{itemize}  Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server
 \item Univariate factorization over the rationals, algebraic number fields and various finite fields  
   
 \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{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}  \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  C language without type declaration, with list processing
 \end{itemize}  
   
 \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{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 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}  Main target : coefficient swells in characteristic 0 cases
 \item Reconfigured as Risa/Asir with the parser and Boehm's conservative GC.  
   
 \item Developed univariate and multivariate factorizers over the rationals.  Main tool : modular method
 \end{itemize}  \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}  \begin{itemize}
 \item Started implementation of Groebner basis computation  
   
 User language $\Rightarrow$ rewritten in C (by Murao) $\Rightarrow$  \item contains PARI library ({\tt http://www.parigp-home.de}) from the very beginning
 trace lifting  
   
 \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{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{History of development : 1994-1996}  \fbox{Capability for polynomial computation}
   
 \begin{itemize}  \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}  \begin{itemize}
 \item Shimoyama-Yokoyama algorithm  \item Univariate : over {\bf Q}, algebraic number fields and finite fields
   
   \item Multivariate : over {\bf Q}
 \end{itemize}  \end{itemize}
   
 \item Improvement of Buchberger algorithm  \item Groebner basis computation
   
 \begin{itemize}  \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{itemize}  \end{itemize}
   
 \end{slide}  \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 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  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}  \item 2000-current
 \item To implement Schoof-Elkies-Atkin algorithm  
   
 Counting rational points on elliptic curves --- not free  Multivariate factorization over small finite fields (in progress)
   
 But related functions are freely available  
 \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
   
 \item Functions for distributed computation were rewritten  User language $\Rightarrow$ C version; trace lifting [TRAV88]
 \end{itemize}  
   
 \item Risa/Asir on Windows  \item 1994-1996
   
 \begin{itemize}  Trace lifting with homogenization
 \item Requirement from a company for which Noro worked  
   
 Written in Visual C++  Omitting GB check by compatible prime [NOYO99]
 \end{itemize}  
   
 \item Test implementation of $F_4$  Modular change of ordering/RUR [NOYO99]
   
 \begin{itemize}  Primary ideal decomposition [SHYO96]
 \item Over $GF(p)$ : pretty good  
   
 \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{itemize}  
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{History of development : 2000-current}  \fbox{Performance --- Factorizer}
 \begin{itemize}  
 \item The source code is freely available  
   
 \begin{itemize}  \begin{itemize}
 \item Noro moved from Fujitsu to Kobe university.  \item 4 years ago
   
 \item Fujitsu kindly permitted to make Risa/Asir open source.  Over {\bf Q} : fine compared with existing software
 \end{itemize}  like REDUCE, Mathematica, maple
   
 \item OpenXM  Univariate, over algebraic number fields :
   fine because of some tricks for polynomials
   derived from norms.
   
 \begin{itemize}  \item Current
 \item Revising the specification : OX-RFC100, 101, (102)  
   
 \item OX-RFC102 : ommunications between servers via MPI  Multivariate : moderate
   
   Univariate : completely obsoleted by M. van Hoeij's new algorithm
   [HOEI00]
 \end{itemize}  \end{itemize}
   
 \item Rings of differential operators  \end{slide}
   
 \begin{itemize}  \begin{slide}{}
 \item Buchberger algorithm  \fbox{Timing data --- Factorization}
   
 \item $b$-function computation  \underline{Univariate; over {\bf Q}}
   
 Minimal polynomial computation by modular method  $N_i$ : a norm of a poly, $\deg(N_i) = i$
 \end{itemize}  \begin{center}
 \end{itemize}  \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}{}  \begin{slide}{}
 \fbox{Status of each component --- Factorizer}  \fbox{Performance --- Groebner basis related computation}
   
 \begin{itemize}  \begin{itemize}
 \item 10 years ago  \item 7 years ago
   
 its performace was fine compared with existing software  Trace lifting : rather fine but coefficient swells often occur
 like REDUCE, Maple, Mathematica.  
   
   Homogenization+trace lifting : robust and fast in the above cases
   
 \item 4 years ago  \item 4 years ago
   
 Univarate factorization over algebraic number fields was  Modular RUR was comparable with Rouillier's implementation.
 still fine because of some tricks on factoring polynomials  
 derived from norms.  
   
   DRL basis of {\it McKay}:
   
   5 days on Risa/Asir, 53 seconds on Faug\`ere FGb
 \item Current  \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  Buchberger in Singular ({\tt http://www.singular.uni-kl.de})
 \end{itemize}  : faster than Risa/Asir
   
   
   $\Leftarrow$ efficient monomial and polynomial computation
   
   \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}
   
   \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}  \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  \item Computation of $b$-function
 efficient.  
   
 Homogenization+trace lifting made it possible to compute  fast but not satisfactory
 wider range of Groebner bases.  
   
 \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  $\Rightarrow$ Risa/Asir can be a front-end of efficient servers
 faster than Risa/Asir, because Singular seems to have efficient  
 monomial and polynomial representation.  
   
 \end{itemize}  \end{itemize}
   
   \begin{center}
   \underline{In both cases, OpenXM interface is important}
   \end{center}
 \end{slide}  \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}{}  \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}  \fbox{Ground fields}
   
 \begin{itemize}  \begin{itemize}
Line 328  Berlekamp-Zassenhaus
Line 554  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 340  DDF + Cantor-Zassenhaus; FFT for large finite fields
Line 566  DDF + Cantor-Zassenhaus; FFT for large finite fields
   
 Classical EZ algorithm  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}
   
 \end{itemize}  \end{itemize}
Line 365  Guess of a groebner basis by detecting zero reduction 
Line 593  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
   
 An efficient implementation of Leibniz rule  Key : an efficient implementation of Leibniz rule
 \end{itemize}  \end{itemize}
   
 \end{itemize}  \end{itemize}
Line 381  An efficient implementation of Leibniz rule
Line 609  An efficient implementation of Leibniz rule
 \begin{itemize}  \begin{itemize}
 \item Over small finite fields ($GF(p)$, $p < 2^{30}$)  \item Over small finite fields ($GF(p)$, $p < 2^{30}$)
 \begin{itemize}  \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}  \end{itemize}
   
 \item Over the rationals  \item Over the rationals
Line 391  but less efficient than FGb by Faugere
Line 619  but less efficient than FGb by Faugere
 \begin{itemize}  \begin{itemize}
 \item Very naive implementation  \item Very naive implementation
   
   Modular computation + CRT + Checking the result at each degree
   
 \item Less efficient than Buchberger algorithm  \item Less efficient than Buchberger algorithm
   
 except for one example  except for one example (={\it McKay})
 \end{itemize}  \end{itemize}
   
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{slide}{}  \begin{slide}{}
 \fbox{Change of ordering for zero-dimimensional ideals}  \fbox{Change of ordering for zero-dimensional ideals}
   
 \begin{itemize}  \begin{itemize}
 \item Any ordering to lex ordering  \item Any ordering to lex ordering
Line 448  An ideal whose radical is prime
Line 678  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 494  evaluated by {\tt eval()}
Line 724  evaluated by {\tt eval()}
   
 The knapsack factorization is available via {\tt pari(factor,{\it poly})}  The knapsack factorization is available via {\tt pari(factor,{\it poly})}
 \end{itemize}  \end{itemize}
   
   
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
 \begin{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}  \fbox{OpenXM server interface in Risa/Asir}
   
 \begin{itemize}  \begin{itemize}
Line 544  The launcher launches a server on the same host.
Line 742  The launcher launches a server on the same host.
   
 \item Server  \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}  \end{itemize}
   
 \item Subroutine call  \item Subroutine call
   
 Risa/Asir subroutine library provides interfaces corresponding to  In Risa/Asir subroutine library {\tt libasir.a}:
 pushing and popping data and executing stack commands.  
   OpenXM functionalities are implemented as function calls
   
   pushing and popping data, executing stack commands etc.
 \end{itemize}  \end{itemize}
 \end{slide}  \end{slide}
   
Line 565  Pushing and popping data, sending commands etc.
Line 768  Pushing and popping data, sending commands etc.
   
 \item Convenient functions  \item Convenient functions
   
 Launching servers, calling remote functions,  Launching servers,
  interrupting remote executions etc.  
   
 \item Parallel distributed computation is easy  Calling remote functions,
   
 Simple parallelization is practically important  Resetting remote executions etc.
   
 Competitive computation is easily realized  \item Parallel distributed computation
 \end{itemize}  
 \end{slide}  
   
   Simple parallelization is practically important
   
 \begin{slide}{}  Competitive computation is easily realized ($\Rightarrow$ demo)
 \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{itemize}
 \end{slide}  \end{slide}
   
 \begin{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}}  \fbox{Executing functions on a server (I) --- {\tt SM\_executeFunction}}
   
 \begin{enumerate}  \begin{enumerate}
 \item (C $\rightarrow$ S) Arguments are sent in binary encoded form.  \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 function name is sent as {\sl Cstring}.
 \item (C $\rightarrow$ S) A command {\tt SM\_executeFunction} is sent.  \item (C $\rightarrow$ S) A command {\tt SM\_executeFunction} is sent.
 \item The result is pushed to the stack.  \item The result is pushed to the stack.
Line 651  conversion are necessary.
Line 803  conversion are necessary.
 \fbox{Executing functions on a server (II) --- {\tt SM\_executeString}}  \fbox{Executing functions on a server (II) --- {\tt SM\_executeString}}
   
 \begin{enumerate}  \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}.  user language is sent as {\sl Cstring}.
 \item (C $\rightarrow$ S) A command {\tt SM\_executeString} is sent.  \item (C $\rightarrow$ S) A command {\tt SM\_executeString} is sent.
 \item The result is pushed to the stack.  \item The result is pushed to the stack.
Line 663  $\Rightarrow$ Communication may be slow, but the clien
Line 815  $\Rightarrow$ Communication may be slow, but the clien
 enough to read the result.  enough to read the result.
 \end{slide}  \end{slide}
   
 \begin{slide}{}  %\begin{slide}{}
 \fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ }  %\fbox{History of development : ---1994}
   %
 \begin{verbatim}  %\begin{itemize}
 /* competitive Gbase computation over GF(M) */  %\item --1989
 /* Cf. A.28 in SINGULAR Manual */  %
 /* Process list is specified as an option : grvsf4(...|proc=P) */  %Several subroutines were developed for a Prolog program.
 def grvsf4(G,V,M,O)  %
 {  %\item 1989--1992
   P = getopt(proc);  %
   if ( type(P) == -1 ) return dp_f4_mod_main(G,V,M,O);  %\begin{itemize}
   P0 = P[0]; P1 = P[1]; P = [P0,P1];  %\item Reconfigured as Risa/Asir with a parser and Boehm's conservative GC
   map(ox_reset,P);  %
   ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O);  %\item Developed univariate and multivariate factorizers over the rationals.
   ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O);  %\end{itemize}
   map(ox_push_cmd,P,262); /* 262 = OX_popCMO */  %
   F = ox_select(P); R = ox_get(F[0]);  %\item 1992--1994
   if ( F[0] == P0 ) { Win = "F4"; Lose = P1;}  %
   else { Win = "Buchberger"; Lose = P0; }  %\begin{itemize}
   ox_reset(Lose); /* simply resets the loser */  %\item Started implementation of Buchberger algorithm
   return [Win,R];  %
 }  %Written in user language $\Rightarrow$ rewritten in C (by Murao)
 \end{verbatim}  %
   %$\Rightarrow$ trace lifting [TRAV88]
 \end{slide}  %
   %\item Univariate factorization over algebraic number fields
 \begin{slide}{}  %
 \end{slide}  %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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