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\title{\tc A computer algebra system Risa/Asir and OpenXM}
\author{Masayuki Noro\\ Kobe University, Japan}
\begin{document}
\setlength{\parskip}{10pt}
\maketitle
%\begin{slide}{}
%\begin{center}
%\fbox{\fbc \large Part I : OpenXM and Risa/Asir --- overview and history}
%\end{center}
%\end{slide}
%\begin{slide}{}
%\fbox{\fbc Integration of mathematical software systems}
%
%\begin{itemize}
%\item Data integration
%
%\begin{itemize}
%\item OpenMath ({\urlc \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 ({\urlc \tt http://www.openxm.org})
%\end{itemize}
%
%Both are necessary for practical implementation
%
%\end{itemize}
%\end{slide}
\begin{slide}{}
\fbox{\fbc A computer algebra system Risa/Asir}
\begin{itemize}
\item {\itc Software mainly for polynomial computation}
Polynomial factorization, Groebner basis computation
\item {\itc User language with C-like syntax}
C language without type declaration, with list processing
\item {\itc Builtin {\tt gdb}-like debugger for user programs}
\item {\itc Open source} ({\urlc \tt http://www.math.kobe-u.ac.jp/Asir/asir.html})
The source and binaries are available via ftp or CVS
See {\urlc \tt http://www.openxm.org} to get the latest version
\item {\itc OpenXM interface}
\begin{itemize}
\item OpenXM ({\urlc \tt http://www.openxm.org})
An infrastructure for exchanging mathematical data
\item Risa/Asir is a main client in OpenXM package
\item {\tt ox\_asir} is an OpenXM server
\item {\tt libasir.a} provides OpenXM interface via function call
\end{itemize}
\end{itemize}
\end{slide}
\begin{slide}{}
\fbox{\fbc Goal of developing Risa/Asir}
\begin{itemize}
\item {\itc Testing new algorithms}
\begin{itemize}
\item Development started in Fujitsu labs
Polynomial factorization, Groebner basis related computation,
cryptosystems , quantifier elimination , $\ldots$
\end{itemize}
\item {\itc To be a general purpose, open system}
Since 1997, we have been developing OpenXM package
containing various servers and clients
Risa/Asir is a component of OpenXM
\item {\itc Environment for parallel and distributed computation}
\end{itemize}
\end{slide}
%\begin{slide}{}
%\fbox{\fbc 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{slide}{}
\fbox{\fbc History of development : Polynomial factorization}
\begin{itemize}
\item {\itc 1989}
Start of Risa/Asir with Boehm's conservative GC
({\urlc \tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc})
\item {\itc 1989-1992}
Univariate and multivariate factorizers over {\bf Q}
\item {\itc 1992-1994}
Univariate factorization over algebraic number fields
Intensive use of successive extension, non-squarefree norms
Application to splitting field and Galois group computation
\item {\itc 1996-1998}
Univariate factorization over large finite fields
Motivated by a reseach project in Fujitsu on cryptography
\item {\itc 2000-current}
Multivariate factorization over small finite fields (in progress)
\end{itemize}
\end{slide}
\begin{slide}{}
\fbox{\fbc History of development : Groebner basis}
\begin{itemize}
\item {\itc 1992-1994}
User language $\Rightarrow$ C version; trace lifting [TRAV88]
\item {\itc 1994-1996}
Trace lifting with homogenization
Omitting GB check by compatible prime [NOYO99]
Modular change of ordering/RUR[ROUI96] [NOYO99]
Primary ideal decomposition [SHYO96]
\item {\itc 1996-1998}
Efficient content reduction during NF computation [NORO97]
Solved {\it McKay} system for the first time
\item {\itc 1998-2000}
Test implementation of $F_4$ [FAUG99]
\item {\itc 2000-current}
Buchberger algorithm in Weyl algebra
Efficient $b$-function computation[OAKU97] by a modular method
\end{itemize}
\end{slide}
\begin{slide}{}
\fbox{\fbc Timing data --- Factorization}
\underline{\itc Univariate; over {\bf Q}} (on Pentium III, 1GHz; unit : second)
$N_{i,j}$ : a norm of a polynomial, $\deg(N_i) = i$ with $j$ modular factor
\begin{center}
\begin{tabular}{|c||c|c|c|c|} \hline
& $N_{105,23}$ & $N_{120,20}$ & $N_{168,24}$ & $N_{210,54}$ \\ \hline
{\tc Asir} & {\tc 0.86} & {\tc 59} & {\tc 840} & {\tc 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}
\underline{\itc 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
variables & 3 & 5 & 5 & 5 & 4 \\ \hline
monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline
{\tc Asir} & {\tc 0.2} & {\tc 4.7} & {\tc 14} & {\tc 17} & {\tc 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}
%--- : not tested
\end{slide}
\begin{slide}{}
\fbox{\fbc Timing data --- DRL Groebner basis computation}
\underline{\itc 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
{\tc Asir $Buchberger$} & {\tc 31} & {\tc 1687} & {\tc 2.6} & {\tc 27} & {\tc 294} & {\tc 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
{\tc Asir $F_4$} & {\tc 5.3} & {\tc 129} & {\tc 0.5} & {\tc 4.5} & {\tc 31} & {\tc 273} & {\tc 2641} \\ \hline
FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline
\end{tabular}
\end{center}
\underline{\itc 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
{\tc Asir $Buchberger$} & {\tc 389} & {\tc 594} & {\tc 29} & {\tc 299} & {\tc 34950} \\ \hline
Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline
CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline
{\tc Asir $F_4$} & {\tc 989} & {\tc 456} & {\tc 90} & {\tc 991} & {\tc 4939} \\ \hline
FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
\end{tabular}
\end{center}
--- : not tested
\end{slide}
\begin{slide}{}
\fbox{\fbc Summary of performance}
\begin{itemize}
\item {\itc Factorizer}
\begin{itemize}
\item Multivariate : reasonable performance
\item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00]
\end{itemize}
\item {\itc Groebner basis computation}
\begin{itemize}
\item Buchberger
Singular shows nice perfomance
Trace lifting is efficient in some cases over {\bf Q}
\item $F_4$
FGb is much faster than Risa/Asir
But we observe that {\it McKay} is computed efficiently by $F_4$
\end{itemize}
\end{itemize}
\end{slide}
\begin{slide}{}
\fbox{\fbc What is the merit to use Risa/Asir?}
\begin{itemize}
\item {\itc Total performance is not excellent, but not bad}
\item {\itc A completely open system}
The whole source is available
\item {\itc It serves as a test bench to try new ideas}
Interactive debugger is very useful
\item {\itc Interface compliant to OpenXM RFC-100}
The interface is fully documented
\end{itemize}
\end{slide}
%\begin{slide}{}
%\fbox{\fbc 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{\fbc 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{\fbc OpenXM (Open message eXchange protocol for Mathematics) }
\begin{itemize}
\item {\itc An environment for parallel distributed computation}
Both for interactive, non-interactive environment
\item {\itc OpenXM RFC-100 = Client-server architecture}
Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server
OX (OpenXM) message : command and data
\item {\itc Data}
Encoding : CMO (Common Mathematical Object format)
Serialized representation of mathematical object
--- Main idea was borrowed from OpenMath
({\urlc \tt http://www.openmath.org})
\item {\itc Command}
stack machine command --- server is a stackmachine
+ server's own command sequences --- hybrid server
\end{itemize}
\end{slide}
\begin{slide}{}
\fbox{\fbc 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 buch_vs_f4_mod(...|proc=P) */
def buch_vs_f4_mod(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); /* resets the both servers */
ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O); /* for F4 */
ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O); /* for Buchberger */
map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
F = ox_select(P); /* waits a server to return the result */
R = ox_get(F[0]); /* gets the result from the winner */
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{\fbc Real speedup by parallelism --- polynomial multiplication}
{\itc Product of dense univariate polynomials with 3000bit coefficients}
{\itc Algorithm : FFT+Chinese remainder (by Shoup)}
\epsfxsize=20cm
\epsffile{3k.ps}
{\itc Communication cost}
\begin{itemize}
\item $O(n{\color{red}\log L})$ with server-server communication (OX-RFC102)
\item $O(n{\color{red}L})$ without server-server communication (OX-RFC100)
\end{itemize}
($L$: number of processes, $n$: degree)
\end{slide}
\begin{slide}{}
\fbox{\fbc 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 Hoeij, 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}
\end{document}