=================================================================== RCS file: /home/cvs/OpenXM/doc/Papers/Attic/dag-noro.tex,v retrieving revision 1.2 retrieving revision 1.3 diff -u -p -r1.2 -r1.3 --- OpenXM/doc/Papers/Attic/dag-noro.tex 2001/10/10 06:32:10 1.2 +++ OpenXM/doc/Papers/Attic/dag-noro.tex 2001/10/12 02:22:17 1.3 @@ -1,18 +1,9 @@ -% $OpenXM: OpenXM/doc/Papers/dag-noro.tex,v 1.1 2001/10/03 08:32:58 noro Exp $ -\documentstyle[epsf]{slides} -\newtheorem{df}{Definition} -\newtheorem{pr}[df]{Proposition} -\newtheorem{lm}[df]{Lemma} -\newtheorem{th}[df]{Theorem} -\newtheorem{co}[df]{Corollary} -\newtheorem{al}[df]{Algorithm} -\newtheorem{re}[df]{Remark} -\newtheorem{ex}[df]{Example} -\newtheorem{mt}[df]{Method} -\newtheorem{nt}[df]{Notation} -\newtheorem{as}[df]{Assumption} -\newtheorem{pro}[df]{Procedure} -\newtheorem{prob}[df]{Probrem} +% $OpenXM: OpenXM/doc/Papers/dag-noro.tex,v 1.2 2001/10/10 06:32:10 noro Exp $ +\documentclass{slides} +\usepackage{color} +\usepackage{rgb} +\usepackage{graphicx} +\usepackage{epsfig} \newcommand{\qed}{$\Box$} \newcommand{\mred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}}} \newcommand{\tmred}[1]{\smash{\mathop{\hbox{\rightarrowfill}}\limits_{\scriptstyle #1}\limits^{\scriptstyle *}}} @@ -25,12 +16,481 @@ \columnsep 0.33in \topmargin -1in -\title{OpenXM and a computer algebra system Risa/Asir} +\title{A computer algebra system Risa/Asir and OpenXM} -\author{Masayuki Noro\\ Kobe University} +\author{Masayuki Noro\\ Kobe University, Japan} \begin{document} \setlength{\parskip}{10pt} \maketitle -\blackandwhite{dagb-noro.tex} -\end{document} +%\begin{slide}{} +%\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 Data integration +% +%\begin{itemize} +%\item OpenMath ({\tt http://www.openmath.org}) , MP [GRAY98] +%\end{itemize} +% +%Standards for representing mathematical objects +% +%\item Control integration +% +%\begin{itemize} +%\item MCP [WANG99], OMEI [LIAO01] +%\end{itemize} +% +%Protocols for remote subroutine calls or session management +% +%\item Combination of two integrations +% +%\begin{itemize} +%\item MathLink, OpenMath+MCP, MP+MCP +% +%and OpenXM ({\tt http://www.openxm.org}) +%\end{itemize} +% +%Both are necessary for practical implementation +% +%\end{itemize} +%\end{slide} +\begin{slide}{} +\fbox{A computer algebra system Risa/Asir} + +({\tt http://www.math.kobe-u.ac.jp/Asir/asir.html}) + +\begin{itemize} +\item Software mainly for polynomial computation + +\item User language with C-like syntax + +C language without type declaration, with list processing + +\item Builtin {\tt gdb}-like debugger for user programs + +\item Open source + +Whole source tree is available via CVS + +The latest version : see {\tt http://www.openxm.org} + +\item OpenXM interface + +\begin{itemize} +\item OpenXM + +An infrastructure for exchanging mathematical data +\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{Goal of developing Risa/Asir} + +\begin{itemize} +\item Testing new algorithms + +\begin{itemize} +\item Development started in Fujitsu labs + +Polynomial factorization, Groebner basis related computation, +cryptosystems , quantifier elimination , $\ldots$ +\end{itemize} + +\item 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 Environment for parallel and distributed computation + +\end{itemize} +\end{slide} + +%\begin{slide}{} +%\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{slide}{} +\fbox{History of development : Polynomial factorization} + +\begin{itemize} +\item 1989 + +Start of Risa/Asir with Boehm's conservative GC + +({\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc}) + +\item 1989-1992 + +Univariate and multivariate factorizers over {\bf Q} + +\item 1992-1994 + +Univariate factorization over algebraic number fields + +Intensive use of successive extension, non-squarefree norms + +\item 1996-1998 + +Univariate factorization over large finite fields + +Motivated by a reseach project in Fujitsu on cryptography + +\item 2000-current + +Multivariate factorization over small finite fields (in progress) +\end{itemize} +\end{slide} + +\begin{slide}{} +\fbox{History of development : Groebner basis} + +\begin{itemize} +\item 1992-1994 + +User language $\Rightarrow$ C version; trace lifting [TRAV88] + +\item 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 1996-1998 + +Efficient content reduction during NF computation [NORO97] +Solved {\it McKay} system for the first time + +\item 1998-2000 + +Test implementation of $F_4$ [FAUG99] + +\item 2000-current + +Buchberger algorithm in Weyl algebra + +Efficient $b$-function computation[OAKU97] by a modular method +\end{itemize} +\end{slide} + +\begin{slide}{} +\fbox{Timing data --- Factorization} + +\underline{Univariate; over {\bf Q}} + +$N_i$ : a norm of a polynomial, $\deg(N_i) = i$ +\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} + +\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 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{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 & $>$ 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} + +\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{Summary of performance} + +\begin{itemize} +\item Factorizer + +\begin{itemize} +\item Multivariate : reasonable performance + +\item Univariate : obsoleted by M. van Hoeij's new algorithm [HOEI00] +\end{itemize} + +\item 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{What is the merit to use Risa/Asir?} + +\begin{itemize} +\item Total performance is not excellent, but not bad + +\item A completely open system + +The whole source is available + +\item Interface compliant to OpenXM RFC-100 + +The interface is fully documented + +\item It serves as a test bench to try new ideas + +Interactive debugger is very useful +\end{itemize} + +\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{OpenXM (Open message eXchange protocol for Mathematics) } + +\begin{itemize} +\item An environment for parallel distributed computation + +Both for interactive, non-interactive environment + +\item OpenXM RFC-100 = Client-server architecture + +Client $\Leftarrow$ OX (OpenXM) message $\Rightarrow$ Server + +OX (OpenXM) message : command and data + +\item Data + +Encoding : CMO (Common Mathematical Object format) + +Serialized representation of mathematical object + +--- Main idea was borrowed from OpenMath + +({\tt http://www.openmath.org}) + +\item Command + +stack machine command --- server is a stackmachine + ++ server's own command sequences --- hybrid server +\end{itemize} +\end{slide} + +\begin{slide}{} +\fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ } + +\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 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}