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 \title{OpenXM and a computer algebra system Risa/Asir}  \title{\tc A computer algebra system Risa/Asir and OpenXM}
   
 \author{Masayuki Noro\\ Kobe University}  \author{Masayuki Noro\\ Kobe University, Japan}
 \begin{document}  \begin{document}
 \setlength{\parskip}{10pt}  \setlength{\parskip}{10pt}
 \maketitle  \maketitle
 \blackandwhite{dagb-noro.tex}  
 \end{document}  
   
   %\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}
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