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1.4     ! noro        1: % $OpenXM: OpenXM/doc/Papers/dag-noro-proc.tex,v 1.3 2001/11/26 08:41:14 noro Exp $
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                     66: \begin{document}
                     67: %
                     68: \title*{A Computer Algebra System Risa/Asir and OpenXM}
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                     71: \toctitle{A Computer Algebra System Risa/Asir and OpenXM}
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                     75: \titlerunning{Risa/Asir and OpenXM}
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                     79: \author{Masayuki Noro\inst{1}}
                     80: %
                     81: %\authorrunning{Masayuki Noro}
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                     83: % please abbreviate author list for running head
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                     86: \institute{Kobe University, Rokko, Kobe 657-8501, Japan}
                     87:
                     88: \maketitle              % typesets the title of the contribution
                     89:
                     90: \begin{abstract}
                     91: OpenXM \cite{OPENXM} is an infrastructure for exchanging mathematical
                     92: data.  It defines a client-server architecture for parallel and
                     93: distributed computation.  Risa/Asir is software for polynomial
                     94: computation. It has been developed for testing new algorithms, and now
                     95: it acts as both a client and a server in the OpenXM package. In this
                     96: article we present an overview of Risa/Asir and its performances on
                     97: several functions.  We also show Risa/Asir's OpenXM interfaces and
                     98: examples of usages of them.
                     99: \end{abstract}
                    100:
                    101: \section{A computer algebra system Risa/Asir}
                    102:
                    103: \subsection{What is Risa/Asir?}
                    104:
                    105: Risa/Asir \cite{RISA} is software mainly for polynomial
                    106: computation. Its major functions are polynomial factorization and
                    107: Groebner basis computation, whose core parts are implemented as
                    108: builtin functions.  Some higher algorithms such as primary ideal
                    109: decomposition or Galois group computation are built on them by the
                    110: user language.  The user language is called Asir language. Asir
                    111: language can be regarded as C language without type declaration of
                    112: variables, with list processing, and with automatic garbage
                    113: collection. A builtin {\tt gdb}-like user language debugger is
                    114: available. It is open source and the source code and binaries are
                    115: available via ftp or CVS.
                    116: Risa/Asir is not only an standalone computer algebra system but also a
                    117: main component in OpenXM package \cite{OPENXM}, which is a collection
                    118: of software comliant to OpenXM protocol specification.  OpenXM is an
                    119: infrastructure for exchanging mathematical data and Risa/Asir has
                    120: three kind of OpenXM intefaces : an inteface as a server, as a cllient
                    121: and as a subroutine library. We will explain them in the later
                    122: section.
                    123:
                    124: Our goals of developing Risa/Asir are as follows:
                    125:
                    126: \begin{enumerate}
                    127: \item Providing a test bed of new algorithms
                    128:
                    129: Risa/Asir has been a platform for testing experimental algorithms in
                    130: polynomial factorization, computation related to Groebner basis,
                    131: cryptography and quantifier elimination. As to Groebner basis, we have
                    132: been mainly interested in problems over {\bf Q} and we tried applying
                    133: various modular techniques to overcome difficulties caused by huge
                    134: intermediate coefficients. We have had several results and they have
                    135: been implemented in Risa/Asir.
                    136:
                    137: \item Gereral purpose open system
                    138:
                    139: We need a lot of functions to make Risa/Asir a general purpose
                    140: computer algebra system.  In recent years we can obtain various high
                    141: performance applications or libraries as free software. We wrapped
                    142: such software as OpenXM servers and we started to release a collection
                    143: of such servers and cleints as OpenXM package in 1997. Risa/Asir is
                    144: now a main client in the package.
                    145:
                    146: \item Environment for parallel and distributed computation
                    147:
                    148: The origin of OpenXM is a protocol for doing parallel distributed
                    149: compuatations by connecting multiple Risa/Asir. OpenXM is also
                    150: designed to provide an enviroment efficient parallel distributed
                    151: computation. Currently only client-server communication is possible,
                    152: but we are preparing a specification OpenXM-RFC 102 allowing
                    153: client-client communication, which will enable us to execute
                    154: wider range of parallel algorithms efficiently.
                    155: \end{enumerate}
                    156:
                    157: \subsection{Groebner basis and the related computation}
                    158:
                    159: Currently Risa/Asir can only deal with polynomial ring. Operations on
                    160: modules over polynomial rings have not yet supported.  However, both
                    161: commutative polynomial rings and Weyl algebra are supported and one
                    162: can compute Groebner basis in both rings over the rationals, fields of
                    163: rational functions and finite fields. In the early stage of our
                    164: development, our effort was mainly devoted to improve the efficiency
                    165: of computation over the rationals. Our main tool is modular
                    166: computation. For Buchberger algorithm we adopted the trace lifting
                    167: algorithm by Traverso \cite{TRAV} and elaborated it by applying our
                    168: theory on a correspondence between Groebner basis and its modular
                    169: image \cite{NOYO}. We also combine the trace lifting with
                    170: homogenization to stabilize selection strategies, which enables us to
                    171: compute several examples efficiently which is hard to compute without
                    172: such a combination.  Our modular method can be applied to the change
                    173: of ordering algorithm and rational univariate representation.  We also
                    174: made a test implementation of $F_4$ algorithm \cite{F4}. Later we will
                    175: show timing data on Groebner basis computation.
                    176:
                    177: \subsection{Polynomial factorization}
                    178:
                    179: Here we briefly review functions on polynomial factorization.  For
                    180: univariate factorization over {\bf Q}, the classical
                    181: Berlekamp-Zassenhaus algorithm is implemented.  Efficient algorithms
                    182: recently proposed have not yet implemented.  For Univariate factorizer
                    183: over algebraic number fields, Trager's algorithm \cite{TRAGER} is
                    184: implemented with some modifications.  Its major applications are
                    185: splitting field and Galois group computation of polynomials over the
                    186: rationals. For such purpose a tower of simple extensions are suitable
                    187: because factors represented over a simple extension often have huge
                    188: coefficients \cite{ANY}.  For univariate factorization over finite
                    189: fields, equal degree factorization + Cantor-Zassenhaus algorithm is
                    190: implemented. We can use various representation of finite fields:
                    191: $GF(p)$ with a machine integer prime $p$, $GF(p)$, $GF(p^n)$ with any
                    192: odd prime $p$, $GF(2^n)$ with a bit representation of polynomials over
                    193: $GF(2)$ and $GF(p^n)$ with small $p^n$ represented by a primitive
                    194: root.  For multivariate factorization over the rationals, the
                    195: classical EZ(Extented Zassenhaus) type algorithm is implemented.
                    196:
                    197: \subsection{Other functions}
                    198: By applying Groebner basis computation and polynomial factorization,
                    199: we have implemented several higher level algorithms. A typical
                    200: application is primary ideal decomposition of polynomial ideals over
                    201: {\bf Q}, which needs both functions.  Shimoyama-Yokoyama algorithm
                    202: \cite{SY} for primary decompsition is written in the user language.
                    203: Splitting field and Galois group computation are closely related and
                    204: are also important applications of polynomial factorization.  Our
                    205: implementation of Galois group computation algorithm \cite{ANY}
                    206: requires splitting field computation, which is written in the
                    207: user language.
                    208:
                    209: \section{Techniques for efficient Groebner basis computation over {\bf Q}}
                    210: \label{gbtech}
                    211:
                    212: In this section we review several practical techniques to improve
                    213: Groebner basis computation over {\bf Q}, which are easily
                    214: implemented but may not be well known.
                    215: We use the following notations.
                    216: \begin{description}
                    217: \item $\phi_p$ : the canonical projection from ${\bf Z}$ onto $GF(p)$
                    218: \item $HT(f)$ : the head term of a polynomail with respect to a term order
                    219: \item $HC(f)$ : the head coefficient of a polynomail with respect to a term order
                    220: \end{description}
                    221:
                    222: \subsection{Combination of homogenization and trace lifting}
                    223:
                    224: Traverso's trace lifting algorithm can be
                    225: formulated in an abstract form as follows \cite{FPARA}.
                    226: \begin{tabbing}
                    227: Input : a finite subset $F \subset {\bf Z}[X]$\\
                    228: Output : a Groebner basis $G$ of $Id(F)$ with respect to a term order $<$\\
                    229: do \= \\
                    230: \> $p \leftarrow$ a new prime\\
                    231: \>Guess \= a Groebner basis candidate $G \subset Id(F)$
                    232: such that $\phi_p(G)$ \\
                    233: \>\> is a Groebner basis of $Id(\phi_p(F))$ in ${GF(p)}[X]$\\
                    234: \>Check that $G$ is a Groebner basis of $Id(G)$ and $F \subset Id(G)$\\
                    235: \>If $G$ passes the check return $G$\\
                    236: end do
                    237: \end{tabbing}
                    238: We can apply various methods for {\tt guess} part of the above
                    239: algorithm.  Originally we guess the candidate by replacing zero normal
                    240: form checks over {\bf Q} with those over $GF(p)$ in the Buchberger
                    241: algorithm, which we call {\it tl\_guess}. In Asir one can specify
                    242: another method {\it tl\_h\_guess\_dh}, which is a combination of
                    243: {\it tl\_guess} and homogenization.
                    244: \begin{tabbing}
                    245: $tl\_h\_guess\_dh(F,p)$\\
                    246: Input : $F\subset {\bf Z}[X]$, a prime $p$\\
                    247: Output : a Groebner basis candidate\\
                    248: $F_h \leftarrow$ the homogenization of $F$\\
                    249: $G_h \leftarrow tl\_guess(F_h,p)$ under an appropriate term order\\
                    250: $G \leftarrow$ the dehomogenization of $G_h$\\
                    251: $G \leftarrow G \setminus \{g \in G| \exists h \in G \setminus \{g\}$
                    252: such that $HT(h)|HT(g)$ \}
                    253: \end{tabbing}
                    254: The input is homogenized to suppress intermediate coefficient swells
                    255: of intermediate basis elements.  The number of zero normal forms may
                    256: increase by the homogenization, but they are detected over
                    257: GF(p). Finally, by dehomogenizing the candidate we can expect that
                    258: lots of redundant elements can be removed.  We will show later that this is
                    259: surely efficient for some input polynomial sets.
                    260:
                    261: \subsection{Minimal polynomial computation by modular method}
                    262: Let $I$ be a zero-dimensional ideal in $R={\bf Q}[x_1,\ldots,x_n]$.
                    263: Then the minimal polynomial $m(x_i)$ of a variable $x_i$ in $R/I$ can
                    264: be computed by a partial FGLM \cite{FGLM}, but it often takes long
                    265: time if one searches $m(x_i)$ incrementally over {\bf Q}.  In this
                    266: case we can apply a simple modular method to compute the minimal
                    267: polynomial.
                    268: \begin{tabbing}
                    269: Input : a Groebner basis $G$ of $I$, a variable $x_i$\\
                    270: Output : the minimal polynomial of $x$ in $R/I$\\
                    271: do \= \\
                    272: \> $p \leftarrow$ a new prime such that $p \not{|} HC(g)$ for all $g \in G$\\
                    273: \> $m_p \leftarrow$ the minimal polynomial of $x_i$ in $GF(p)[x_1,\ldots,x_n]/Id(\phi_p(G))$\\
                    274: \> If there exists $m(x_i) \in I$ such that $\phi_p(m) = m_p$ and $\deg(m)=\deg(m_p)$\\
                    275: \> then return $m(x_i)$\\
                    276: end do
                    277: \end{tabbing}
                    278: In this algorithm, $m_p$ can be obtained by a partial FGLM over
                    279: $GF(p)$ because $\phi_p(G)$ is a Groebner basis. Once we know the
                    280: candidate of $\deg(m(x_i))$, $m(x_i)$ can be determined by solving a
                    281: system of linear equations via the method of indeterminate
                    282: coefficient. Arguments on \cite{NOYO} ensures that $m(x_i)$ is what we
                    283: want if it exists. Note that the full FGLM can also be computed by the
                    284: same method.
                    285:
                    286: \subsection{Integer contents reduction}
                    287:
                    288: In some cases the cost to remove integer contents during nomal form
                    289: computations is dominant. We can remove the content of an integral
                    290: polynomial $f$ efficiently by the following method \cite{REPL}.
                    291: \begin{tabbing}
                    292: Input : an integral polynomial $f$\\
                    293: Output : a pair $(\cont(f),f/\cont(f))$\\
                    294: $g_0 \leftarrow$ an estimate of $\cont(f)$ such that $\cont(f)|g_0$\\
                    295: Write $f$ as $f = g_0q+r$ by division with remainder for each coefficient\\
                    296: If $r = 0$ then return $(g_0,q)$\\
                    297: else return $(g,g_0/g \cdot q + r/g)$, where $g = \GCD(g_0,\cont(r))$
                    298: \end{tabbing}
                    299: By serataing the set of coefficients of $f$ into two subsets and by
                    300: computing GCD of sums in the elements in the subsets we can estimate
                    301: $g_0$ with high accuracy. Then other components are easily computed.
                    302:
                    303: %\subsection{Demand loading of reducers}
                    304: %An execution of the Buchberer algorithm may produce vary large number
                    305: %of intermediate basis elements. In Asir, we can specify that such
                    306: %basis elements should be put on disk to enlarge free memory space.
                    307: %This does not reduce the efficiency so much because all basis elements
                    308: %are not necessarily used in a single normal form computation, and the
                    309: %cost for reading basis elements from disk is often negligible because
                    310: %of the cost for coefficient computations.
                    311:
                    312: \section{Risa/Asir performance}
                    313:
                    314: We show timing data on Risa/Asir for polynomial factorization
                    315: and Groebner basis computation. The measurements were made on
                    316: a PC with PentiumIII 1GHz and 1Gbyte of main memory. Timings
                    317: are given in seconds. In the tables `---' means it was not
                    318: measured.
                    319:
                    320: \subsection{Groebner basis computation}
                    321:
                    322: Table \ref{gbmod} and Table \ref{gbq} shows timing data for Groebner
                    323: basis compuation over $GF(32003)$ and over {\bf Q} respectively.
                    324: $C_n$ is the cyclic $n$ system and $K_n$ is the Katsura $n$ system,
                    325: both are famous bench mark problems.  We also measured the timing for
                    326: $McKay$ system over {\bf Q} \cite{REPL}.  the term order is graded
                    327: reverse lexicographic order.  In the both tables, the first three rows
                    328: are timings for the Buchberger algorithm, and the last two rows are
                    329: timings for $F_4$ algorithm. As to the Buchberger algorithm over
                    330: $GF(32003)$, Singular\cite{SINGULAR} shows the best performance among
                    331: the three systems. $F_4$ implementation in Risa/Asir is faster than
                    332: the Buchberger algorithm implementation in Singluar, but it is still
                    333: several times slower than $F_4$ implemenataion in FGb \cite{FGB}.  In
                    334: Table \ref{gbq}, $C_7$ and $McKay$ can be computed by the Buchberger
                    335: algorithm with the methods described in Section \ref{gbtech}.  It is
                    336: obvious that $F_4$ implementation in Risa/Asir over {\bf Q} is too
                    337: immature. Nevertheless the timing of $McKay$ is greatly reduced.
1.3       noro      338: Fig. \ref{f4vsbuch} explains why $F_4$ is efficient in this case.
                    339: The figure shows that
                    340: the Buchberger algorithm produces normal forms with
                    341: huge coefficients for S-polynomals after the 250-th one,
                    342: which are the computations in degree 16.
                    343: However, we know that the reduced basis elements have
                    344: much smaller coefficients after removing contents.
                    345: As $F_4$ algorithm automatically produces the reduced ones,
                    346: the degree 16 computation is quite easy in $F_4$.
1.1       noro      347:
                    348: \begin{table}[hbtp]
                    349: \begin{center}
                    350: \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
                    351:                & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline
                    352: Asir $Buchberger$      & 31 & 1687  & 2.6  & 27 & 294  & 4309 & --- \\ \hline
                    353: Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline
                    354: CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021  & --- \\ \hline\hline
                    355: Asir $F_4$     & 5.3 & 129 & 0.5  & 4.5 & 31  & 273 & 2641 \\ \hline
                    356: FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline
                    357: \end{tabular}
                    358: \end{center}
                    359: \caption{Groebner basis computation over $GF(32003)$}
                    360: \label{gbmod}
                    361: \end{table}
                    362:
                    363: \begin{table}[hbtp]
                    364: \begin{center}
                    365: \begin{tabular}{|c||c|c|c|c|c|} \hline
                    366:                & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline
                    367: Asir $Buchberger$      & 389 & 594 & 29 & 299 & 34950 \\ \hline
                    368: Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline
                    369: CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline
                    370: Asir $F_4$     &  989 & 456 & 90 & 991 & 4939 \\ \hline
                    371: FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
                    372: \end{tabular}
                    373: \end{center}
                    374: \caption{Groebner basis computation over {\bf Q}}
                    375: \label{gbq}
                    376: \end{table}
                    377:
                    378: \begin{figure}[hbtp]
                    379: \begin{center}
                    380: \epsfxsize=12cm
1.4     ! noro      381: \epsffile{../compalg/ps/blenall.ps}
1.1       noro      382: \end{center}
                    383: \caption{Maximal coefficient bit length of intermediate bases}
                    384: \label{f4vsbuch}
                    385: \end{figure}
                    386:
                    387: \subsection{Polynomial factorization}
                    388:
1.3       noro      389: %Table \ref{unifac} shows timing data for univariate factorization over
                    390: %{\bf Q}.  $N_{i,j}$ is an irreducible polynomial which are hard to
                    391: %factor by the classical algorithm. $N_{i,j}$ is a norm of a polynomial
                    392: %and $\deg(N_i) = i$ with $j$ modular factors. Risa/Asir is
                    393: %disadvantageous in factoring polynomials of this type because the
                    394: %algorithm used in Risa/Asir has exponential complexity. In contrast,
                    395: %CoCoA 4\cite{COCOA} and NTL-5.2\cite{NTL} show nice performances
                    396: %because they implement recently developed algorithms.
                    397: %
                    398: %\begin{table}[hbtp]
                    399: %\begin{center}
                    400: %\begin{tabular}{|c||c|c|c|c|} \hline
                    401: %              & $N_{105,23}$ & $N_{120,20}$ & $N_{168,24}$ & $N_{210,54}$ \\ \hline
                    402: %Asir  & 0.86  & 59 & 840 & hard \\ \hline
                    403: %Asir NormFactor & 1.6         & 2.2& 6.1& hard \\ \hline
                    404: %%Singular& hard?      & hard?& hard? & hard? \\ \hline
                    405: %CoCoA 4 & 0.2         & 7.1   & 16 & 0.5 \\ \hline\hline
                    406: %NTL-5.2       & 0.16  & 0.9   & 1.4 & 0.4 \\ \hline
                    407: %\end{tabular}
                    408: %\end{center}
                    409: %\caption{Univariate factorization over {\bf Q}}
                    410: %\label{unifac}
                    411: %\end{table}
1.1       noro      412:
                    413: Table \ref{multifac} shows timing data for multivariate
                    414: factorization over {\bf Q}.
                    415: $W_{i,j,k}$ is a product of three multivariate polynomials
                    416: $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file
                    417: {\tt fctrdata} in Asir library directory. It is also included
                    418: in Risa/Asir source tree and located in {\tt asir2000/lib}.
                    419: For these examples Risa/Asir shows reasonable performance
                    420: compared with other famous systems.
                    421:
                    422: \begin{table}[hbtp]
                    423: \begin{center}
                    424: \begin{tabular}{|c||c|c|c|c|c|} \hline
                    425:        & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline
                    426: variables & 3 & 5 & 5 & 5 & 4 \\ \hline
                    427: monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline
                    428: Asir   & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline
                    429: %Singular& $>$15min    & ---   & ---& ---& ---\\ \hline
                    430: CoCoA 4 & 5.2 & $>$15min       & $>$15min & $>$15min & 117 \\ \hline\hline
                    431: Mathematica 4& 0.2     & 16    & 23 & 36 & 1.1 \\ \hline
                    432: Maple 7& 0.5   & 18    & 967  & 48 & 1.3 \\ \hline
                    433: \end{tabular}
                    434: \end{center}
                    435: \caption{Multivariate factorization over {\bf Q}}
                    436: \label{multifac}
                    437: \end{table}
                    438:
1.3       noro      439: As to univariate factorization over {\bf Q},
                    440: the univariate factorizer implements only classical
                    441: algorithms and its behaviour is what one expects,
                    442: that is, it shows average performance in cases
                    443: where there are little eraneous factors, but
                    444: shows poor performance for hard to factor polynomials.
                    445:
1.1       noro      446: \section{OpenXM and Risa/Asir OpenXM interfaces}
                    447:
                    448: \subsection{OpenXM overview}
                    449:
                    450: OpenXM stands for Open message eXchange protocol for Mathematics.
                    451: Form the viewpoint of protocol design, it is a child of OpenMath
                    452: \cite{OPENMATH}.  However our approch is somewhat different. Our main
                    453: purpose is to provide an environment for integrating {\it existing}
                    454: mathematical software systems. OpenXM RFC-100 \cite{RFC100} defines a
                    455: client-server architecture.  Under this specification, a client
                    456: invokes an OpenXM (OX) server.  The client can send OpenXM (OX)
                    457: messages to the server.  OX messages consist of {\it data} and {\it
                    458: command}. Data is encoded according to the common mathematical object
                    459: (CMO) format which defines serialized representation of mathematical
                    460: objects.  An OX server is a stackmachine. If data is sent as an OX
                    461: message, the server pushes the data onto its stack. There is a common
                    462: set of stackmachine commands and all OX server understands its subset.
                    463: The command set includes commands for manipulating the stack and
                    464: requests for execution of a procedure. In addition, a server may
                    465: accept its own command sequences if the server wraps some interactive
                    466: software. That is the server may be a hybrid server.
                    467:
                    468: OpenXM RFC-100 also defines methods for session management. In particular
                    469: the method to reset a server is carefully designed and it provides
                    470: a robust way of using servers both for interactive and non-interactive
                    471: purposes.
                    472:
                    473: \subsection{OpenXM client interface of {\tt asir}}
                    474:
                    475: Risa/Asir is a main client in OpenXM package.  The application {\tt
                    476: asir} can access to OpenXM servers via several builtin interface
                    477: functions. and various inferfaces to existing OpenXM servers are
                    478: prepared as user defined functions written in Asir language.  We show
                    479: a typical OpenXM session.
                    480:
                    481: \begin{verbatim}
                    482: [1] P = ox_launch();  /* invoke an OpenXM asir server */
                    483: 0
                    484: [2] ox_push_cmo(P,x^10-y^10);
                    485: /* push a polynomial onto the stack */
                    486: 0
                    487: [3] ox_execute_function(P,"fctr",1);  /* call factorizer */
                    488: 0
                    489: [4] ox_pop_cmo(P);  /* get the result from the stack */
                    490: [[1,1],[x^4+y*x^3+y^2*x^2+y^3*x+y^4,1],
                    491: [x^4-y*x^3+y^2*x^2-y^3*x+y^4,1],[x-y,1],[x+y,1]]
                    492: [5] ox_cmo_rpc(P,"fctr,",x^10000-2^10000*y^10000);
                    493: /* call factorizer; an utility function */
                    494: 0
                    495: [6] ox_reset(P); /* reset the computation in the server */
                    496: 1
                    497: [7] ox_shutdown(P); /* shutdown the server */
                    498: 0
                    499: \end{verbatim}
                    500:
                    501: \subsection{OpenXM server {\tt ox\_asir}}
                    502:
                    503: An application {\tt ox\_asir} is a wrapper of {\tt asir} and provides
                    504: all the functions of {\tt asir} to OpenXM clients. It completely
                    505: implements the OpenXM reset protocol and also provides remote
                    506: debugging of user programs running on the server. We show a program
                    507: for checking whether a polynomial set is a Groebner basis or not. A
                    508: client executes {\tt gbcheck()} and servers execute {\tt
                    509: sp\_nf\_for\_gbcheck()} which is a simple normal form computation of a
                    510: S-polynomial. First of all the client collects all critical pairs
                    511: necessary for the check. Then the client requests normal form
                    512: computations to idling servers. If there are no idling servers the
                    513: clients waits for some servers to return results by {\tt
                    514: ox\_select()}, which is a wrapper of UNIX {\tt select()}. If we have
                    515: large number of critcal pairs to be processed, we can expect
                    516: good load balancing by {\tt ox\_select()}.
                    517:
                    518: \begin{verbatim}
                    519: def gbcheck(B,V,O,Procs) {
                    520:   map(ox_reset,Procs);
                    521:   dp_ord(O); D = map(dp_ptod,B,V);
                    522:   L = dp_gr_checklist(D); DP = L[0]; Plist = L[1];
                    523:   /* register DP in servers */
                    524:   map(ox_cmo_rpc,Procs,"register_data_for_gbcheck",vtol(DP));
                    525:   /* discard return value in stack */
                    526:   map(ox_pop_cmo,Procs);
                    527:   Free = Procs; Busy = [];
                    528:   while ( Plist != [] || Busy != []  )
                    529:     if ( Free == [] || Plist == [] ) {
                    530:       /* someone is working; wait for data */
                    531:       Ready = ox_select(Busy);
                    532:          /* update Busy list and Free list */
                    533:       Busy = setminus(Busy,Ready); Free = append(Ready,Free);
                    534:       for ( ; Ready != []; Ready = cdr(Ready) )
                    535:         if ( ox_get(car(Ready)) != 0 ) {
                    536:                  /* a normal form is non zero */
                    537:           map(ox_reset,Procs); return 0;
                    538:         }
                    539:     } else {
                    540:          /* update Busy list and Free list */
                    541:       Id = car(Free); Free = cdr(Free); Busy = cons(Id,Busy);
                    542:          /* take a pair */
                    543:          Pair = car(Plist); Plist = cdr(Plist);
                    544:          /* request a normal form computation */
                    545:       ox_cmo_rpc(Id,"sp_nf_for_gbcheck",Pair);
                    546:       ox_push_cmd(Id,262); /* 262 = OX_popCMO */
                    547:     }
                    548:   map(ox_reset,Procs); return 1;
                    549: }
                    550: \end{verbatim}
                    551:
                    552: \subsection{Asir OpenXM library {\tt libasir.a}}
                    553:
                    554: Asir OpenXM library {\tt libasir.a} includes functions simulating the
                    555: stack machine commands supported in {\tt ox\_asir}.  By linking {\tt
                    556: libasir.a} an application can use the same functions as in {\tt
1.3       noro      557: ox\_asir} without accessing to {\tt ox\_asir} via TCP/IP. There is
                    558: also a stack and library functions to manipulate it. In order to make
                    559: full use of this interface, one has to prepare conversion functions
                    560: between CMO and the data structures proper to the application.
                    561: A function {\tt asir\_ox\_pop\_string()} is provided to convert
                    562: CMO to a human readable form, which may be sufficient for a simple
                    563: use of this interface.
1.1       noro      564:
                    565: \section{Concluding remarks}
                    566: We have shown the current status of Risa/Asir and its OpenXM
                    567: interfaces. As a result of our policy of development, it is true that
                    568: Risa/Asir does not have abundant functions. However it is a completely
                    569: open system and its total performance is not bad. As OpenXM interface
                    570: specification is completely documented, we can add another function to
                    571: Risa/Asir by wrapping an existing software system as an OX server and
                    572: vice versa. User program debugger can be used both for local and
                    573: remote debugging. By combining the debugger and the function to reset
                    574: servers, one will be able to enjoy parallel and distributed
                    575: computation.
                    576: %
                    577: \begin{thebibliography}{7}
                    578: %
                    579: \addcontentsline{toc}{section}{References}
                    580:
                    581: \bibitem{ANY}
                    582: Anay, H., Noro, M., Yokoyama, K. (1996)
                    583: Computation of the Splitting fields and the Galois Groups of Polynomials.
                    584: Algorithms in Algebraic geometry and Applications,
                    585: Birkh\"auser (Proceedings of MEGA'94), 29--50.
                    586:
                    587: \bibitem{FPARA}
                    588: Jean-Charles Faug\`ere (1994)
                    589: Parallelization of Groebner basis.
                    590: Proceedings of PASCO'94, 124--132.
                    591:
                    592: \bibitem{F4}
                    593: Jean-Charles Faug\`ere (1999)
                    594: A new efficient algorithm for computing Groebner bases  ($F_4$).
                    595: Journal of Pure and Applied Algebra (139) 1-3 , 61--88.
                    596:
                    597: \bibitem{FGLM}
                    598: Faug\`ere, J.-C. et al. (1993)
                    599: Efficient computation of zero-dimensional Groebner bases by change of ordering.
                    600: Journal of Symbolic Computation 16, 329--344.
                    601:
                    602: \bibitem{RFC100}
                    603: M. Maekawa, et al. (2001)
                    604: The Design and Implementation of OpenXM-RFC 100 and 101.
                    605: Proceedings of ASCM2001, World Scientific, 102--111.
                    606:
                    607: \bibitem{RISA}
                    608: Noro, M. et al. (1994-2001)
                    609: A computer algebra system Risa/Asir.
                    610: {\tt http://www.openxm.org}, {\tt http://www.math.kobe-u.ac.jp/Asir/asir.html}.
                    611:
                    612: \bibitem{REPL}
                    613: Noro, M., McKay, J. (1997)
                    614: Computation of replicable functions on Risa/Asir.
                    615: Proceedings of PASCO'97, ACM Press, 130--138.
                    616:
                    617: \bibitem{NOYO}
                    618: Noro, M., Yokoyama, K. (1999)
                    619: A Modular Method to Compute the Rational Univariate
                    620: Representation of Zero-Dimensional Ideals.
                    621: Journal of Symbolic Computation, 28, 1, 243--263.
                    622:
                    623: \bibitem{OPENXM}
                    624: OpenXM committers (2000-2001)
                    625: OpenXM package.
                    626: {\tt http://www.openxm.org}.
                    627:
                    628: \bibitem{SY}
                    629: Shimoyama, T., Yokoyama, K. (1996)
                    630: Localization and Primary Decomposition of Polynomial Ideals.
                    631: Journal of Symbolic Computation, 22, 3, 247--277.
                    632:
                    633: \bibitem{TRAGER}
                    634: Trager, B.M. (1976)
                    635: Algebraic Factoring and Rational Function Integration.
                    636: Proceedings of SYMSAC 76, 219--226.
                    637:
                    638: \bibitem{TRAV}
                    639: Traverso, C. (1988)
                    640: Groebner trace algorithms.
                    641: LNCS {\bf 358} (Proceedings of ISSAC'88), Springer-Verlag, 125--138.
                    642:
                    643: \bibitem{COCOA}
                    644: {\tt http://cocoa.dima.unige.it/}.
                    645:
                    646: \bibitem{FGB}
                    647: {\tt http://www-calfor.lip6.fr/\~\,jcf/}.
                    648:
                    649: \bibitem{NTL}
                    650: {\tt http://www.shoup.net/}.
                    651:
                    652: \bibitem{OPENMATH}
                    653: {\tt http://www.openmath.org/}.
                    654:
                    655: \bibitem{SINGULAR}
                    656: {\tt http://www.singular.uni-kl.de/}.
                    657:
                    658: \end{thebibliography}
                    659:
                    660: %INDEX%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                    661: \clearpage
                    662: \addcontentsline{toc}{section}{Index}
                    663: \flushbottom
                    664: \printindex
                    665: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                    666:
                    667: \end{document}
                    668: