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Line 93  data.  It defines a client-server architecture for par
Line 93  data.  It defines a client-server architecture for par
 distributed computation.  Risa/Asir is software for polynomial  distributed computation.  Risa/Asir is software for polynomial
 computation. It has been developed for testing new algorithms, and now  computation. It has been developed for testing new algorithms, and now
 it acts as both a client and a server in the OpenXM package. In this  it acts as both a client and a server in the OpenXM package. In this
 article we present an overview of Risa/Asir and its performances on  article we present an overview of Risa/Asir and review several
 several functions.  We also show Risa/Asir's OpenXM interfaces and  techniques for improving performances of Groebner basis computation.
 examples of usages of them.  We also show Risa/Asir's OpenXM interfaces and their usages by
   examples.
 \end{abstract}  \end{abstract}
   
 \section{A computer algebra system Risa/Asir}  \section{A computer algebra system Risa/Asir}
Line 105  examples of usages of them.
Line 106  examples of usages of them.
 Risa/Asir \cite{RISA} is software mainly for polynomial  Risa/Asir \cite{RISA} is software mainly for polynomial
 computation. Its major functions are polynomial factorization and  computation. Its major functions are polynomial factorization and
 Groebner basis computation, whose core parts are implemented as  Groebner basis computation, whose core parts are implemented as
 builtin functions.  Some higher algorithms such as primary ideal  built-in functions.  Some higher algorithms such as primary ideal
 decomposition or Galois group computation are built on them by the  decomposition or Galois group computation are built on them by the
 user language.  The user language is called Asir language. Asir  user language called Asir language. Asir language can be regarded as C
 language can be regarded as C language without type declaration of  language without type declaration of variables, with list processing,
 variables, with list processing, and with automatic garbage  and with automatic garbage collection. A built-in {\tt gdb}-like user
 collection. A builtin {\tt gdb}-like user language debugger is  language debugger is available. It is open source and the source code
 available. It is open source and the source code and binaries are  and binaries are available via {\tt ftp} or {\tt CVS}.  Risa/Asir is
 available via ftp or CVS.  not only a standalone computer algebra system but also a main
 Risa/Asir is not only an standalone computer algebra system but also a  component in OpenXM package \cite{OPENXM}, which is a collection of
 main component in OpenXM package \cite{OPENXM}, which is a collection  various software compliant to OpenXM protocol specification.  OpenXM
 of software comliant to OpenXM protocol specification.  OpenXM is an  is an infrastructure for exchanging mathematical data and Risa/Asir
 infrastructure for exchanging mathematical data and Risa/Asir has  has three kind of OpenXM interfaces : client interfaces, an OpenXM
 three kind of OpenXM intefaces : an inteface as a server, as a cllient  server, and a subroutine library. Our goals of developing Risa/Asir
 and as a subroutine library. We will explain them in the later  are as follows:
 section.  
   
 Our goals of developing Risa/Asir are as follows:  
   
 \begin{enumerate}  \begin{enumerate}
 \item Providing a test bed of new algorithms  \item Providing a platform for testing new algorithms
   
 Risa/Asir has been a platform for testing experimental algorithms in  Risa/Asir has been a platform for testing experimental algorithms in
 polynomial factorization, computation related to Groebner basis,  polynomial factorization, computation related to Groebner basis,
Line 134  various modular techniques to overcome difficulties ca
Line 132  various modular techniques to overcome difficulties ca
 intermediate coefficients. We have had several results and they have  intermediate coefficients. We have had several results and they have
 been implemented in Risa/Asir.  been implemented in Risa/Asir.
   
 \item Gereral purpose open system  \item General purpose open system
   
 We need a lot of functions to make Risa/Asir a general purpose  We need a lot of functions to make Risa/Asir a general purpose
 computer algebra system.  In recent years we can obtain various high  computer algebra system.  In recent years we can obtain various high
 performance applications or libraries as free software. We wrapped  performance applications or libraries as free software. We wrapped
 such software as OpenXM servers and we started to release a collection  such software as OpenXM servers and we started to release a collection
 of such servers and cleints as OpenXM package in 1997. Risa/Asir is  of such servers and clients as the OpenXM package in 1997. Risa/Asir
 now a main client in the package.  is now a main client in the package.
   
 \item Environment for parallel and distributed computation  \item Environment for parallel and distributed computation
   
 The origin of OpenXM is a protocol for doing parallel distributed  The origin of OpenXM is a protocol for doing parallel distributed
 compuatations by connecting multiple Risa/Asir. OpenXM is also  computations by connecting multiple Risa/Asir's over TCP/IP. OpenXM is
 designed to provide an enviroment efficient parallel distributed  also designed to provide an environment efficient parallel distributed
 computation. Currently only client-server communication is possible,  computation. Currently only client-server communication is available,
 but we are preparing a specification OpenXM-RFC 102 allowing  but we are preparing a specification OpenXM-RFC 102 allowing
 client-client communication, which will enable us to execute  client-client communication, which will enable us to execute wider
 wider range of parallel algorithms efficiently.  range of parallel algorithms efficiently.
 \end{enumerate}  \end{enumerate}
   
 \subsection{Groebner basis and the related computation}  \subsection{Groebner basis and the related computation}
Line 183  recently proposed have not yet implemented.  For Univa
Line 181  recently proposed have not yet implemented.  For Univa
 over algebraic number fields, Trager's algorithm \cite{TRAGER} is  over algebraic number fields, Trager's algorithm \cite{TRAGER} is
 implemented with some modifications.  Its major applications are  implemented with some modifications.  Its major applications are
 splitting field and Galois group computation of polynomials over the  splitting field and Galois group computation of polynomials over the
 rationals. For such purpose a tower of simple extensions are suitable  rationals \cite{ANY}. For such purpose a tower of simple extensions
 because factors represented over a simple extension often have huge  are suitable because factors represented over a simple extension often
 coefficients \cite{ANY}.  For univariate factorization over finite  have huge coefficients.  For univariate factorization over finite
 fields, equal degree factorization + Cantor-Zassenhaus algorithm is  fields, equal degree factorization and Cantor-Zassenhaus algorithm are
 implemented. We can use various representation of finite fields:  implemented. We can use various representation of finite fields:
 $GF(p)$ with a machine integer prime $p$, $GF(p)$, $GF(p^n)$ with any  $GF(p)$ with a machine integer prime $p$, $GF(p)$ and $GF(p^n)$ with
 odd prime $p$, $GF(2^n)$ with a bit representation of polynomials over  any odd prime $p$, $GF(2^n)$ with a bit-array representation of
 $GF(2)$ and $GF(p^n)$ with small $p^n$ represented by a primitive  polynomials over $GF(2)$ and $GF(p^n)$ with small $p^n$ represented by
 root.  For multivariate factorization over the rationals, the  a primitive root.  For multivariate factorization over the rationals,
 classical EZ(Extented Zassenhaus) type algorithm is implemented.  the classical EZ(Extended Zassenhaus) type algorithm is implemented.
   
 \subsection{Other functions}  \subsection{Other functions}
 By applying Groebner basis computation and polynomial factorization,  By applying Groebner basis computation and polynomial factorization,
 we have implemented several higher level algorithms. A typical  we have implemented several higher level algorithms. A typical
 application is primary ideal decomposition of polynomial ideals over  application is primary ideal decomposition of polynomial ideals over
 {\bf Q}, which needs both functions.  Shimoyama-Yokoyama algorithm  {\bf Q}, which needs both functions.  Shimoyama-Yokoyama algorithm
 \cite{SY} for primary decompsition is written in the user language.  \cite{SY} for primary decomposition is written in the user language.
 Splitting field and Galois group computation are closely related and  Splitting field and Galois group computation \cite{ANY} are closely
 are also important applications of polynomial factorization.  Our  related and are also important applications of polynomial
 implementation of Galois group computation algorithm \cite{ANY}  factorization.
 requires splitting field computation, which is written in the  
 user language.  
   
 \section{Techniques for efficient Groebner basis computation over {\bf Q}}  \section{Techniques for efficient Groebner basis computation over {\bf Q}}
 \label{gbtech}  \label{gbtech}
Line 214  Groebner basis computation over {\bf Q}, which are eas
Line 210  Groebner basis computation over {\bf Q}, which are eas
 implemented but may not be well known.  implemented but may not be well known.
 We use the following notations.  We use the following notations.
 \begin{description}  \begin{description}
   \item $Id(F)$ : a polynomial ideal generated by $F$
 \item $\phi_p$ : the canonical projection from ${\bf Z}$ onto $GF(p)$  \item $\phi_p$ : the canonical projection from ${\bf Z}$ onto $GF(p)$
 \item $HT(f)$ : the head term of a polynomail with respect to a term order  \item $HT(f)$ : the head term of a polynomial with respect to a term order
 \item $HC(f)$ : the head coefficient of a polynomail with respect to a term order  \item $HC(f)$ : the head coefficient of a polynomial with respect to a term order
 \end{description}  \end{description}
   
 \subsection{Combination of homogenization and trace lifting}  \subsection{Combination of homogenization and trace lifting}
Line 235  such that $\phi_p(G)$ \\
Line 232  such that $\phi_p(G)$ \\
 \>If $G$ passes the check return $G$\\  \>If $G$ passes the check return $G$\\
 end do  end do
 \end{tabbing}  \end{tabbing}
 We can apply various methods for {\tt guess} part of the above  We can apply various methods for {\it guess} part of the above
 algorithm.  Originally we guess the candidate by replacing zero normal  algorithm.  In the original algorithm we guess the candidate by
 form checks over {\bf Q} with those over $GF(p)$ in the Buchberger  replacing zero normal form checks over {\bf Q} with those over $GF(p)$
 algorithm, which we call {\it tl\_guess}. In Asir one can specify  in the Buchberger algorithm, which we call {\it tl\_guess}. In Asir
 another method {\it tl\_h\_guess\_dh}, which is a combination of  one can specify another method {\it tl\_h\_guess\_dh}, which is a
 {\it tl\_guess} and homogenization.  combination of {\it tl\_guess} and homogenization.
 \begin{tabbing}  \begin{tabbing}
 $tl\_h\_guess\_dh(F,p)$\\  $tl\_h\_guess\_dh(F,p)$\\
 Input : $F\subset {\bf Z}[X]$, a prime $p$\\  Input : $F\subset {\bf Z}[X]$, a prime $p$\\
Line 254  such that $HT(h)|HT(g)$ \}
Line 251  such that $HT(h)|HT(g)$ \}
 The input is homogenized to suppress intermediate coefficient swells  The input is homogenized to suppress intermediate coefficient swells
 of intermediate basis elements.  The number of zero normal forms may  of intermediate basis elements.  The number of zero normal forms may
 increase by the homogenization, but they are detected over  increase by the homogenization, but they are detected over
 GF(p). Finally, by dehomogenizing the candidate we can expect that  $GF(p)$. Finally, by dehomogenizing the candidate we can expect that
 lots of redundant elements can be removed.  We will show later that this is  lots of redundant elements can be removed.  We will show later that this is
 surely efficient for some input polynomial sets.  surely efficient for some input polynomial sets.
   
Line 285  same method.
Line 282  same method.
   
 \subsection{Integer contents reduction}  \subsection{Integer contents reduction}
   
 In some cases the cost to remove integer contents during nomal form  In some cases the cost to remove integer contents during normal form
 computations is dominant. We can remove the content of an integral  computations is dominant. We can remove the content of an integral
 polynomial $f$ efficiently by the following method \cite{REPL}.  polynomial $f$ efficiently by the following method \cite{REPL}.
 \begin{tabbing}  \begin{tabbing}
Line 296  Write $f$ as $f = g_0q+r$ by division with remainder f
Line 293  Write $f$ as $f = g_0q+r$ by division with remainder f
 If $r = 0$ then return $(g_0,q)$\\  If $r = 0$ then return $(g_0,q)$\\
 else return $(g,g_0/g \cdot q + r/g)$, where $g = \GCD(g_0,\cont(r))$  else return $(g,g_0/g \cdot q + r/g)$, where $g = \GCD(g_0,\cont(r))$
 \end{tabbing}  \end{tabbing}
 By serataing the set of coefficients of $f$ into two subsets and by  By separating the set of coefficients of $f$ into two subsets and by
 computing GCD of sums in the elements in the subsets we can estimate  computing GCD of sums in the elements in the subsets we can estimate
 $g_0$ with high accuracy. Then other components are easily computed.  $g_0$ with high accuracy. Then other components are easily computed.
   
 %\subsection{Demand loading of reducers}  %\subsection{Demand loading of reducers}
 %An execution of the Buchberer algorithm may produce vary large number  %An execution of the Buchberger algorithm may produce vary large number
 %of intermediate basis elements. In Asir, we can specify that such  %of intermediate basis elements. In Asir, we can specify that such
 %basis elements should be put on disk to enlarge free memory space.  %basis elements should be put on disk to enlarge free memory space.
 %This does not reduce the efficiency so much because all basis elements  %This does not reduce the efficiency so much because all basis elements
Line 311  $g_0$ with high accuracy. Then other components are ea
Line 308  $g_0$ with high accuracy. Then other components are ea
   
 \section{Risa/Asir performance}  \section{Risa/Asir performance}
   
 We show timing data on Risa/Asir for polynomial factorization  We show timing data on Risa/Asir for Groebner basis computation
 and Groebner basis computation. The measurements were made on  and polynomial factorization. The measurements were made on
 a PC with PentiumIII 1GHz and 1Gbyte of main memory. Timings  a PC with PentiumIII 1GHz and 1Gbyte of main memory. Timings
 are given in seconds. In the tables `---' means it was not  are given in seconds. In the tables `---' means it was not
 measured.  measured.
   
 \subsection{Groebner basis computation}  \subsection{Groebner basis computation}
   
 Table \ref{gbmod} and Table \ref{gbq} shows timing data for Groebner  Table \ref{gbmod} and Table \ref{gbq} show timing data for Groebner
 basis compuation over $GF(32003)$ and over {\bf Q} respectively.  basis computation over $GF(32003)$ and over {\bf Q} respectively.
 $C_n$ is the cyclic $n$ system and $K_n$ is the Katsura $n$ system,  $C_n$ is the cyclic $n$ system and $K_n$ is the Katsura $n$ system,
 both are famous bench mark problems.  We also measured the timing for  both are famous bench mark problems \cite{BENCH}.  We also measured
 $McKay$ system over {\bf Q} \cite{REPL}.  the term order is graded  the timing for $McKay$ system over {\bf Q} \cite{REPL}.  the term
 reverse lexicographic order.  In the both tables, the first three rows  order is graded reverse lexicographic order.  In the both tables, the
 are timings for the Buchberger algorithm, and the last two rows are  first three rows are timings for the Buchberger algorithm, and the
 timings for $F_4$ algorithm. As to the Buchberger algorithm over  last two rows are timings for $F_4$ algorithm. As to the Buchberger
 $GF(32003)$, Singular\cite{SINGULAR} shows the best performance among  algorithm over $GF(32003)$, Singular\cite{SINGULAR} shows the best
 the three systems. $F_4$ implementation in Risa/Asir is faster than  performance among the three systems. $F_4$ implementation in Risa/Asir
 the Buchberger algorithm implementation in Singluar, but it is still  is faster than the Buchberger algorithm implementation in Singular,
 several times slower than $F_4$ implemenataion in FGb \cite{FGB}.  In  but it is still several times slower than $F_4$ implementation in FGb
 Table \ref{gbq}, $C_7$ and $McKay$ can be computed by the Buchberger  \cite{FGB}.  In Table \ref{gbq}, $C_7$ and $McKay$ can be computed by
 algorithm with the methods described in Section \ref{gbtech}.  It is  the Buchberger algorithm with the methods described in Section
 obvious that $F_4$ implementation in Risa/Asir over {\bf Q} is too  \ref{gbtech}.  It is obvious that $F_4$ implementation in Risa/Asir
 immature. Nevertheless the timing of $McKay$ is greatly reduced.  over {\bf Q} is too immature. Nevertheless the timing of $McKay$ is
 Fig. \ref{f4vsbuch} explains why $F_4$ is efficient in this case.  greatly reduced.  Fig. \ref{f4vsbuch} explains why $F_4$ is efficient
 The figure shows that  in this case.  The figure shows that the Buchberger algorithm produces
 the Buchberger algorithm produces normal forms with  normal forms with huge coefficients for S-polynomials after the 250-th
 huge coefficients for S-polynomals after the 250-th one,  one, which are the computations in degree 16.  However, we know that
 which are the computations in degree 16.  the reduced basis elements have much smaller coefficients after
 However, we know that the reduced basis elements have  removing contents.  As $F_4$ algorithm automatically produces the
 much smaller coefficients after removing contents.  reduced ones, the degree 16 computation is quite easy in $F_4$.
 As $F_4$ algorithm automatically produces the reduced ones,  
 the degree 16 computation is quite easy in $F_4$.  
   
 \begin{table}[hbtp]  \begin{table}[hbtp]
 \begin{center}  \begin{center}
Line 362  FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \
Line 357  FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \
   
 \begin{table}[hbtp]  \begin{table}[hbtp]
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|c|} \hline
                 & $C_7$ & $Homog. C_7$ & $K_7$ & $K_8$ & $McKay$ \\ \hline                  & $C_7$ & $Homog. C_7$ & $C_8$ & $K_7$ & $K_8$ & $McKay$ \\ \hline
 Asir $Buchberger$       & 389 & 594 & 29 & 299 & 34950 \\ \hline  Asir $Buchberger$       & 389 & 594 & 54000 & 29 & 299 & 34950 \\ \hline
 Singular & --- & 15247 & 7.6 & 79 & $>$ 20h \\ \hline  Singular & --- & 15247 & --- & 7.6 & 79 & $>$ 20h \\ \hline
 CoCoA 4 & --- & 13227 & 57 & 709 & --- \\ \hline\hline  CoCoA 4 & --- & 13227 & --- & 57 & 709 & --- \\ \hline\hline
 Asir $F_4$      &  989 & 456 & 90 & 991 & 4939 \\ \hline  Asir $F_4$      &  989 & 456 & --- & 90 & 991 & 4939 \\ \hline
 FGb(estimated)  & 8 &11 & 0.6 & 5 & 10 \\ \hline  FGb(estimated)  & 8 &11 & 288 &  0.6 & 5 & 10 \\ \hline
 \end{tabular}  \end{tabular}
 \end{center}  \end{center}
 \caption{Groebner basis computation over {\bf Q}}  \caption{Groebner basis computation over {\bf Q}}
Line 384  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
Line 379  FGb(estimated) & 8 &11 & 0.6 & 5 & 10 \\ \hline
 \label{f4vsbuch}  \label{f4vsbuch}
 \end{figure}  \end{figure}
   
   Table \ref{minipoly} shows timing data for the minimal polynomial
   computation over {\bf Q}. Singular provides a function {\tt finduni}
   for computing the minimal polynomial in each variable in ${\bf
   Q}[x_1,\ldots,x_n]/I$ for zero dimensional ideal $I$. The modular
   method used in Asir is efficient when the resulting minimal
   polynomials have large coefficients and we can verify the fact from Table
   \ref{minipoly}.
   \begin{table}[hbtp]
   \begin{center}
   \begin{tabular}{|c||c|c|c|c|c|} \hline
                   & $C_6$ & $C_7$ & $K_6$ & $K_7$ & $K_8$ \\ \hline
   Singular & 0.9 & 846 & 307 & 60880 & ---  \\ \hline
   Asir & 1.5 & 182 & 12 & 164 & 3420  \\ \hline
   \end{tabular}
   \end{center}
   \caption{Minimal polynomial computation}
   \label{minipoly}
   \end{table}
   
 \subsection{Polynomial factorization}  \subsection{Polynomial factorization}
   
 %Table \ref{unifac} shows timing data for univariate factorization over  %Table \ref{unifac} shows timing data for univariate factorization over
Line 418  $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file
Line 432  $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file
 in Risa/Asir source tree and located in {\tt asir2000/lib}.  in Risa/Asir source tree and located in {\tt asir2000/lib}.
 For these examples Risa/Asir shows reasonable performance  For these examples Risa/Asir shows reasonable performance
 compared with other famous systems.  compared with other famous systems.
   
 \begin{table}[hbtp]  \begin{table}[hbtp]
 \begin{center}  \begin{center}
 \begin{tabular}{|c||c|c|c|c|c|} \hline  \begin{tabular}{|c||c|c|c|c|c|} \hline
Line 435  Maple 7& 0.5  & 18  & 967  & 48 & 1.3 \\ \hline
Line 448  Maple 7& 0.5  & 18  & 967  & 48 & 1.3 \\ \hline
 \caption{Multivariate factorization over {\bf Q}}  \caption{Multivariate factorization over {\bf Q}}
 \label{multifac}  \label{multifac}
 \end{table}  \end{table}
   
 As to univariate factorization over {\bf Q},  As to univariate factorization over {\bf Q},
 the univariate factorizer implements only classical  the univariate factorizer implements only classical
 algorithms and its behaviour is what one expects,  algorithms and its behavior is what one expects,
 that is, it shows average performance in cases  that is, it shows average performance in cases
 where there are little eraneous factors, but  where there are little extraneous factors, but
 shows poor performance for hard to factor polynomials.  shows poor performance for hard to factor polynomials.
   
 \section{OpenXM and Risa/Asir OpenXM interfaces}  \section{OpenXM and Risa/Asir OpenXM interfaces}
Line 448  shows poor performance for hard to factor polynomials.
Line 460  shows poor performance for hard to factor polynomials.
 \subsection{OpenXM overview}  \subsection{OpenXM overview}
   
 OpenXM stands for Open message eXchange protocol for Mathematics.  OpenXM stands for Open message eXchange protocol for Mathematics.
 Form the viewpoint of protocol design, it is a child of OpenMath  From the viewpoint of protocol design, it can be regarded as a child
 \cite{OPENMATH}.  However our approch is somewhat different. Our main  of OpenMath \cite{OPENMATH}.  However our approach is somewhat
 purpose is to provide an environment for integrating {\it existing}  different. Our main purpose is to provide an environment for
 mathematical software systems. OpenXM RFC-100 \cite{RFC100} defines a  integrating {\it existing} mathematical software systems. OpenXM
 client-server architecture.  Under this specification, a client  RFC-100 \cite{RFC100} defines a client-server architecture.  Under
 invokes an OpenXM (OX) server.  The client can send OpenXM (OX)  this specification, a client invokes an OpenXM ({\it OX}) server.  The
 messages to the server.  OX messages consist of {\it data} and {\it  client can send OpenXM ({\it OX}) messages to the server.  OX messages
 command}. Data is encoded according to the common mathematical object  consist of {\it data} and {\it command}. Data is encoded according to
 (CMO) format which defines serialized representation of mathematical  the common mathematical object ({\it CMO}) format which defines
 objects.  An OX server is a stackmachine. If data is sent as an OX  serialized representation of mathematical objects.  An OX server is a
 message, the server pushes the data onto its stack. There is a common  stackmachine. If data is sent as an OX message, the server pushes the
 set of stackmachine commands and all OX server understands its subset.  data onto its stack. There is a common set of stackmachine commands
 The command set includes commands for manipulating the stack and  and each OX server understands its subset. The command set includes
 requests for execution of a procedure. In addition, a server may  stack manipulating commands and requests for execution of a procedure.
 accept its own command sequences if the server wraps some interactive  In addition, a server may accept its own command sequences if the
 software. That is the server may be a hybrid server.  server wraps some interactive software. That is the server may be a
   hybrid server.
   
 OpenXM RFC-100 also defines methods for session management. In particular  OpenXM RFC-100 also defines methods for session management. In particular
 the method to reset a server is carefully designed and it provides  the method to reset a server is carefully designed and it provides
Line 473  purposes.
Line 486  purposes.
 \subsection{OpenXM client interface of {\tt asir}}  \subsection{OpenXM client interface of {\tt asir}}
   
 Risa/Asir is a main client in OpenXM package.  The application {\tt  Risa/Asir is a main client in OpenXM package.  The application {\tt
 asir} can access to OpenXM servers via several builtin interface  asir} can access to OpenXM servers via several built-in interface
 functions. and various inferfaces to existing OpenXM servers are  functions. and various interfaces to existing OpenXM servers are
 prepared as user defined functions written in Asir language.  We show  prepared as user defined functions written in Asir language.
 a typical OpenXM session.  We show a typical OpenXM session.
   
 \begin{verbatim}  \begin{verbatim}
 [1] P = ox_launch();  /* invoke an OpenXM asir server */  [1] P = ox_launch();  /* invoke an OpenXM asir server */
Line 503  a typical OpenXM session.
Line 516  a typical OpenXM session.
 An application {\tt ox\_asir} is a wrapper of {\tt asir} and provides  An application {\tt ox\_asir} is a wrapper of {\tt asir} and provides
 all the functions of {\tt asir} to OpenXM clients. It completely  all the functions of {\tt asir} to OpenXM clients. It completely
 implements the OpenXM reset protocol and also provides remote  implements the OpenXM reset protocol and also provides remote
 debugging of user programs running on the server. We show a program  debugging of user programs running on the server. As an example we
 for checking whether a polynomial set is a Groebner basis or not. A  show a program for checking whether a polynomial set is a Groebner
 client executes {\tt gbcheck()} and servers execute {\tt  basis or not. A client executes {\tt gbcheck()} and servers execute
 sp\_nf\_for\_gbcheck()} which is a simple normal form computation of a  {\tt sp\_nf\_for\_gbcheck()} which is a simple normal form computation
 S-polynomial. First of all the client collects all critical pairs  of a S-polynomial. First of all the client collects all critical pairs
 necessary for the check. Then the client requests normal form  necessary for the check. Then the client requests normal form
 computations to idling servers. If there are no idling servers the  computations to idling servers. If there are no idling servers the
 clients waits for some servers to return results by {\tt  clients waits for some servers to return results by {\tt
 ox\_select()}, which is a wrapper of UNIX {\tt select()}. If we have  ox\_select()}, which is a wrapper of UNIX {\tt select()}. If we have
 large number of critcal pairs to be processed, we can expect  large number of critical pairs to be processed, we can expect good
 good load balancing by {\tt ox\_select()}.  load balancing by {\tt ox\_select()}.
   
 \begin{verbatim}  \begin{verbatim}
 def gbcheck(B,V,O,Procs) {  def gbcheck(B,V,O,Procs) {
Line 555  Asir OpenXM library {\tt libasir.a} includes functions
Line 568  Asir OpenXM library {\tt libasir.a} includes functions
 stack machine commands supported in {\tt ox\_asir}.  By linking {\tt  stack machine commands supported in {\tt ox\_asir}.  By linking {\tt
 libasir.a} an application can use the same functions as in {\tt  libasir.a} an application can use the same functions as in {\tt
 ox\_asir} without accessing to {\tt ox\_asir} via TCP/IP. There is  ox\_asir} without accessing to {\tt ox\_asir} via TCP/IP. There is
 also a stack and library functions to manipulate it. In order to make  also a stack, which can be manipulated by library functions. In
 full use of this interface, one has to prepare conversion functions  order to make full use of this interface, one has to prepare
 between CMO and the data structures proper to the application.  conversion functions between CMO and the data structures proper to the
 A function {\tt asir\_ox\_pop\_string()} is provided to convert  application.  A function {\tt asir\_ox\_pop\_string()} is provided to
 CMO to a human readable form, which may be sufficient for a simple  convert CMO to a human readable form, which may be sufficient for a
 use of this interface.  simple use of this interface.
   
 \section{Concluding remarks}  \section{Concluding remarks}
 We have shown the current status of Risa/Asir and its OpenXM  We have shown the current status of Risa/Asir and its OpenXM
 interfaces. As a result of our policy of development, it is true that  interfaces. As a result of our policy of development, it is true that
 Risa/Asir does not have abundant functions. However it is a completely  Risa/Asir does not have abundant functions. However it is a completely
 open system and its total performance is not bad. As OpenXM interface  open system and its total performance is not bad. Especially on
 specification is completely documented, we can add another function to  Groebner basis computation over {\bf Q}, many techniques for improving
 Risa/Asir by wrapping an existing software system as an OX server and  practical performances have been implemented. As the OpenXM interface
 vice versa. User program debugger can be used both for local and  specification is completely documented, we can easily add another
 remote debugging. By combining the debugger and the function to reset  function to Risa/Asir by wrapping an existing software system as an OX
 servers, one will be able to enjoy parallel and distributed  server, and vice versa. User program debugger can be used both for
 computation.  local and remote debugging. By combining the debugger and the function
   to reset servers, one will be able to enjoy parallel and distributed
   computation with OpenXM facilities.
 %  %
 \begin{thebibliography}{7}  \begin{thebibliography}{7}
 %  %
Line 640  Traverso, C. (1988)
Line 655  Traverso, C. (1988)
 Groebner trace algorithms.  Groebner trace algorithms.
 LNCS {\bf 358} (Proceedings of ISSAC'88), Springer-Verlag, 125--138.  LNCS {\bf 358} (Proceedings of ISSAC'88), Springer-Verlag, 125--138.
   
   \bibitem{BENCH}
   {\tt http://www.math.uic.edu/\~\,jan/demo.html}.
   
 \bibitem{COCOA}  \bibitem{COCOA}
 {\tt http://cocoa.dima.unige.it/}.  {\tt http://cocoa.dima.unige.it/}.
   
 \bibitem{FGB}  \bibitem{FGB}
 {\tt http://www-calfor.lip6.fr/\~\,jcf/}.  {\tt http://www-calfor.lip6.fr/\~\,jcf/}.
   
 \bibitem{NTL}  %\bibitem{NTL}
 {\tt http://www.shoup.net/}.  %{\tt http://www.shoup.net/}.
   
 \bibitem{OPENMATH}  \bibitem{OPENMATH}
 {\tt http://www.openmath.org/}.  {\tt http://www.openmath.org/}.
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