Annotation of OpenXM/doc/Papers/dag-noro-proc.tex, Revision 1.7
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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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1.7 ! noro 79: \author{Masayuki Noro}
1.1 noro 80: %
81: %\authorrunning{Masayuki Noro}
82: % if there are more than two authors,
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}
1.7 ! noro 91: Risa/Asir is software for polynomial computation. It has been
! 92: developed for testing experimental polynomial algorithms, and now it
! 93: acts also as a main component in the OpenXM package \cite{OPENXM}.
! 94: OpenXM is an infrastructure for exchanging mathematical
1.1 noro 95: data. It defines a client-server architecture for parallel and
1.7 ! noro 96: distributed computation. In this article we present an overview of
! 97: Risa/Asir and review several techniques for improving performances of
! 98: Groebner basis computation over {\bf Q}. We also show Risa/Asir's
! 99: OpenXM interfaces and their usages.
1.1 noro 100: \end{abstract}
101:
102: \section{A computer algebra system Risa/Asir}
103:
104: \subsection{What is Risa/Asir?}
105:
106: Risa/Asir \cite{RISA} is software mainly for polynomial
107: computation. Its major functions are polynomial factorization and
108: Groebner basis computation, whose core parts are implemented as
1.5 noro 109: built-in functions. Some higher algorithms such as primary ideal
1.1 noro 110: decomposition or Galois group computation are built on them by the
1.5 noro 111: user language called Asir language. Asir language can be regarded as C
112: language without type declaration of variables, with list processing,
113: and with automatic garbage collection. A built-in {\tt gdb}-like user
1.7 ! noro 114: language debugger is available. Risa/Asir is open source and the
! 115: source code and binaries are available via {\tt ftp} or {\tt CVS}.
! 116: Risa/Asir is not only a standalone computer algebra system but also a
! 117: main component in OpenXM package \cite{OPENXM}, which is a collection
! 118: of various software compliant to OpenXM protocol specification.
! 119: OpenXM is an infrastructure for exchanging mathematical data and
! 120: Risa/Asir has three kinds of OpenXM interfaces : as a client, as a
! 121: server, and as a subroutine library. Our goals of developing Risa/Asir
1.5 noro 122: are as follows:
1.1 noro 123:
124: \begin{enumerate}
1.5 noro 125: \item Providing a platform for testing new algorithms
1.1 noro 126:
127: Risa/Asir has been a platform for testing experimental algorithms in
1.7 ! noro 128: polynomial factorization, Groebner basis computation,
1.1 noro 129: cryptography and quantifier elimination. As to Groebner basis, we have
130: been mainly interested in problems over {\bf Q} and we tried applying
131: various modular techniques to overcome difficulties caused by huge
132: intermediate coefficients. We have had several results and they have
1.7 ! noro 133: been implemented in Risa/Asir with other known methods.
1.1 noro 134:
1.5 noro 135: \item General purpose open system
1.1 noro 136:
137: We need a lot of functions to make Risa/Asir a general purpose
1.7 ! noro 138: computer algebra system. In recent years we can make use of various high
1.1 noro 139: performance applications or libraries as free software. We wrapped
140: such software as OpenXM servers and we started to release a collection
1.5 noro 141: of such servers and clients as the OpenXM package in 1997. Risa/Asir
142: is now a main client in the package.
1.1 noro 143:
144: \item Environment for parallel and distributed computation
145:
1.7 ! noro 146: The ancestor of OpenXM is a protocol designed for doing parallel
! 147: distributed computations by connecting multiple Risa/Asir's over
! 148: TCP/IP. OpenXM is also designed to provide an environment for
! 149: efficient parallel distributed computation. Currently only
! 150: client-server communication is available, but we are preparing a
! 151: specification OpenXM-RFC 102 allowing client-client communication,
! 152: which will enable us to execute wider range of parallel algorithms
! 153: requiring collective operations efficiently.
1.1 noro 154: \end{enumerate}
155:
156: \subsection{Groebner basis and the related computation}
157:
158: Currently Risa/Asir can only deal with polynomial ring. Operations on
159: modules over polynomial rings have not yet supported. However, both
160: commutative polynomial rings and Weyl algebra are supported and one
1.7 ! noro 161: can compute Groebner basis in both rings over {\bf Q}, fields of
1.1 noro 162: rational functions and finite fields. In the early stage of our
163: development, our effort was mainly devoted to improve the efficiency
1.7 ! noro 164: of computation over {\bf Q}. Our main tool is modular
1.1 noro 165: computation. For Buchberger algorithm we adopted the trace lifting
166: algorithm by Traverso \cite{TRAV} and elaborated it by applying our
167: theory on a correspondence between Groebner basis and its modular
168: image \cite{NOYO}. We also combine the trace lifting with
169: homogenization to stabilize selection strategies, which enables us to
1.7 ! noro 170: compute several examples efficiently which are hard to compute without
1.1 noro 171: such a combination. Our modular method can be applied to the change
1.7 ! noro 172: of ordering algorithm\cite{FGLM} and rational univariate
! 173: representation \cite{RUR}. We also made a test implementation of
! 174: $F_4$ algorithm \cite{F4}. In the later section we will show timing
! 175: data on Groebner basis computation.
1.1 noro 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
1.7 ! noro 182: recently proposed have not yet implemented. For univariate
! 183: factorization over algebraic number fields, Trager's algorithm
! 184: \cite{TRAGER} is implemented with some modifications. Its major
! 185: applications are splitting field and Galois group computation of
! 186: polynomials over {\bf Q} \cite{ANY}. For such purpose a tower of
! 187: simple extensions are suitable because factors represented over a
! 188: simple extension often have huge coefficients. For univariate
! 189: factorization over finite fields, equal degree factorization and
! 190: Cantor-Zassenhaus algorithm are implemented. We can use various
! 191: representation of finite fields: $GF(p)$ with a machine integer prime
! 192: $p$, $GF(p)$ and $GF(p^n)$ with any odd prime $p$, $GF(2^n)$ with a
! 193: bit-array representation of polynomials over $GF(2)$ and $GF(p^n)$
! 194: with small $p^n$ represented by a primitive root. For multivariate
! 195: factorization over {\bf Q}, the classical EZ(Extended
! 196: Zassenhaus) type algorithm is implemented.
1.1 noro 197:
198: \subsection{Other functions}
199: By applying Groebner basis computation and polynomial factorization,
200: we have implemented several higher level algorithms. A typical
201: application is primary ideal decomposition of polynomial ideals over
202: {\bf Q}, which needs both functions. Shimoyama-Yokoyama algorithm
1.5 noro 203: \cite{SY} for primary decomposition is written in the user language.
204: Splitting field and Galois group computation \cite{ANY} are closely
205: related and are also important applications of polynomial
206: factorization.
1.1 noro 207:
208: \section{Techniques for efficient Groebner basis computation over {\bf Q}}
209: \label{gbtech}
210:
211: In this section we review several practical techniques to improve
212: Groebner basis computation over {\bf Q}, which are easily
213: implemented but may not be well known.
214: We use the following notations.
215: \begin{description}
1.7 ! noro 216: \item $Id(F)$ : a polynomial ideal generated by a polynomial set $F$
1.1 noro 217: \item $\phi_p$ : the canonical projection from ${\bf Z}$ onto $GF(p)$
1.5 noro 218: \item $HT(f)$ : the head term of a polynomial with respect to a term order
219: \item $HC(f)$ : the head coefficient of a polynomial with respect to a term order
1.1 noro 220: \end{description}
221:
222: \subsection{Combination of homogenization and trace lifting}
1.7 ! noro 223: \label{gbhomo}
1.1 noro 224:
225: Traverso's trace lifting algorithm can be
1.7 ! noro 226: formulated in an abstract form as follows (c.f. \cite{FPARA}).
1.1 noro 227: \begin{tabbing}
228: Input : a finite subset $F \subset {\bf Z}[X]$\\
229: Output : a Groebner basis $G$ of $Id(F)$ with respect to a term order $<$\\
230: do \= \\
231: \> $p \leftarrow$ a new prime\\
232: \>Guess \= a Groebner basis candidate $G \subset Id(F)$
233: such that $\phi_p(G)$ \\
234: \>\> is a Groebner basis of $Id(\phi_p(F))$ in ${GF(p)}[X]$\\
235: \>Check that $G$ is a Groebner basis of $Id(G)$ and $F \subset Id(G)$\\
236: \>If $G$ passes the check return $G$\\
237: end do
238: \end{tabbing}
1.5 noro 239: We can apply various methods for {\it guess} part of the above
240: algorithm. In the original algorithm we guess the candidate by
241: replacing zero normal form checks over {\bf Q} with those over $GF(p)$
242: in the Buchberger algorithm, which we call {\it tl\_guess}. In Asir
243: one can specify another method {\it tl\_h\_guess\_dh}, which is a
244: combination of {\it tl\_guess} and homogenization.
1.1 noro 245: \begin{tabbing}
246: $tl\_h\_guess\_dh(F,p)$\\
247: Input : $F\subset {\bf Z}[X]$, a prime $p$\\
248: Output : a Groebner basis candidate\\
249: $F_h \leftarrow$ the homogenization of $F$\\
250: $G_h \leftarrow tl\_guess(F_h,p)$ under an appropriate term order\\
251: $G \leftarrow$ the dehomogenization of $G_h$\\
252: $G \leftarrow G \setminus \{g \in G| \exists h \in G \setminus \{g\}$
253: such that $HT(h)|HT(g)$ \}
254: \end{tabbing}
255: The input is homogenized to suppress intermediate coefficient swells
256: of intermediate basis elements. The number of zero normal forms may
257: increase by the homogenization, but they are detected over
1.5 noro 258: $GF(p)$. Finally, by dehomogenizing the candidate we can expect that
1.7 ! noro 259: lots of redundant elements can be removed.
1.1 noro 260:
261: \subsection{Minimal polynomial computation by modular method}
1.7 ! noro 262:
1.1 noro 263: Let $I$ be a zero-dimensional ideal in $R={\bf Q}[x_1,\ldots,x_n]$.
264: Then the minimal polynomial $m(x_i)$ of a variable $x_i$ in $R/I$ can
265: be computed by a partial FGLM \cite{FGLM}, but it often takes long
266: time if one searches $m(x_i)$ incrementally over {\bf Q}. In this
267: case we can apply a simple modular method to compute the minimal
268: polynomial.
269: \begin{tabbing}
270: Input : a Groebner basis $G$ of $I$, a variable $x_i$\\
271: Output : the minimal polynomial of $x$ in $R/I$\\
272: do \= \\
273: \> $p \leftarrow$ a new prime such that $p \not{|} HC(g)$ for all $g \in G$\\
274: \> $m_p \leftarrow$ the minimal polynomial of $x_i$ in $GF(p)[x_1,\ldots,x_n]/Id(\phi_p(G))$\\
275: \> If there exists $m(x_i) \in I$ such that $\phi_p(m) = m_p$ and $\deg(m)=\deg(m_p)$\\
276: \> then return $m(x_i)$\\
277: end do
278: \end{tabbing}
279: In this algorithm, $m_p$ can be obtained by a partial FGLM over
280: $GF(p)$ because $\phi_p(G)$ is a Groebner basis. Once we know the
281: candidate of $\deg(m(x_i))$, $m(x_i)$ can be determined by solving a
282: system of linear equations via the method of indeterminate
1.7 ! noro 283: coefficient, and it can be solved efficiently by $p$-adic method.
! 284: Arguments on \cite{NOYO} ensures that $m(x_i)$ is what we want if it
! 285: exists. Note that the full FGLM can also be computed by the same
! 286: method.
1.1 noro 287:
288: \subsection{Integer contents reduction}
1.7 ! noro 289: \label{gbcont}
1.1 noro 290:
1.5 noro 291: In some cases the cost to remove integer contents during normal form
1.1 noro 292: computations is dominant. We can remove the content of an integral
293: polynomial $f$ efficiently by the following method \cite{REPL}.
294: \begin{tabbing}
295: Input : an integral polynomial $f$\\
296: Output : a pair $(\cont(f),f/\cont(f))$\\
297: $g_0 \leftarrow$ an estimate of $\cont(f)$ such that $\cont(f)|g_0$\\
1.7 ! noro 298: Write $f$ as $f = g_0q+r$ by division with remainder by $g_0$ for each coefficient\\
1.1 noro 299: If $r = 0$ then return $(g_0,q)$\\
300: else return $(g,g_0/g \cdot q + r/g)$, where $g = \GCD(g_0,\cont(r))$
301: \end{tabbing}
1.5 noro 302: By separating the set of coefficients of $f$ into two subsets and by
1.7 ! noro 303: computing GCD of sums of the elements in each subset we can estimate
1.1 noro 304: $g_0$ with high accuracy. Then other components are easily computed.
305:
306: %\subsection{Demand loading of reducers}
1.5 noro 307: %An execution of the Buchberger algorithm may produce vary large number
1.1 noro 308: %of intermediate basis elements. In Asir, we can specify that such
309: %basis elements should be put on disk to enlarge free memory space.
310: %This does not reduce the efficiency so much because all basis elements
311: %are not necessarily used in a single normal form computation, and the
312: %cost for reading basis elements from disk is often negligible because
313: %of the cost for coefficient computations.
314:
315: \section{Risa/Asir performance}
316:
1.5 noro 317: We show timing data on Risa/Asir for Groebner basis computation
318: and polynomial factorization. The measurements were made on
1.1 noro 319: a PC with PentiumIII 1GHz and 1Gbyte of main memory. Timings
320: are given in seconds. In the tables `---' means it was not
321: measured.
322:
323: \subsection{Groebner basis computation}
324:
1.5 noro 325: Table \ref{gbmod} and Table \ref{gbq} show timing data for Groebner
326: basis computation over $GF(32003)$ and over {\bf Q} respectively.
1.1 noro 327: $C_n$ is the cyclic $n$ system and $K_n$ is the Katsura $n$ system,
1.5 noro 328: both are famous bench mark problems \cite{BENCH}. We also measured
329: the timing for $McKay$ system over {\bf Q} \cite{REPL}. the term
330: order is graded reverse lexicographic order. In the both tables, the
331: first three rows are timings for the Buchberger algorithm, and the
332: last two rows are timings for $F_4$ algorithm. As to the Buchberger
333: algorithm over $GF(32003)$, Singular\cite{SINGULAR} shows the best
334: performance among the three systems. $F_4$ implementation in Risa/Asir
335: is faster than the Buchberger algorithm implementation in Singular,
336: but it is still several times slower than $F_4$ implementation in FGb
1.7 ! noro 337: \cite{FGB}. In Table \ref{gbq}, Risa/Asir computed $C_7$ and $McKay$
! 338: by the Buchberger algorithm with the methods described in Section
! 339: \ref{gbhomo} and \ref{gbcont}. It is obvious that $F_4$
! 340: implementation in Risa/Asir over {\bf Q} is too immature. Nevertheless
! 341: the timing of $McKay$ is greatly reduced. Fig. \ref{f4vsbuch}
! 342: explains why $F_4$ is efficient in this case. The figure shows that
! 343: the Buchberger algorithm produces normal forms with huge coefficients
! 344: for S-polynomials after the 250-th one, which are the computations in
! 345: degree 16. However, we know that the reduced basis elements have much
! 346: smaller coefficients after removing contents. As $F_4$ algorithm
! 347: automatically produces the reduced ones, the degree 16 computation is
! 348: quite easy in $F_4$.
1.1 noro 349:
350: \begin{table}[hbtp]
351: \begin{center}
352: \begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
353: & $C_7$ & $C_8$ & $K_7$ & $K_8$ & $K_9$ & $K_{10}$ & $K_{11}$ \\ \hline
354: Asir $Buchberger$ & 31 & 1687 & 2.6 & 27 & 294 & 4309 & --- \\ \hline
355: Singular & 8.7 & 278 & 0.6 & 5.6 & 54 & 508 & 5510 \\ \hline
356: CoCoA 4 & 241 & $>$ 5h & 3.8 & 35 & 402 &7021 & --- \\ \hline\hline
357: Asir $F_4$ & 5.3 & 129 & 0.5 & 4.5 & 31 & 273 & 2641 \\ \hline
358: FGb(estimated) & 0.9 & 23 & 0.1 & 0.8 & 6 & 51 & 366 \\ \hline
359: \end{tabular}
360: \end{center}
361: \caption{Groebner basis computation over $GF(32003)$}
362: \label{gbmod}
363: \end{table}
364:
365: \begin{table}[hbtp]
366: \begin{center}
1.5 noro 367: \begin{tabular}{|c||c|c|c|c|c|c|} \hline
368: & $C_7$ & $Homog. C_7$ & $C_8$ & $K_7$ & $K_8$ & $McKay$ \\ \hline
369: Asir $Buchberger$ & 389 & 594 & 54000 & 29 & 299 & 34950 \\ \hline
370: Singular & --- & 15247 & --- & 7.6 & 79 & $>$ 20h \\ \hline
371: CoCoA 4 & --- & 13227 & --- & 57 & 709 & --- \\ \hline\hline
372: Asir $F_4$ & 989 & 456 & --- & 90 & 991 & 4939 \\ \hline
373: FGb(estimated) & 8 &11 & 288 & 0.6 & 5 & 10 \\ \hline
1.1 noro 374: \end{tabular}
375: \end{center}
376: \caption{Groebner basis computation over {\bf Q}}
377: \label{gbq}
378: \end{table}
379:
380: \begin{figure}[hbtp]
381: \begin{center}
382: \epsfxsize=12cm
1.6 noro 383: %\epsffile{../compalg/ps/blenall.ps}
384: \epsffile{blen.ps}
1.1 noro 385: \end{center}
386: \caption{Maximal coefficient bit length of intermediate bases}
387: \label{f4vsbuch}
388: \end{figure}
389:
1.5 noro 390: Table \ref{minipoly} shows timing data for the minimal polynomial
391: computation over {\bf Q}. Singular provides a function {\tt finduni}
392: for computing the minimal polynomial in each variable in ${\bf
393: Q}[x_1,\ldots,x_n]/I$ for zero dimensional ideal $I$. The modular
394: method used in Asir is efficient when the resulting minimal
395: polynomials have large coefficients and we can verify the fact from Table
396: \ref{minipoly}.
397: \begin{table}[hbtp]
398: \begin{center}
399: \begin{tabular}{|c||c|c|c|c|c|} \hline
400: & $C_6$ & $C_7$ & $K_6$ & $K_7$ & $K_8$ \\ \hline
401: Singular & 0.9 & 846 & 307 & 60880 & --- \\ \hline
402: Asir & 1.5 & 182 & 12 & 164 & 3420 \\ \hline
403: \end{tabular}
404: \end{center}
405: \caption{Minimal polynomial computation}
406: \label{minipoly}
407: \end{table}
408:
1.1 noro 409: \subsection{Polynomial factorization}
410:
1.3 noro 411: %Table \ref{unifac} shows timing data for univariate factorization over
412: %{\bf Q}. $N_{i,j}$ is an irreducible polynomial which are hard to
413: %factor by the classical algorithm. $N_{i,j}$ is a norm of a polynomial
414: %and $\deg(N_i) = i$ with $j$ modular factors. Risa/Asir is
415: %disadvantageous in factoring polynomials of this type because the
416: %algorithm used in Risa/Asir has exponential complexity. In contrast,
417: %CoCoA 4\cite{COCOA} and NTL-5.2\cite{NTL} show nice performances
418: %because they implement recently developed algorithms.
419: %
420: %\begin{table}[hbtp]
421: %\begin{center}
422: %\begin{tabular}{|c||c|c|c|c|} \hline
423: % & $N_{105,23}$ & $N_{120,20}$ & $N_{168,24}$ & $N_{210,54}$ \\ \hline
424: %Asir & 0.86 & 59 & 840 & hard \\ \hline
425: %Asir NormFactor & 1.6 & 2.2& 6.1& hard \\ \hline
426: %%Singular& hard? & hard?& hard? & hard? \\ \hline
427: %CoCoA 4 & 0.2 & 7.1 & 16 & 0.5 \\ \hline\hline
428: %NTL-5.2 & 0.16 & 0.9 & 1.4 & 0.4 \\ \hline
429: %\end{tabular}
430: %\end{center}
431: %\caption{Univariate factorization over {\bf Q}}
432: %\label{unifac}
433: %\end{table}
1.1 noro 434:
435: Table \ref{multifac} shows timing data for multivariate
436: factorization over {\bf Q}.
437: $W_{i,j,k}$ is a product of three multivariate polynomials
438: $Wang[i]$, $Wang[j]$, $Wang[k]$ given in a data file
439: {\tt fctrdata} in Asir library directory. It is also included
440: in Risa/Asir source tree and located in {\tt asir2000/lib}.
441: For these examples Risa/Asir shows reasonable performance
442: compared with other famous systems.
443: \begin{table}[hbtp]
444: \begin{center}
445: \begin{tabular}{|c||c|c|c|c|c|} \hline
446: & $W_{1,2,3}$ & $W_{4,5,6}$ & $W_{7,8,9}$ & $W_{10,11,12}$ & $W_{13,14,15}$ \\ \hline
447: variables & 3 & 5 & 5 & 5 & 4 \\ \hline
448: monomials & 905 & 41369 & 51940 & 30988 & 3344 \\ \hline\hline
449: Asir & 0.2 & 4.7 & 14 & 17 & 0.4 \\ \hline
450: %Singular& $>$15min & --- & ---& ---& ---\\ \hline
451: CoCoA 4 & 5.2 & $>$15min & $>$15min & $>$15min & 117 \\ \hline\hline
452: Mathematica 4& 0.2 & 16 & 23 & 36 & 1.1 \\ \hline
453: Maple 7& 0.5 & 18 & 967 & 48 & 1.3 \\ \hline
454: \end{tabular}
455: \end{center}
456: \caption{Multivariate factorization over {\bf Q}}
457: \label{multifac}
458: \end{table}
1.3 noro 459: As to univariate factorization over {\bf Q},
460: the univariate factorizer implements only classical
1.5 noro 461: algorithms and its behavior is what one expects,
1.3 noro 462: that is, it shows average performance in cases
1.5 noro 463: where there are little extraneous factors, but
1.7 ! noro 464: shows poor performance for hard to factor polynomials with
! 465: many extraneous factors.
1.3 noro 466:
1.1 noro 467: \section{OpenXM and Risa/Asir OpenXM interfaces}
468:
469: \subsection{OpenXM overview}
470:
471: OpenXM stands for Open message eXchange protocol for Mathematics.
1.5 noro 472: From the viewpoint of protocol design, it can be regarded as a child
473: of OpenMath \cite{OPENMATH}. However our approach is somewhat
474: different. Our main purpose is to provide an environment for
475: integrating {\it existing} mathematical software systems. OpenXM
476: RFC-100 \cite{RFC100} defines a client-server architecture. Under
477: this specification, a client invokes an OpenXM ({\it OX}) server. The
478: client can send OpenXM ({\it OX}) messages to the server. OX messages
479: consist of {\it data} and {\it command}. Data is encoded according to
480: the common mathematical object ({\it CMO}) format which defines
481: serialized representation of mathematical objects. An OX server is a
482: stackmachine. If data is sent as an OX message, the server pushes the
483: data onto its stack. There is a common set of stackmachine commands
484: and each OX server understands its subset. The command set includes
485: stack manipulating commands and requests for execution of a procedure.
486: In addition, a server may accept its own command sequences if the
487: server wraps some interactive software. That is the server may be a
488: hybrid server.
1.1 noro 489:
490: OpenXM RFC-100 also defines methods for session management. In particular
491: the method to reset a server is carefully designed and it provides
492: a robust way of using servers both for interactive and non-interactive
493: purposes.
494:
495: \subsection{OpenXM client interface of {\tt asir}}
496:
497: Risa/Asir is a main client in OpenXM package. The application {\tt
1.5 noro 498: asir} can access to OpenXM servers via several built-in interface
499: functions. and various interfaces to existing OpenXM servers are
500: prepared as user defined functions written in Asir language.
501: We show a typical OpenXM session.
1.1 noro 502:
503: \begin{verbatim}
504: [1] P = ox_launch(); /* invoke an OpenXM asir server */
505: 0
506: [2] ox_push_cmo(P,x^10-y^10);
507: /* push a polynomial onto the stack */
508: 0
509: [3] ox_execute_function(P,"fctr",1); /* call factorizer */
510: 0
511: [4] ox_pop_cmo(P); /* get the result from the stack */
512: [[1,1],[x^4+y*x^3+y^2*x^2+y^3*x+y^4,1],
513: [x^4-y*x^3+y^2*x^2-y^3*x+y^4,1],[x-y,1],[x+y,1]]
514: [5] ox_cmo_rpc(P,"fctr,",x^10000-2^10000*y^10000);
1.7 ! noro 515: /* call factorizer; a utility function */
1.1 noro 516: 0
517: [6] ox_reset(P); /* reset the computation in the server */
518: 1
519: [7] ox_shutdown(P); /* shutdown the server */
520: 0
521: \end{verbatim}
522:
523: \subsection{OpenXM server {\tt ox\_asir}}
524:
525: An application {\tt ox\_asir} is a wrapper of {\tt asir} and provides
526: all the functions of {\tt asir} to OpenXM clients. It completely
1.7 ! noro 527: implements the OpenXM reset protocol and also allows remote
1.5 noro 528: debugging of user programs running on the server. As an example we
529: show a program for checking whether a polynomial set is a Groebner
530: basis or not. A client executes {\tt gbcheck()} and servers execute
531: {\tt sp\_nf\_for\_gbcheck()} which is a simple normal form computation
1.7 ! noro 532: of an S-polynomial. First of all the client collects all critical pairs
1.1 noro 533: necessary for the check. Then the client requests normal form
534: computations to idling servers. If there are no idling servers the
535: clients waits for some servers to return results by {\tt
536: ox\_select()}, which is a wrapper of UNIX {\tt select()}. If we have
1.5 noro 537: large number of critical pairs to be processed, we can expect good
538: load balancing by {\tt ox\_select()}.
1.1 noro 539:
540: \begin{verbatim}
541: def gbcheck(B,V,O,Procs) {
542: map(ox_reset,Procs);
543: dp_ord(O); D = map(dp_ptod,B,V);
544: L = dp_gr_checklist(D); DP = L[0]; Plist = L[1];
545: /* register DP in servers */
546: map(ox_cmo_rpc,Procs,"register_data_for_gbcheck",vtol(DP));
547: /* discard return value in stack */
548: map(ox_pop_cmo,Procs);
549: Free = Procs; Busy = [];
550: while ( Plist != [] || Busy != [] )
551: if ( Free == [] || Plist == [] ) {
552: /* someone is working; wait for data */
553: Ready = ox_select(Busy);
554: /* update Busy list and Free list */
555: Busy = setminus(Busy,Ready); Free = append(Ready,Free);
556: for ( ; Ready != []; Ready = cdr(Ready) )
557: if ( ox_get(car(Ready)) != 0 ) {
558: /* a normal form is non zero */
559: map(ox_reset,Procs); return 0;
560: }
561: } else {
562: /* update Busy list and Free list */
563: Id = car(Free); Free = cdr(Free); Busy = cons(Id,Busy);
564: /* take a pair */
565: Pair = car(Plist); Plist = cdr(Plist);
566: /* request a normal form computation */
567: ox_cmo_rpc(Id,"sp_nf_for_gbcheck",Pair);
568: ox_push_cmd(Id,262); /* 262 = OX_popCMO */
569: }
570: map(ox_reset,Procs); return 1;
571: }
572: \end{verbatim}
573:
574: \subsection{Asir OpenXM library {\tt libasir.a}}
575:
1.7 ! noro 576: Asir OpenXM library {\tt libasir.a} contains functions simulating the
1.1 noro 577: stack machine commands supported in {\tt ox\_asir}. By linking {\tt
578: libasir.a} an application can use the same functions as in {\tt
1.3 noro 579: ox\_asir} without accessing to {\tt ox\_asir} via TCP/IP. There is
1.7 ! noro 580: also a stack, which can be manipulated by the library functions. In
1.5 noro 581: order to make full use of this interface, one has to prepare
582: conversion functions between CMO and the data structures proper to the
1.7 ! noro 583: application itself. A function {\tt asir\_ox\_pop\_string()} is
! 584: provided to convert CMO to a human readable form, which may be
! 585: sufficient for a simple use of this interface.
1.1 noro 586:
587: \section{Concluding remarks}
588: We have shown the current status of Risa/Asir and its OpenXM
589: interfaces. As a result of our policy of development, it is true that
590: Risa/Asir does not have abundant functions. However it is a completely
1.5 noro 591: open system and its total performance is not bad. Especially on
592: Groebner basis computation over {\bf Q}, many techniques for improving
593: practical performances have been implemented. As the OpenXM interface
594: specification is completely documented, we can easily add another
595: function to Risa/Asir by wrapping an existing software system as an OX
1.7 ! noro 596: server, and other clients can call functions in Risa/Asir by
! 597: implementing the OpenXM client interface. With the remote debugging
! 598: and the function to reset servers, one will be able to enjoy parallel
! 599: and distributed computation with OpenXM facilities.
1.1 noro 600: %
601: \begin{thebibliography}{7}
602: %
603: \addcontentsline{toc}{section}{References}
604:
605: \bibitem{ANY}
606: Anay, H., Noro, M., Yokoyama, K. (1996)
607: Computation of the Splitting fields and the Galois Groups of Polynomials.
608: Algorithms in Algebraic geometry and Applications,
609: Birkh\"auser (Proceedings of MEGA'94), 29--50.
610:
611: \bibitem{FPARA}
612: Jean-Charles Faug\`ere (1994)
613: Parallelization of Groebner basis.
614: Proceedings of PASCO'94, 124--132.
615:
616: \bibitem{F4}
617: Jean-Charles Faug\`ere (1999)
618: A new efficient algorithm for computing Groebner bases ($F_4$).
619: Journal of Pure and Applied Algebra (139) 1-3 , 61--88.
620:
621: \bibitem{FGLM}
622: Faug\`ere, J.-C. et al. (1993)
623: Efficient computation of zero-dimensional Groebner bases by change of ordering.
624: Journal of Symbolic Computation 16, 329--344.
625:
626: \bibitem{RFC100}
627: M. Maekawa, et al. (2001)
628: The Design and Implementation of OpenXM-RFC 100 and 101.
629: Proceedings of ASCM2001, World Scientific, 102--111.
630:
631: \bibitem{RISA}
632: Noro, M. et al. (1994-2001)
633: A computer algebra system Risa/Asir.
634: {\tt http://www.openxm.org}, {\tt http://www.math.kobe-u.ac.jp/Asir/asir.html}.
635:
636: \bibitem{REPL}
637: Noro, M., McKay, J. (1997)
638: Computation of replicable functions on Risa/Asir.
639: Proceedings of PASCO'97, ACM Press, 130--138.
640:
641: \bibitem{NOYO}
642: Noro, M., Yokoyama, K. (1999)
643: A Modular Method to Compute the Rational Univariate
644: Representation of Zero-Dimensional Ideals.
645: Journal of Symbolic Computation, 28, 1, 243--263.
646:
647: \bibitem{OPENXM}
648: OpenXM committers (2000-2001)
649: OpenXM package.
650: {\tt http://www.openxm.org}.
1.7 ! noro 651:
! 652: \bibitem{RUR}
! 653: Rouillier, R. (1996)
! 654: R\'esolution des syst\`emes z\'ero-dimensionnels.
! 655: Doctoral Thesis(1996), University of Rennes I, France.
1.1 noro 656:
657: \bibitem{SY}
658: Shimoyama, T., Yokoyama, K. (1996)
659: Localization and Primary Decomposition of Polynomial Ideals.
660: Journal of Symbolic Computation, 22, 3, 247--277.
661:
662: \bibitem{TRAGER}
663: Trager, B.M. (1976)
664: Algebraic Factoring and Rational Function Integration.
665: Proceedings of SYMSAC 76, 219--226.
666:
667: \bibitem{TRAV}
668: Traverso, C. (1988)
669: Groebner trace algorithms.
670: LNCS {\bf 358} (Proceedings of ISSAC'88), Springer-Verlag, 125--138.
671:
1.5 noro 672: \bibitem{BENCH}
673: {\tt http://www.math.uic.edu/\~\,jan/demo.html}.
674:
1.1 noro 675: \bibitem{COCOA}
676: {\tt http://cocoa.dima.unige.it/}.
677:
678: \bibitem{FGB}
679: {\tt http://www-calfor.lip6.fr/\~\,jcf/}.
680:
1.5 noro 681: %\bibitem{NTL}
682: %{\tt http://www.shoup.net/}.
1.1 noro 683:
684: \bibitem{OPENMATH}
685: {\tt http://www.openmath.org/}.
686:
687: \bibitem{SINGULAR}
688: {\tt http://www.singular.uni-kl.de/}.
689:
690: \end{thebibliography}
691:
692: %INDEX%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
693: \clearpage
694: \addcontentsline{toc}{section}{Index}
695: \flushbottom
696: \printindex
697: %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
698:
699: \end{document}
700:
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