% $OpenXM: OpenXM/doc/Papers/dagb-noro.tex,v 1.3 2001/10/04 08:16:26 noro Exp $
\setlength{\parskip}{10pt}

\begin{slide}{}
\begin{center}
\fbox{\large Part I : Overview and history of Risa/Asir}
\end{center}
\end{slide}

\begin{slide}{}
\fbox{A computer algebra system Risa/Asir}

\begin{itemize}
\item Old style software for polynomial computation

\begin{itemize}
\item Domain specification is not necessary prior to computation
\item automatic conversion of inputs into internal canonical forms
\end{itemize}

\item User language with C-like syntax

\begin{itemize}
\item No type declaration of variables
\item Builtin debugger for user programs
\end{itemize}

\item Open source

\begin{itemize}
\item Whole source tree is available via CVS
\end{itemize}

\item OpenXM ((Open message eXchange protocol for Mathematics) interface

\begin{itemize}
\item As a client : can call procedures on other OpenXM servers
\item As a server : offers all its functionalities to OpenXM clients
\item As a library : OpenXM functionality is available via subroutine calls
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{Major functionalities}

\begin{itemize}
\item Fundamental polynomial arithmetics

\begin{itemize}
\item Internal form of a polynomial : recursive representaion or distributed
representation
\end{itemize}

\item Polynomial factorization

\begin{itemize}
\item Univariate factorization over the rationals, algebraic number fields and various finite fields

\item Multivariate factorization over the rationals
\end{itemize}

\item Groebner basis computation

\begin{itemize}
\item Buchberger and $F_4$ [Faug\'ere] algorithm

\item Change of ordering/RUR [Rouillier] of 0-dimensional ideals

\item Primary ideal decomposition

\item Computation of $b$-function
\end{itemize}

\item PARI [PARI] library interface 

\item Paralell distributed computation under OpenXM 
\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{History of development : ---1994}

\begin{itemize}
\item --1989

Several subroutines were developed for a Prolog program.

\item 1989--1992

\begin{itemize}
\item Reconfigured as Risa/Asir with a parser and Boehm's conservative GC [Boehm]

\item Developed univariate and multivariate factorizers over the rationals.
\end{itemize}

\item 1992--1994

\begin{itemize}
\item Started implementation of Buchberger algorithm

Written in user language $\Rightarrow$ rewritten in C (by Murao)

$\Rightarrow$ trace lifting [Traverso]

\item Univariate factorization over algebraic number fields

Intensive use of successive extension, non-squarefree norms
\end{itemize}
\end{itemize}

\end{slide}

\begin{slide}{}
\fbox{History of development : 1994-1996}

\begin{itemize}
\item Free distribution of binary versions from Fujitsu

\item Primary ideal decomposition

\begin{itemize}
\item Shimoyama-Yokoyama algorithm [SY]
\end{itemize}

\item Improvement of Buchberger algorithm

\begin{itemize}
\item Trace lifting+homogenization

\item Omitting check by compatible prime

\item Modular change of ordering, Modular RUR

These are joint works with Yokoyama [NY]
\end{itemize}
\end{itemize}

\end{slide}

\begin{slide}{}
\fbox{History of development : 1996-1998}

\begin{itemize}
\item Distributed compuatation

\begin{itemize}
\item A prototype of OpenXM
\end{itemize}

\item Improvement of Buchberger algorithm

\begin{itemize}
\item Content reduction during nomal form computation

\item Its parallelization by the above facility

\item Computation of odd order replicable functions [Noro]

Risa/Asir : it took 5days to compute a DRL basis ({\it McKay})

Faug\`ere FGb : computation of the DRL basis 53sec
\end{itemize}


\item Univariate factorization over large finite fields

\begin{itemize}
\item To implement Schoof-Elkies-Atkin algorithm 

Counting rational points on elliptic curves

--- not free But related functions are freely available
\end{itemize}
\end{itemize}

\end{slide}

\begin{slide}{}
\fbox{History of development : 1998-2000}
\begin{itemize}
\item OpenXM

\begin{itemize}
\item OpenXM specification was written by Noro and Takayama

Borrowed idea on encoding, phrase book from OpenMath [OpenMath]

\item Functions for distributed computation were rewritten
\end{itemize}

\item Risa/Asir on Windows

\begin{itemize}
\item Requirement from a company for which Noro worked

Written in Visual C++
\end{itemize}

\item Test implementation of $F_4$

\begin{itemize}
\item Implemented according to [Faug\`ere]

\item Over $GF(p)$ : pretty good

\item Over the rationals : not so good except for {\it McKay}
\end{itemize}
\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{History of development : 2000-current}
\begin{itemize}
\item The source code is freely available

\begin{itemize}
\item Noro moved from Fujitsu to Kobe university

Started Kobe branch [Risa/Asir]
\end{itemize}

\item OpenXM [OpenXM]

\begin{itemize}
\item Revising the specification : OX-RFC100, 101, (102)

\item OX-RFC102 : communications between servers via MPI
\end{itemize}

\item Weyl algebra

\begin{itemize}
\item Buchberger algorithm [Takayama]

\item $b$-function computation [Oaku]

Minimal polynomial computation by modular method
\end{itemize}
\end{itemize}

\end{slide}

\begin{slide}{}
\fbox{Status of each component --- Factorizer}

\begin{itemize}
\item 10 years ago

its performace was fine compared with existing software
like REDUCE, Mathematica.

\item 4 years ago

Univarate factorization over algebraic number fields was
still fine because of some tricks on factoring polynomials
derived from norms.

\item Current

Multivariate : not so bad

Univariate : completely obsolete by M. van Hoeij's new algorithm
[Hoeij]
\end{itemize}

\end{slide}

\begin{slide}{}
\fbox{Status of each component --- Groebner basis related functions}

\begin{itemize}
\item 8 years ago

The performace was poor with only the sugar strategy.

\item 7 years ago

Rather fine with trace lifting but Faug\`ere's (old)Gb was more
efficient.  

Homogenization+trace lifting made it possible to compute
wider range of Groebner bases.

\item 4 years ago

Modular RUR was comparable with Rouillier's implementation.

\item Current

FGb seems much more efficient than our $F_4$ implementation.

Singular [Singular] is also several times
faster than Risa/Asir, because Singular seems to have efficient
monomial and polynomial representation.

\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{OpenXM}

\begin{itemize}
\item An environment for parallel distributed computation

Both for interactive, non-interactive environment

\item Message passing

OX (OpenXM) message : command and data

\item Hybrid command execution

\begin{itemize}
\item Stack machine command

push, pop, function execution, $\ldots$

\item accepts its own command sequences

{\tt execute\_string} --- easy to use
\end{itemize}

\item Data is represented as CMO

CMO (Common Mathematical Object format)

--- Serialized representation of mathematical object

{\sl Integer32}, {\sl Cstring}, {\sl List}, {\sl ZZ}, $\ldots$
\end{itemize}
\end{slide}


\begin{slide}{}
\fbox{OpenXM and OpenMath}

\begin{itemize}
\item OpenMath

\begin{itemize}
\item A standard for representing mathematical objects

\item CD (Content Dictionary) : assigns semantics to symbols

\item Phrasebook : convesion between internal and OpenMath objects.

\item Encoding : format for actual data exchange
\end{itemize}

\item OpenXM

\begin{itemize}
\item Specification for encoding and exchanging messages

\item It also specifies behavior of servers and session management
\end{itemize}

\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{OpenXM server interface in Risa/Asir}

\begin{itemize}
\item TCP/IP stream

\begin{itemize}
\item Launcher

A client executes a launcher on a host.

The launcher launches a server on the same host.

\item Server

A server reads from the descriptor 3, write to the descriptor 4.

\end{itemize}

\item Subroutine call

Risa/Asir subroutine library provides interfaces corresponding to
pushing and popping data and executing stack commands.
\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{OpenXM client interface in Risa/Asir}

\begin{itemize}
\item Primitive interface functions

Pushing and popping data, sending commands etc.

\item Convenient functions

Launching servers, calling remote functions,
 interrupting remote executions etc.

\item Parallel distributed computation is easy

Simple parallelization is practically important

Competitive computation is easily realized 
\end{itemize}
\end{slide}


%\begin{slide}{}
%\fbox{CMO = Serialized representation of mathematical object}
%
%\begin{itemize}
%\item primitive data
%\begin{eqnarray*}
%\mbox{Integer32} &:& ({\tt CMO\_INT32}, {\sl int32}\ \mbox{n}) \\
%\mbox{Cstring}&:& ({\tt CMO\_STRING},{\sl int32}\,  \mbox{ n}, {\sl string}\, \mbox{s}) \\
%\mbox{List} &:& ({\tt CMO\_LIST}, {\sl int32}\, len, ob[0], \ldots,ob[m-1])
%\end{eqnarray*}
%
%\item numbers and polynomials
%\begin{eqnarray*}
%\mbox{ZZ}         &:& ({\tt CMO\_ZZ},{\sl int32}\, {\rm f}, {\sl byte}\, \mbox{a[1]}, \ldots
%{\sl byte}\, \mbox{a[$|$f$|$]} ) \\
%\mbox{Monomial32}&:& ({\tt CMO\_MONOMIAL32}, n, \mbox{e[1]}, \ldots, \mbox{e[n]}, \mbox{Coef}) \\
%\mbox{Coef}&:& \mbox{ZZ} | \mbox{Integer32} \\
%\mbox{Dpolynomial}&:& ({\tt CMO\_DISTRIBUTED\_POLYNOMIAL},\\
%                  & & m, \mbox{DringDefinition}, \mbox{Monomial32}, \ldots)\\
%\mbox{DringDefinition}
%                  &:& \mbox{DMS of N variables} \\
%                  & & ({\tt CMO\_RING\_BY\_NAME}, name) \\
%                  & & ({\tt CMO\_DMS\_GENERIC}) \\
%\end{eqnarray*}
%\end{itemize}
%\end{slide}
%
%\begin{slide}{}
%\fbox{Stack based communication}
%
%\begin{itemize}
%\item Data arrived a client
%
%Pushed to the stack
%
%\item Result
%
%Pushd to the stack
%
%Written to the stream when requested by a command
%
%\item The reason why we use the stack
%
%\begin{itemize}
%\item Stack = I/O buffer for (possibly large) objects
%
%Multiple requests can be sent before their exection
%
%A server does not get stuck in sending results
%\end{itemize}
%\end{itemize}
%\end{slide}

\begin{slide}{}
\fbox{Executing functions on a server (I) --- {\tt SM\_executeFunction}}

\begin{enumerate}
\item (C $\rightarrow$ S) Arguments are sent in binary encoded form.
\item (C $\rightarrow$ S) The number of aruments is sent as {\sl Integer32}.
\item (C $\rightarrow$ S) A function name is sent as {\sl Cstring}.
\item (C $\rightarrow$ S) A command {\tt SM\_executeFunction} is sent.
\item The result is pushed to the stack.
\item (C $\rightarrow$ S) A command {\tt SM\_popCMO} is sent.
\item (S $\rightarrow$ C) The result is sent in binary encoded form.
\end{enumerate}

$\Rightarrow$ Communication is fast, but functions for binary data
conversion are necessary.
\end{slide}

\begin{slide}{}
\fbox{Executing functions on a server (II) --- {\tt SM\_executeString}}

\begin{enumerate}
\item (C $\rightarrow$ S) A character string represeting a request in a server's
user language is sent as {\sl Cstring}.
\item (C $\rightarrow$ S) A command {\tt SM\_executeString} is sent.
\item The result is pushed to the stack.
\item (C $\rightarrow$ S) A command {\tt SM\_popString} is sent.
\item (S $\rightarrow$ C) The result is sent in readable form.
\end{enumerate}

$\Rightarrow$ Communication may be slow, but the client parser may be
enough to read the result.
\end{slide}

\begin{slide}{}
\fbox{Example of distributed computation --- $F_4$ vs. $Buchberger$ }

\begin{verbatim}
/* competitive Gbase computation over GF(M) */
/* Cf. A.28 in SINGULAR Manual */
/* Process list is specified as an option : grvsf4(...|proc=P) */
def grvsf4(G,V,M,O)
{
  P = getopt(proc);
  if ( type(P) == -1 ) return dp_f4_mod_main(G,V,M,O);
  P0 = P[0]; P1 = P[1]; P = [P0,P1];
  map(ox_reset,P);
  ox_cmo_rpc(P0,"dp_f4_mod_main",G,V,M,O);
  ox_cmo_rpc(P1,"dp_gr_mod_main",G,V,0,M,O);
  map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
  F = ox_select(P); R = ox_get(F[0]);
  if ( F[0] == P0 ) { Win = "F4"; Lose = P1;}
  else { Win = "Buchberger"; Lose = P0; }
  ox_reset(Lose); /* simply resets the loser */
  return [Win,R];
}
\end{verbatim}
\end{slide}

\begin{slide}{}
\fbox{References}

[Bernardin] L. Bernardin, On square-free factorization of 
multivariate polynomials over a finite field, Theoretical
Computer Science 187 (1997), 105-116.

[Boehm] {\tt http://www.hpl.hp.com/personal/Hans\_Boehm/gc}

[Faug\`ere] J.C. Faug\`ere,
A new efficient algorithm for computing Groebner bases  ($F_4$),
Journal of Pure and Applied Algebra (139) 1-3 (1999), 61-88.

[Hoeij] M. van Heoij, Factoring polynomials and the knapsack problem,
to appear in Journal of Number Theory (2000).

[NY] M. Noro, K. Yokoyama, 
A Modular Method to Compute the Rational Univariate
Representation of Zero-Dimensional Ideals.
J. Symb. Comp. {\bf 28}/1 (1999), 243-263.

[Oaku] T. Oaku, Algorithms for $b$-functions, restrictions and algebraic
local cohomology groups of $D$-modules.
Advancees in Applied Mathematics, 19 (1997), 61-105.
\end{slide}

\begin{slide}{}

[OpenMath] {\tt http://www.openmath.org}

[OpenXM] {\tt http://www.openxm.org}

[PARI] {\tt http://www.parigp-home.de}

[Risa/Asir] {\tt http://www.math.kobe-u.ac.jp/Asir/asir.html}

[Rouillier] F. Rouillier,
R\'esolution des syst\`emes z\'ero-dimensionnels. 
Doctoral Thesis(1996), University of Rennes I, France.

[SY] T. Shimoyama, K. Yokoyama, Localization and Primary Decomposition of Polynomial Ideals.  J. Symb. Comp. {\bf 22} (1996), 247-277.

[Singular] {\tt http://www.singular.uni-kl.de}

[Traverso] C. Traverso, \gr trace algorithms. Proc. ISSAC '88 (LNCS 358), 125-138.

\end{slide}

\begin{slide}{}
\begin{center}
\fbox{\large Part II : Algorithms and implementations in Risa/Asir}
\end{center}
\end{slide}

\begin{slide}{}
\fbox{Ground fields}

\begin{itemize}
\item The rational number field
\item Algebraic number fields

represented by successive extensions
\item Finite fields
\begin{itemize}
\item $GF(p)$ ($p<2^{30}$); represented by a single word
\item $GF(p^n)$ ($p^n < 2^{20}$); represented by a primitive root
\item $GF(2^n)$; represented by a bit string
\item $GF(p)$ ($p$ : arbitrary prime)
\item $GF(p^n)$ ($p$ : arbitrary odd prime)
\end{itemize}

\item Real and complex number fields

\begin{itemize}
\item Double float
\item PARI bigfloat
\end{itemize}

\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{Polynomial factorization}
\begin{itemize}
\item Univariate factorization

\begin{itemize}
\item Over the rationals

Berlekamp-Zassenhaus

(classical; knapsack has not yet implemented)

\item Over algebraic number fields

Trager's algorithm + some improvement

\item Over finite fieds

DDF + Cantor-Zassenhaus; FFT for large finite fields
\end{itemize}

\item Multivariate factorization

\begin{itemize}
\item Over the rationals

Classical EZ algorithm

\item Over small finite fieds

Modified Bernardin's square free algorithm [Bernardin],

possibly Hensel lifting over extension fields
\end{itemize}

\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{Groebner basis computation --- Buchberger algorithm}
\begin{itemize}
\item Polynomial rings
\begin{itemize}
\item Over finite fields

Any finite field is available as a ground field

\item Over the rationals

Guess of a groebner basis by detecting zero reduction by modular computation
+ check (trace lifting)

Homogenization+guess+dehomogenization+check
\end{itemize}

\item Weyl Algebra

\begin{itemize}
\item Groebner basis of a left ideal

Key : an efficient implementation of Leibniz rule
\end{itemize}

\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{$F_4$ algorithm}
\begin{itemize}
\item Over small finite fields ($GF(p)$, $p < 2^{30}$)
\begin{itemize}
\item More efficient than our Buchberger algorithm implementation

but less efficient than FGb by Faugere
\end{itemize}

\item Over the rationals

\begin{itemize}
\item Very naive implementation

Modular computation + CRT + Checking the result at each degree

\item Less efficient than Buchberger algorithm

except for one example (={\it McKay})
\end{itemize}

\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{Change of ordering for zero-dimensional ideals}

\begin{itemize}
\item Any ordering to lex ordering

\begin{itemize}
\item Modular change of ordering

Guess of the support by modular FGLM

+ solving linear systems by Hensel lifting

\end{itemize}

\item RUR (generalized shape lemma)

\begin{itemize}
\item Modular RUR (only implemented on the shape base case)

Almost the same as modular change of ordering
\end{itemize}

\end{itemize}
\end{slide}

\begin{slide}{}
\fbox{Primary decomposition --- Shimoyama-Yokoyama algorithm}

\begin{itemize}
\item Only implemented over the rationals

Finite field version will soon be available

\item Pseudo primary ideal

An ideal whose radical is prime

\item Prime decomp. of the radical $\Rightarrow$ pseudo primary decomp.

\item Extraction of embedded components

\end{itemize}

\end{slide}

\begin{slide}{}
\fbox{Computation of $b$-function}

$D=K\langle x,\partial \rangle$ : Weyl algebra

$b(s)$ : a polynomial of the smallest degree s.t.
there exists $P(s) \in D[s]$, $P(s)f^{s+1}=b(s)f^s$

$b(s)$ : $b$-function of a polynomial $f$

$\Rightarrow$ $b(s)$ can be computed by Oaku's algorithm

On Risa/Asir : $b(s)$ is computed by

A groebner basis $\Rightarrow$ guess of $\deg(b(s))$ by modular
computation $\Rightarrow$ solving a linear system
\end{slide}

\begin{slide}{}
\fbox{Interface to PARI library}

\begin{itemize}
\item Data conversion

\begin{itemize}

\item Only primitive data can be passed to PARI

Rational number, bignum, bigfloat, polynomial,
vector, matrix

\item Results are converted to Risa objects

\end{itemize}

\item Evaluation of transcendental functions

\begin{itemize}
\item Most of univariate functions in PARI can be 
evaluated by {\tt eval()}
\end{itemize}

\item Calling PARI functions

\begin{itemize}
\item One can call PARI functions by {\tt pari({\it parifunction},{\it args})}

The knapsack factorization is available via {\tt pari(factor,{\it poly})}
\end{itemize}
\end{itemize}
\end{slide}

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