Computer Algebra Seminar No.25 (計算代数セミナー, 第25回) at Kyushu Univ.
(九州大学)
Aug 25 (Wed), 2004. 8月25日 (水曜日), 10:00 --- 17:00
場所:
理学部３号館３３１１室 (変更です!).
Room No.3311 (3F), Third Building of Faculty of Sciences
All talks will be presented in English.
Koji Nakagawa (九州大学) [中川 康二]
Converting Printed Mathematical Document into a Content Base Format
[印刷された数学文書からその内容をとりだすことについて]
,
10:00--10:45
OCR (Optical Character Recognition) technology makes it possible to
digitize printed mathematical documents into presentation-base XML
languages. In order to use the data stored in the XML languages
usefully (e.g. search), the data should be converted into content base
formats that show logical structures. In this research we convert a
presentation-base XML language KML, obtained from an integrated OCR
environment Infty, to a content-base XML language OMDoc.
Tadashi Takahashi (神戸大学) [高橋正]
Groebner Bases and Invariants
[グレブナ基底と不変量]
,
11:00--11:45
Invariants of rational singularities are well known. They obtained by
using actions of quotient ring. We can get the invariants of
singularities by using their Groebner Basis. We will show them in this
presentation.
Bruno Buchberger (京都大学、RISC-Linz)
Algorithmic Algorithm Synthesis: Case Study Groebner Bases
[アルゴリズム合成のアルゴリズム: グレブナ基底の場合]
,
13:30--15:00
Recently, we introduced a new method, called "Lazy Thinking", for the
algorithmic invention of algorithms. The method starts from a given
formal specifications of a problem in predicate logic and tries out
various algorithm schemes from a library of possible algorithm
schemes. An algorithm scheme is a recursive definition of an
algorithm in terms of unspecified auxiliary functions. For each
algorithm scheme in the library, the method tries an (automated)
correctness proof w.r.t. to the given problem specification.
Normally, this proof will fail because nothing is known about the
unspecificied subalgorithms. From the failing proof, the method
automatically generates specifications for the unknown subalgorithms
that will guarantee that the correctness proof can be completed. Now,
there are two possibilities: Either we know algorithms that meet the
specifications generated for the subalgorithms. Then we are done,
i.e. we have synthesized an algorithm for the original problem. Or we
apply the lazy thinking method recursively to the specifications of
the unknown subalgorithms until we arrive at problem specifications
that can be met by known algorithms.
We implemented the method in the Theorema system, which is a software
system for mathematical theory exploration. At the beginning, for
consolidating the method, we successfully tried out the lazy thinking
algorithm synthesis method for synthesizing various sorting
algorithms. Recently, we were able to show that the method is powerful
enough to synthesize the author's Groebner bases algorithm, which is
deemed to be a nontrivial algorithm in the area of computer algebra
and, hence, may well serve as a benchmark for the logical power of
algorithm synthesis methods.
In the talk, we will explain the method and, in particular, the synthesis
of the Groebner bases algorithm in detail. We will finally draw some
conclusions for the methodology of algorithmic mathematics, for the future
of mathematical software systems, and for the future education of
mathematics and computer science students.
I will show how this method can be successfully applied to the
automated invention of my Groebner bases algorithm starting from a
formal specification of the Groebner bases problem.
This talk is dedicated to the memory of Mikhail Mescsheryakov,
director of LCTA at JINR, where I was a visiting researcher in 1970 /
71.
Mitsushi Fujimoto (福岡教育大学) [藤本光史]
On polynomial curves in the affine plane
[アフィン平面の多項式曲線について]
,
15:15--16:00
A curve that can be parametrized by polynomials is called a polynomial
curve. It is well-known that a polynomial curve has only one place at
infinity.
In 1977, Sathaye indicated the Abhyankar's question for curves with
one place at infinity.
Let $C$ be a curve with one place at infinity, and $\Omega$ be the
semigroup generated by pole orders of $C$ at infinity. Is there a
polynomial curve associated with $\Omega$? In 1990, Sathaye-Stenerson
suggested a candidate of counter example for this question. However,
they could not give the answer to the question since the root
computation for a huge polynomial system was required. We found a
counter example for the Abhyankar's question using Groebner basis
computation. In this talk, we give the details.
Seiichi Toyoda (九州大学) [豊田晴一]
joint work with Masakazu Suzuki, Mitsushi Fujimoto
An Implementation of Symbolic Integration on Risa/Asir
[Risa/Asirへの記号積分法の実装]
,
16:15--17:00
We implemented symbolic integration on Risa/Asir. Algorithms which we
adopted are heuristic. We think that heuristic algorithms are useful
in education. At present, our package works for a part of elementary
functions. In our talk, we would like to present our algorithms,
implementation, educational use and future works.
---------------------------------------------------------------
The computer algebra seminar is co-organized by
M.Noro, noro@math.kobe-u.ac.jp
N.Takayama, takayama@math.kobe-u.ac.jp
K.Yokoyama, yokoyama@math.kyushu-u.ac.jp
For more info see http://www.math.kobe-u.ac.jp/seminar.html/compalg.html