皆様
リュブリャナ大学のAndrej Bauer教授の講演のお知らせです。
ふるってご参加ください。
問合せ先:
石原 哉
北陸先端科学技術大学院大学 情報科学研究科
e-mail: ishihara(a)jaist.ac.jp
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* JAIST Logic Seminar Series *
* This seminar is held as a part of the EU FP7 Marie Curie Actions
IRSES project COMPUTAL (http://computal.uni-trier.de/).
Date: Friday 20 September, 2013, 15:00-17:00
Place: JAIST, Collaboration room 6 (I-57g)
(Access: http://www.jaist.ac.jp/english/location/access.html)
Speaker: Professor Andrej Bauer (University of Ljubljana, Slovenia)
Title: A higher inductive construction of the real numbers
Abstract:
Homotopy type theory [1] is a foundation of mathematics based on
Martin-Löf type
theory extended with the Univalence axiom and higher inductive types. The
latter allow us to present mathematical objects as inductive types generated
by point and path constructors. For instance, the circle is represented by a
type with a point and a loop satisfying the "circle induction" principle.
In the first part I shall review higher inductive types.
In the second part of the talk I shall look at a higher inductive
construction
of real numbers. Ordinary higher inductive types do not seem to suffice --
instead we have to use a higher inductive-inductive definition which
simultaneously constructs the reals and a proximity relation on them. We
thus
obtain a type of real numbers which satisfies a principle of "real
induction".
We can prove properties of reals by induction!
From a logical point of view the construction is interesting because
it avoids the
axiom of choice, impredicativity (powersets), and excluded middle.
Nevertheless, the resulting theory still allows us to develop the basics of
real analysis as usual.
The work described in the talk is joint work of the members of the Univalent
foundations project [2].
References:
[1] Univalent Foundations Project: "Homotopy type theory: Univalent
foundations of Mathematics",
http://homotopytypetheory.org/book/
[2] Univalent Foundations Project, Institute for Advanced Study, Princeton,
http://www.math.ias.edu/sp/univalent
--
Professor Hajime Ishihara
School of Information Science
Japan Advanced Institute of Science and Technology
1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan
Tel: +81-761-51-1206
Fax: +81-761-51-1149
ishihara(a)jaist.ac.jp
http://www.jaist.ac.jp/~ishihara
Dr. Andreas Abel at NII Logic Seminar
Date: October 11, 2013, 14:00--16:00
Place: National Institute of Informatics, Room 1210 (12th floor)
場所: 国立情報学研究所 12階 1210室
(半蔵門線,都営地下鉄三田線・新宿線 神保町駅または東西線 竹橋駅より徒歩5分)
(地図 http://www.nii.ac.jp/about/access/)
Speaker: Dr. Andreas Abel (Chalmers and Gothenburg University)
Title: Copatterns: Programming Infinite Structures by Observations
Abstract:
Inductive datatypes provide mechanisms to define finite data such as
finite lists and trees via constructors and allow programmers to
analyze and manipulate finite data via pattern matching. In this
paper, we develop a dual approach for working with infinite data
structures such as streams. Infinite data inhabits coinductive
datatypes which denote greatest fixpoints. Unlike finite data which
is defined by constructors we define infinite data by observations.
Dual to pattern matching, a tool for analyzing finite data, we develop
the concept of copattern matching, which allows us to synthesize
infinite data. This leads to a symmetric language design where
pattern matching on finite and infinite data can be mixed.
We present a core language for programming with infinite structures by
observations together with its operational semantics based on
(co)pattern matching and describe coverage of copatterns. Our
language naturally supports both call-by-name and call-by-value
interpretations and can be seamlessly integrated into existing
languages like Haskell and ML.
We prove type soundness for our language and sketch
how copatterns open new directions for solving problems in the
interaction of coinductive and dependent types.
This is joint work with Brigitte Pientka, McGill University, Montreal.
問合せ先:
龍田 真 (国立情報学研究所)
e-mail: tatsuta(a)nii.ac.jp
http://research.nii.ac.jp/~tatsuta