The Sixth International Symposium
on Domain Theory and its Applications
(ISDT’13)
October 25-29, 2013
Hunan University
Changsha, China
II
Contents
1. Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
2. Conference Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VI
3. Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI
III
Information
The Sixth International symposium on Domain Theory will take place on the campus
of Hunan University in Changsha, China, from October 25 to October 29, 2013 (the
first ISDT was held in Shanghai, October 17-24, 1999, the second ISDT was held in
Chengdu, China, October 22-26, 2001, the third ISDT was held in Xi’an, China, May
10-14, 2004, the fourth ISDT was held in Changsha, China, June 2-6, 2006, and the
fifth ISDT was held in Shanghai, China, September 12-14, 2009).
This conference is intended to be a forum for researchers in domain theory and its
applications. The conference series also aim to broaden its scope of applications in
computer science and mathematics.
Topics of interest
• Topological and logical aspects of domains
• Categories of domains and powerdomains
• Continuous posets and fuzzy domains
• Partial orders, lattice theory and metric spaces
• Types, process algebra and concurrency
• Non-classical and partial logics
• Programming language semantics
• Applications in computer science and mathematics
Invited speakers
The confirmed invited speakers and professors for invited presentations are:
Dana Scott (Keynote), Carnegie Mellon University, USA
Achim Jung, University of Birmingham, UK
Glynn Winskel, University of Cambridge, UK
Pierre-Louis Curien, CNRS & Université Paris Diderot, France
Klaus Keimel, Technische Universitaet Darmstadt, Germany
Jimmie Lawson, Louisiana State University, USA
Michael Mislove, Tulane University, USA
IV
Guo-Qiang Zhang, Case Western Reserve University, USA
Bin Zhao, Shaanxi Normal University, China
Committees
Program Committee:
Ying-Ming Liu (chair), Sichuan University, China
Mao-kang Luo (co-chair), Sichuan University, China
Michael Mislove(co-chair), Tulane University, USA
Guo-Qiang Zhang (co-chair), Case Western Reserve University, USA
Lars Birkedal, IT University of Copenhagen, Denmark
Yixiang Chen, East China Normal University, China
Pierre-Louis Curien, CNRS & Université Paris Diderot, France
Martin Escardo, University of Birmingham, UK
Yuxi Fu, Shanhai Jiao Tong University, China
Ying Jiang, Institute of Software, Chinese Academy of Sciences, China
Klaus Keimel, Technische Universitaet Darmstadt, Germany
Hui Kou, Sichuan University, China
Jimmie Lawson, Louisiana State University, USA
Andrew Moshier, Chapman University, USA
Jan Rutten, CWI & Radboud University Nijmegen, Netherland
Daniele Varacca, Université Paris Diderot, France
Guo-Jun Wang, Shaanxi Normal University, China
Luoshan Xu, Yangzhou University, China
Xiaoquan Xu, Jiangxi Normal University, China
Zhongqiang Yang, Shantou University, China
Dexue Zhang, Sichuan University, China
Bin Zhao, Shaanxi Normal University, China
V
Organizing and Local Arrangement Committee:
Yueyu Zhao (chair) (Hunan University, President)
Yueping Jiang (co-Chair) (Hunan University, Dean, College of Mathematics
and Econometrics)
Qingguo Li (co-Chair) (Hunan University, Dean, Graduate School)
Web site: http://math.hnu.cn/isdt13
VI
Conference Program for ISDT’13
October 25
07:00-07:40
Breakfast
12:00-13:00
Lunch
18:00-19:00
Dinner
08:00-22:00
Registration
Site:
Jixian Hotel
Note:
Most activities except the talks during the conference will take place
in this hotel. The registration desk is situated in the lobby; Breakfast,
Lunch and Dinner will be served in the dinning room on the first floor;
Accommodations will be arranged within the hotel.
Attention:
• Please wear your delegate card when you are present at the symposium, attend relevant activities or have meals.
• Talks are given in the meeting room on the the first floor of the
building of Business School of Hunan University.
• Please show your breakfast ticket before every breakfast in Jixian
Hotel.
VII
October 26
07:00-08:10
Breakfast
Session A
Chair: Qingguo Li
08:30-09:10
Opening ceremony
Welcome to ISDT by President Yueyu Zhao
Welcome to ISDT by Prof. Yingming Liu
09:10-09:30
Group picture
09:30-10:00
Coffee Break
Session B
Chair: Guo-Qiang Zhang
10:00-11:00
Keynote Speaker: Dana Scott
Stochastic Lambda Calculi
11:00-11:40
Contributed talk: Haoran Zhao, Hui Kou
Tω as a stable universal domain
12:00-13:00
Lunch
Session C
Chair: Michael Mislove
14:30-15:10
Invited presentation: Klaus Keimel
On the equivalence of state transformer and predicate
transformer semantics
15:10-15:50
Contributed talk: Wenfeng Zhang, Xiaoquan Xu
Completely precontinuous posets
15:50-16:10
Contributed talk: Xiaoyong Xi, Guohua Wu
A note on coherent domains
16:10-16:30
Coffee break
Session D
Chair: Yixiang Chen
16:30-17:10
Invited presentation: Bin Zhao, Shengwei Han
On the power-set Q-algebras
VIII
17:10-17:50
Contributed talk: Moez A. AbdelGawad
A Domain-Theoretic Model Of Nominally-Typed
Object-oriented Programming
17:50-18:10
Contributed talk: Mucong Zheng, Zhongke Shi,
Guodong Hang
Triple I method of IFMT based on residual implication
18:30-20:00
Banquet
October 27
07:00-08:10
Breakfast
Session A
Chair: Pierre-Louis Curien
08:30-09:30
Invited Speaker: Glynn Winskel
Concurrent strategies
09:30-09:50
Contributed talk: Xiaoquan Xu
Double dual topologies
09:50-10:10
Coffee break
Session B
Chair: Achim Jung
10:10-10:50
Contributed talk: Mingyuan Wu, Qingguo Li,
Xiangnan Zhou
Representations of algebraic domains
and algebraic L-domain by information systems
10:50-11:30
Contributed talk: Ingo Battenfeld
Observationally-induced algebras in Domain Theory
11:30-11:50
Contributed talk: Hengyang Wu, Yixiang Chen
Coalgebras for Fuzzy Transition Systems
12:00-13:00
Lunch
IX
Session C
Chair: Glynn Winskel
14:30-15:10
Invited presentation: Pierre-Louis Curien
mu and mu tilde: two useful combinators
15:10-15:50
Contributed talk: Wei Yao
A duality between ω-categories and algebraic ω-categories
15:50-16:10
Contributed talk: Qingyu He, Luoshan Xu
On Semicontinuous Lattices and Their distributive reflections
16:10-16:30
Coffee break
Session D
Chair: Bin Zhao
16:30-17:10
Invited presentation: Jimmie Lawson
Close Encounters of the Third Kind:
Domain Theory Meets T0 -Spaces Meets Topology
17:10-17:50
Contributed talk: Luoshan Xu, Jing Zhao
Reductions and Saturation Reductions of (Abstract)
Knowledge Bases
17:50-18:10
Contributed talk: Xuechong Guan, Yongming Li
On a Condition for Semirings to Induce Compact
Information Algebras
18:10-19:10
Dinner
October 28
07:00-08:10
Breakfast
Session A
Chair: Klaus Keimel
08:30-09:30
Invited speaker: Achim Jung
Continuous Domain Theory in Logical Form
09:30-09:50
Contributed talk: Lankun Guo, Petko Valtchev,
Qingguo Li, Robert Godin
A note on finitely derived information systems
09:50-10:10
Coffee break
X
Session B
Chair: Jimmie Lawson
10:10-10:50
Invited presentation: Guo-Qiang Zhang
TCS-Thinking in Biomedical Ontology Research
10:50-11:30
Contributed talk: Weng Kin HO
Characterising E-projectives via Co-monads
11:30-11:50
Contributed talk: Mengqiao Huang, Qingguo Li,
Lankun Guo
Formal contexts for Algebraic Domains
11:50-13:00
Lunch
Session C
Chair: Maokang Luo
14:30-15:10
Invited presentation: Michael Mislove
On Random Variable Models of Domains
15:10-15:50
Contributed talk: Wenfeng Zhang, Xiaoquan Xu
Meet Precontinuous Posets
15:50-16:10
Contributed talk: Li-Gang Wu, Lei Fan
Generalized Scott Topology on Sets with Families of Pre-orders
16:10-16:40
Closing ceremony
18:00-19:00
Dinner
October 29
Tour
We will arrange buses for the tourists at 7:30am to Hengshan Mountain
and then get back to Changsha at about 5:00pm.
XI
Abstracts
Invited Speaker
(1)Dana Scott (Keynote)Stochastic Lambda Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
(2)Glynn Winskel
Concurrent strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
(3)Achim Jung
Continuous Domain Theory in Logical Form. . . . . . . . . 3
Invited Presentation
(1)Klaus Keimel
On the equivalence of state transformer
and predicate transformer semantics . . . . . . . . . . . . . . . . . . . . . . 4
(2)Bin Zhao
On the power-set Q-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
(3)Pierre-Louis Curienmu and mu tilde: two useful combinators . . . . . . . . . . . . . . . . . 7
(4)Jimmie Lawson
Close Encounters of the Third Kind:
Domain Theory Meets T0 -Spaces Meets Topology . . . . . . . . . 8
(5)Guo-Qiang Zhang TCS-Thinking in Biomedical Ontology Research . . . . . . . . . 9
(6)Michael Mislove
On Random Variable Models of Domains . . . . . . . . . . . . . . . . 10
Contributed talk
(1)Moez A. AbdelGawadA Domain-Theoretic Model Of Nominally-Typed
Object-Oriented Programming . . . . . . . . . . . . . . . . . . . . . . . . 11
(2)Ingo Battenfeld
Observationally-induced algebras in Domain Theory. . . 12
(3)Xuechong Guan,
On a Condition for Semirings to Induce Compact
Yongming Li
Information Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
(4)Lankun Guo,
Qingguo Li,
A note on finitely derived information systems
Petko Valtchev,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Robert Godin
XII
(5)Weng Kin Ho
Characterising E-projectives via Co-monads . . . . . . . . . . . . . . . 15
(6)Mengqiao Huang,
Qingguo Li,
Formal contexts for Algebraic Domains
Lankun Guo
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
(7)Hengyang Wu,
Yixiang Chen
(8)Li-Gang Wu,
Lei Fan
(9)Mingyuan Wu,
Coalgebras for Fuzzy Transition Systems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Generalized Scott Topology on Sets with Families of Pre-orders
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Representations of algebraic domains and
Qingguo Li,
and algebraic L-domains by information systems
Xiangnan Zhou
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
(10)Xiaoyong Xi,
Guohua Wu
(11)Luoshan Xu,
Jing Zhao
(12)Wei Yao,
Bin Zhao
(13)Wenfeng Zhang,
Xiaoquan Xu
(14)Wenfeng Zhang,
Xiaoquan Xu
(15)Haoran Zhao,
Hui Kou
On the largest cartesian closed category of stable domains
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Reductions and Saturation Reductions of
(Abstract) Knowledge Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A duality between Ω-categories and algebraic Ω-categories
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Completely Precontinuous Posets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Meet Precontinuous Posets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Tω as a stable universal domain
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
(16)Mucong Zheng,
Zhongke Shi,
Triple I method of IFMT based on Residual Implication
Guodong Han
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
(17)Qingyu He,
On Semicontinuous Lattices and Their Distributive Reflections
Luoshan Xu
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
(18)Xiaoquan Xu
A Note on Double Dual Topologies . . . . . . . . . . . . . . . . . . . . . . . . 28
ISDT’13
Changsha
1
Stochastic Lambda Calculi
Dana Scott
Carnegie Mellon University, USA
E-mail: [email protected]
BIOGRAPHY The emeritus Hillman University Professor of Computer Science,
Philosophy, and Mathematical Logic at Carnegie Mellon University.
Editor-in-chief of the ”new journal Logical Methods in Computer Science”.
His research career has spanned computer science, mathematics, and philosophy.
His work on automata theory earned him the ACM Turing Award in 1976, while his
collaborative work with Christopher Strachey in the 1970s laid the foundations of
modern approaches to the semantics of programming languages. The Scott topology
named after him in Domain Theory is an essential tool for the investigation of domain
structures. He has worked also on modal logic, topology, and category theory.
Homepage: http://www.cs.cmu.edu/ scott/
ISDT’13
2
Changsha
Concurrent Strategies
Glynn Winskel
Computer Laboratory
University of Cambridge, Cambridge CB3 0FD, UK
E-mail: [email protected]
BIOGRAPHY Professor of Computer Science at University of Cambridge. Fellow of Emmanuel College.
On the editorial board of ”Mathematical Structures in Computer Science”. On the
editorial board of ”Transactions on Petri Nets and Other Models of Concurrency”.
His research interests are in theoretical computer science and related mathematics.
Especially, semantics of programming languages, mathematical foundations of denotational semantics, domain theory, and logic of programs.
Homepage: http://www.cl.cam.ac.uk/ gw104/
ISDT’13
Changsha
3
Continuous Domain Theory in Logical Form
Achim Jung
School of Computer Science
The University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Email: [email protected]
Abstract
In 1987 Samson Abramsky presented ”Domain Theory in Logical Form” (DTLF)
in the Logic in Computer Science conference. His contribution to the conference proceedings was honoured with the Test-of-Time award 20 years later. In this talk I will
trace a particular line of research that arose from this landmark paper, which tries to
extend DTLF to continuous structures, such as ones used to model probabilistic and
exact real number computation. I will emphasize the motivation that underlies DTLF
and point out the many open problems that remain in this area.
BIOGRAPHY Editor of ”Theoretical Computer Science”, ”Soft Computing”
and ”Electronic Notes in Theoretical Computer Science”.
His research interests are in Domain Theory, Topology, Semantics of Programming
Languages, Probability and Lambda Calculus. He first proposed the notion of FSdomain, and showed the maximality of category of FS-domain and that of L-domain
in Domain category respect to Cartesian closure. The significant work did a great
contribution to the classification of domains.
Homepage: http://www.cs.bham.ac.uk/ axj/
ISDT’13
4
Changsha
On the equivalence of state transformer and predicate
transformer semantics
Klaus Keimel
Fachbereich Mathematik
Technische Universität Darmstadt, Germany
Email: [email protected]
Abstract
Domain Theory provides two kinds of semantics for programs. State transformer
semantics describes the input – output behavior of programs. Predicate transformer
semantics at the contrary assigns to every (desired) property of the output the weakest
precondition on the input which guarantees the (desired) property for the output. In
several situations the two semantics have been proved to be dually equivalent.
For deterministic programs this is a straightforward but extremely useful observation for the verification of programs. It has been extended to nondeterministic programs
by M. B. Smyth. For programs involving probabilistic features this has has been noted
by D. Kozen and in a domain theoretical setting by C. Jones. For situations combining
probabilistic and ordinary nondeterminism, K. Keimel and G. D. Plotkin as well as J.
Goubault-Larrecq have also established this dual equivalence. The methods needed for
the proofs became more and more involved.
The question arises whether such a dual equivalence between predicate and state
transformer semantics may be expected quite generally. The talk will be devoted to
this question. We will show that such a dual equivalence can arise in quite special
situations only.
We will use the continuation monad over a domain of ’observations’ and monads
subordinate to the continuation monad. Notions from universal algebra will play a role.
ISDT’13
Changsha
5
We will see that the commutativity of the monads is the property which is essential
for the desired dual equivalence.
ISDT’13
6
Changsha
On the power-set Q-algebras
Bin Zhao
Department of Mathematics
Shaanxi Normal University, Xi’an 710062, P.R. China
Email: [email protected]
Abstract
The main purpose of this paper is to consider some methods for constructing Qalgebras. Based on the power-set Q-algebras constructed from ordered semigroups we
give a new representation theorem for Q-algebras, and by means of a kind of relations
between ordered semigroups get a general characterization for the strong homomorphisms between power-set Q-algebras. As an important conclusion, we prove that the
homomorphisms between lower-set quantales are completely determined by the decomposed ul-sub-semigroups.
ISDT’13
Changsha
7
mu and mu tilde: two useful combinators
Pierre-Louis Curien
CNRS and Université Paris Diderot
Email: [email protected]
Abstract
In my work with Hugo Herbelin, and then with my student Guillaume Munch,
we explored term syntaxes for sequent calculus (for classical logic). I recently found
that the same style of syntax allows for a neat description of the wiring structures
underlying operads, cooperads . . . , which are used in contemporary algebra to describe various types of algebras and coalgebras. Under this new light, the fundamental
non-determinism of classical logic appears to match the notion of duplicial algebra, proposed by the late mathematician Jean-Louis Loday among a wealth of new algebraic
structures which he unveiled over the last 20 years or so.
ISDT’13
8
Changsha
Close Encounters of the Third Kind: Domain Theory Meets
T0 -Spaces Meets Topology
Jimmie Lawson
Department of Mathematics
Louisiana State University, Baton Rouge, LA 70803
Email: [email protected]
Abstract
Domain theory has been a major influence in the development of a substantial
theory of T0 -spaces. In many settings domain theoretic concepts and results inspire an
investigation of analogues in the T0 -setting, a setting in which this theory often can be
enriched and even deepened. The resulting theory often provides useful constructions
and tools for the typical concerns of domain theory. In another direction general
topology has historically typically limited itself to the study of Hausdorff spaces, but
substantial blocks of the material can be carried over to the T0 -setting. We consider
how important notions such as locally compact, compact, Hausdorff, and polish spaces
have appropriate T0 -analogues. We also consider distinctive topological features of
T0 -spaces.
ISDT’13
Changsha
TCS-Thinking in Biomedical Ontology Research
Guo-Qiang Zhang
Electronic Engineering & Computer Science
Case Western Reserve University, Cleveland, OH 44106-7291, USA
Email: [email protected]
9
10
ISDT’13
Changsha
On Random Variable Models of Domains
Michael Mislove
Department of Mathematics
Tulane University, New Orleans, LA, USA
Email: [email protected]
Abstract
One of the most perplexing problems in domain theory is how to devise a model
for probabilistic choice that works best with other constructs in domain theory. For
example, it’s well known that the probabilistic power domain forms a monad on the
category of domains (and Scott continuous maps), but the question of whether there
is a cartesian closed category of domains that’s left invariant by this monad is a very
hard, unsolved problem. Recent work in the area has turned to random variables as
a possible alternative to constructing models for probabilistic choice on domains. In
this talks, I’ll review the known constructs along this line, and discuss a new approach
that shows promise for providing the kind of example that’s needed.
ISDT’13
Changsha
11
A Domain-Theoretic Model Of Nominally-Typed
Object-Oriented Programming
Moez A. AbdelGawad
Computer Science Department
Rice University, Houston, Texas, USA
Email: [email protected]
Abstract
The majority of contemporary mainstream object-oriented (OO) software is written
using nominally-typed OO programming languages. Extant domain-theoretic models
of OOP developed to analyze OO type systems miss crucial features of these mainstream OO languages, such as nominality. This paper summarizes the construction of
NOOP as a domain-theoretic model of OOP that includes nominal information found
in nominally-typed mainstream OO software. Inclusion of nominal type information
and asserting that type inheritance in statically-typed OO programming languages is
an inherently nominal notion allow readily proving that inheritance and subtyping are
completely identified in these languages. This conclusion is in full agreement with
intuitions of OO developers using these languages, and contrary to the belief that “inheritance is not subtyping”, which came from assuming non-nominal structural models
of OO type systems. NOOP, thus, provides a firmer semantic foundation for analyzing
and progressing nominally-typed mainstream OO programming languages.
12
ISDT’13
Changsha
Observationally-induced algebras in Domain Theory
Ingo Battenfeld
Fakultät für Informatik
TU Dortmund, Dortmund, Germany
Email: [email protected]
Abstract
We investigate the observationally-induced free algebra approach for constructing
computational monads in the categories of classical domain theory. Our investigation
yields that the free algebra construction exists for all finitary algebraic signatures and
computational prototypes. We furthermore investigate the classical powerdomain constructions in the observationally-induced approach. For the Hoare, Smyth and probabilistic powerdomain constructions we build on established results, showing that they
can be recovered observationally-induced. However, the Plotkin powerdomain turns
out to be more problematic. Here we show that with the obvious prototype algebra,
Heckmanns algebra A, one does not get the classical Plotkin powerdomain.
ISDT’13
Changsha
13
On a Condition for Semirings to Induce Compact Information
Algebras
Xuechong Guan1 , Yongming Li2
1
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, China
2
College of Computer Science, Shaanxi Normal University, Xi’an, China
Email: [email protected], [email protected]
Abstract
In this paper we study the relationship between ordering structures on semirings
and semiring-induced valuation algebras. We show that a semiring-induced valuation
algebra is a complete (resp. continuous) lattice if and only if the semiring is complete (resp. continuous) lattice with respect to the reverse order-relation on semirings.
Furthermore, a semiring-induced information algebra is compact, if the dual of the
semiring is an algebraic lattice.
14
ISDT’13
Changsha
A note on finitely derived information systems
Lankun Guo1 , Qingguo Li2 , Petko Valtchev3 , Robert Godin3
1
College of Information Science and Engineering, Hunan University, Changsha, P.R.
China
2
College of Mathematics and Econometrics, Hunan University, Changsha, P.R. China
3
Département d’Informatique, Université du Québec à Montréal, Montréal, Canada
Email: [email protected], [email protected]
Abstract
The notion of information system initially introduced by Scott provides an efficient
approach to represent various kinds of domains. In this note, a new type of information
systems named finitely derived information systems is introduced. For this notion,
the requirement for the consistency predicate used in Scott’s information systems is
simplified, and the reflexive and transitive rules for the entailment relation are preserved
while the finitely derived rule is introduced. A comprehensive investigation is made on
the interrelation between finitely derived information systems and algebraic domains.
It turns out that their corresponding categories are equivalent, which indicates that the
proposed notion of finitely derived information system provides a concrete approach to
representing algebraic domains.
ISDT’13
Changsha
15
Characterising E-projectives via Co-monads
Weng Kin Ho
Mathematics and Mathematics Education, National Institute of Education
Nanyang Technological University, Singapore
Email: [email protected]
Abstract
This paper demonstrates the usefulness of a comonadic approach to give previously
unknown characterisation of projective objects in certain categories over particular subclasses of epimorphisms. This approach is a simple adaptation of a powerful technique
due to M. Escardó which has been used extensively to characterise injective spaces and
locales over various kinds of embeddings, but never previously for projective structures.
Using some examples, we advertise the versatility of this approach – in particular, highlighting its advantage over existing methods on characterisation of projectives, which
is that the comonadic machinery forces upon us the structural properties of projectives without relying on extraneous characterisations of the underlying object of the
co-algebra arising from the comonad.
ISDT’13
16
Changsha
Formal contexts for Algebraic Domains
Mengqiao Huang1 , Qingguo Li2 , Lankun Guo3
1
Economics Department, Hunan International Economics University, Changsha,
China
2
College of Mathematics and Econometrics, Hunan Unversity, Changsha, China
3
College of Mathematics and Econometrics, Hunan Unversity, Changsha, China
Email: [email protected], [email protected]
Abstract
In this paper, we investigate the representation of algebraic domains by means
of Formal Concept Analysis. For a formal context, we can define a large number of
consistent sets. Associated with each consistent set, there is a set of F -approximable
concepts which are selected from the well known approximable concepts. By virtue
of F -approximable concepts, formal contexts and algebraic domains are able to interpret each other. Moreover, by analyzing the finitely consistent sets, the algebraic
bifinite domains, algebraic L−domains are exactly located at the corresponding formal
contexts, respectively.
ISDT’13
Changsha
17
Coalgebras for Fuzzy Transition Systems
Hengyang Wu1 , Yixiang Chen2
1
Information Engineer College, Hangzhou Dianzi University, Hangzhou 310018, China
2
MoE Engineering Research Center for Software/Hardware Co-design Technology and
Application
Shanghai Key Lab for Trustworthy Computing, East China Normal University
Shanghai 200062, China
Email: wuhengy− [email protected]
Abstract
This paper studies a coalgebraic theory of fuzzy transition systems. Main conclusions include: the functor F A for deterministic fuzzy transition systems and the functor
(P ◦ F)A for nondeterministic fuzzy transition systems preserve weak pullbacks, and
the functor F A has a final coalgebra under some restricted conditions. Moreover, we
show how to get a concrete (fuzzy) bisimulation from a coalgebraic bisimulation.
ISDT’13
18
Changsha
Generalized Scott Topology on Sets with Families of
Pre-orders
Li-Gang Wu1 , Lei Fan2
1
Department of Mathematics, School of Science
Beijing University of Civil Engineering and Architecture
Beijing 102616, P.R.China.
2
Department of Educational Technology, Capital Normal University
Beijing 100048, P.R.China.
Email: [email protected], [email protected]
Abstract
Fan, Ji and Wang[5] proposed a class of sets with families of pre-orders (R-posets
for short). They are not only a non-symmetric generalization of sfe[8] but also a special
case of quasi-metric spaces (qms, [10]) and generalized ultrametric spaces (gums, [9]).
In this paper, we define a kind of generalized Scott topology on R-posets and discuss
some basic properties of the topology. Some relevant interesting examples are offered.
It is worth pointing out that an R-monotone functions is R-continuous if and only
if (iff for short) it’s continuous with respect to (w.r.t for short) the generalized Scott
topology.
ISDT’13
Changsha
19
Representations of algebraic domains and algebraic L-domains
by information systems
Mingyuan Wu, Qingguo Li, Xiangnan Zhou
College of Mathematics and Econometrics, Hunan University, Changsha, P.R. China
Email: [email protected], [email protected]
Abstract
Information systems play an important role in characterizing order structures. In
this paper, we introduce the notions of the algebraic information system and algebraic
L-information system. They are of the same logic-oriented style as the information
system introduced by Scott (1982). But the axioms in this paper are briefer than
reported in existing work. We also prove that the two new information systems exactly
represent the algebraic domains and algebraic L-domains respectively. Based on the
notion of approximable mapping between the algebraic information systems and the
algebraic L-information systems, we obtain the result that the corresponding categories
of algebraic information systems and algebraic L-information systems are equivalent to
the category of algebraic domains and algebraic L-domains respectively.
ISDT’13
20
Changsha
On the largest cartesian closed category of stable domains
Xiaoyong Xi1 , Guohua Wu2
1
2
College of Mathematics Sciences, Jiangsu Normal university, Xuzhou, China
Division of Mathematical Sciences, School of Physical and Mathematical Sciences
Nanyang Technological University, Singapore
Email: [email protected], [email protected]
Abstract
^ be the category of algebraic bounded complete domains
Let SABC (resp., SABC)
with conditionally multiplicative mappings, that is, Scott-continuous mappings preserving meets of pairs of compatible elements (resp., stable mappings). Zhang showed
that the category of dI-domains is the largest cartesian closed subcategory of ω-SABC
^ with the exponential being the stable function space, where ω-SABC
and ω-SABC,
^ are full subcategories of SABC and SABC
^ respectively which contain
and ω-SABC
countablly based algebraic bounded complete domains as objects. This paper shows
that:
^ are exactly function
i) The exponentials of any full subcategory of SABC or SABC
spaces;
^ the category of distributive algebraic bounded complete domains, is
ii) SDABC,
^
the largest cartesian closed subcategory of SABC;
The compact elements of function spaces in the category SABC are also studied.
ISDT’13
Changsha
21
Reductions and Saturation Reductions of (Abstract)
Knowledge Bases
Luoshan Xu, Jing Zhao
Mathematics Department, Yangzhou University, Yangzhou 225002, P. R. China
Email: [email protected], [email protected]
Abstract
Rough set theory is a useful tool for dealing with fuzzyness and uncertainty of
knowledge. In rough set theory, knowledge reductions and generatings are important research topics and critical steps of knowledge acquisition. This paper generalize
knowledge bases to abstract knowledge bases and study (abstract) knowledge bases
on infinite universe by considering the problem of existence of finite reductions of infinite knowledge bases. For abstract knowledge bases, the concept of saturations and
saturation reductions are introduced. Global properties of saturations and saturation
reductions of abstract knowledge bases are investigated. It is proved that for a given
abstract knowledge base which is closed w.r.t. arbitrary unions on a finite universe U ,
its saturation augmented the unverse U forms a topology, whereas a counterexample
is constructed to show that this may not be true if U is infinite. Making use of the
saturation of an abstract knowledge base, some sufficient and/or necessary conditions
for existence of finite reductions of an infinite abstract knowledge base are given. It
is proved that for an abstract knowledge base on finite universe, there is one and only
one saturation reduction. Some examples are constructed to reveal various cases of
existence of knowledge reductions. Simple applications of saturation reductions are
also given.
ISDT’13
22
Changsha
A duality between Ω-categories and algebraic Ω-categories
Wei Yao1 , Bin Zhao2
1
School of Science, Hbei University of Science and Technology, Shijiazhuang, China
2
College of Mathematics and Information Science
Shaanxi Normal University, Xi’an, China
Email: [email protected], [email protected]
Abstract
In this paper, we propose a definition of algebraic Ω-categories. Let Ω-POID
denote the category of Ω-categories with Ω-functors between them such that inverse
image of ideals are also ideals, and let Ω-AlgDomG denote the category of algebraic
Ω-categories with Scott continuous functors between them having left Ω-adjoints. We
show that Ω-AlgDomG and Ω-POID are dual equivalent to each other.
ISDT’13
Changsha
23
Completely Precontinuous Posets
Wenfeng Zhang1 , Xiaoquan Xu2
1
2
Department of Mathematics, Sichuan University, Chendu, China
Department of Mathematics, Jiangxi Normal University, Nanchang, China
Email: [email protected], [email protected]
Abstract
In this paper, concepts of strongly way below relations, completely precontinuous
posets, coprimes and Heyting posets are introduced. The main results are: (1) The
strongly way below relations of completely precontinuous posets have the interpolation
property; (2) A poset P is a completely precontinuous poset iff its normal completion
is a completely distributive lattice; (3) An ω-chain complete P is completely precontinuous iff P and P op are precontinuous and its normal completion is distributive iff P
is precontinuous and has enough coprimes; (4) A poset P is completely precontinuous
iff the strongly way below relation is the smallest approximating auxiliary relation on
P iff P is a Heyting poset and there is a smallest approximating auxiliary relation on
P . Finally, given a poset P and an auxiliary relation on P , we characterize those joindense subsets of P whose strongly way-below relation agrees with the given auxiliary
relation.
ISDT’13
24
Changsha
Meet Precontinuous Posets
Wenfeng Zhang1 , Xiaoquan Xu2
1
2
Department of Mathematics, Sichuan University, Chendu, China
Department of Mathematics, Jiangxi Normal University, Nanchang, China
Email: [email protected], [email protected]
Abstract
In this paper, we introduce the concept of meet precontinuous posets, a generalization of meet continuous lattices to posets. The main results are: (1) A poset P is
meet precontinuous iff its normal completion is a meet continuous lattice iff a certain
system γ(P ) which is, in the case of complete lattices, the lattice of all Scott-closed
sets is a complete Heyting algebra; (2) A poset P is precontinuous iff the way below
relation is the smallest approximating auxiliary relation iff P is meet precontinuous
and there is a smallest approximating auxiliary relation on P . Finally, given a poset
P and an auxiliary relation on P , we characterize those join-dense subsets of P whose
way-below relation agrees with the given auxiliary relation.
ISDT’13
Changsha
25
Tω as a stable universal domain
Haoran Zhao, Hui Kou
School of Mathematics, Sichuan University, Chengdu 610064, China
Email: [email protected]
Abstract
In the seventies, G. Plotkin noticed that Tω , the cartesian product of ω copies of the
3 elements flat domain of Boolean, is a universal domain, where “universal” means that
the retracts of Tω in Scott’s continuous semantics are exactly all the ωCC-domains,
which with Scott continuous functions form a cartesian closed category. As usual “ω” is
for “countably based”, and here “CC” is for “conditionally complete”, which essentially
means that any subset which is pairwise bounded has an upper bound. Since Tω is
also an ωDI-domain (an important structure in the stable domain theory), a problem
arises naturally: Is Tω a universal domain for Berry’s stable semantics? The aim of
this paper is to answer this question. We investigate the properties of stable retracts
and introduce a new domain named a conditionally complete DI-domain (a CCDIdomain for short). We show that, (1) a dcpo is a stable retract of Tω if and only if it is
an ωCCDI-domain; (2) the category of ωCCDI-domain (resp. CCDI-domains) with
stable functions is cartesian closed. So, the problem above has an affirmative answer.
ISDT’13
26
Changsha
Triple I method of IFMT based on Residual Implication
Mucong Zheng1 , Zhongke Shi1 , Guodong Han2
1
College of Automation, Northwestern Polytechnical University, Xi’an, China
2
College of Mathematics and Information Science
Shaanxi Normal University, Xi’an, China
Email: [email protected], [email protected], [email protected]
Abstract
Fuzzy modus tollens(FMT) is one of the basic inference models of fuzzy reasoning. The Triple I method is a very important method to solve the problem of FMT.
The aim of this paper is to extend the Triple I method for intuitionistic fuzzy modus
tollens(IFMT). In the paper, the Triple I method of IFMT based on residual implication is presented and the corresponding calculate examples are provided. Then, the
reductivity of the Triple I method is investigated and the sufficient conditions of the
local reductivity for some residual intuitionistic implications are obtained. Lastly, the
α−Triple I method of IFMT is proposed.
ISDT’13
Changsha
27
On Semicontinuous Lattices and Their Distributive
Reflections
Qingyu He, Luoshan Xu
School of Mathematical Science, Yangzhou University, Yangzhou 225002, P. R. China
Email: [email protected], [email protected]
Abstract
In [D. Zhao, Semicontinuous Lattices, Algebra Universalis, 37(1997)], Zhao introduced semicontinuous lattices, and considered relationships between the semicontinuity
of a complete lattice L and the continuity of its distributive reflection Ld introduced in
[Rav, Y., Semiprime ideals in general lattices, J. Pure and Applied Algebra, 56(1989)].
Zhao showed by a counterexample that generally the continuity of Ld does not imply
the semicontinuity of L, and guessed that the inverse is not true either. Besides, he
gave two more questions. In this paper, we confirm the guess of Zhao and answer
these open problems by constructing some counterexamples, indicating that, if Ld is
continuous and the map d : L → Ld is Scott continuous, L is not necessary to be semicontinuous; if Ld and L are continuous, d : L → Ld is not necessary to be continuous.
It is obtained that for a complete lattice L, the distributive reflection Ld is isomorphic
to the lattice of radicals of L in the set inclusion order, generating a method to depict the distributive reflection of a given lattice. Moreover, it is proved that, if every
semiprime element x ∈ L is maximal in d(x) and d is Scott continuous, then Ld is a
continuous lattice, and in this case the semicontinuity of a complete lattice L implies
the continuity of its distributive reflection Ld .
ISDT’13
28
Changsha
A Note on Double Dual Topologies
Xiaoquan Xu
Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China
Email: [email protected]
Abstract
Given a topological space (X, τ ), a set A ⊆ X is said to be saturated if it is the
intersection of open sets. One defines the de Groot dual topology τ d by taking as a
subbase for the closed sets all saturated compact sets in (X, τ ). The joint of these two
topologies is called the patch topology.
In the Problem 540 of Open Problems in Topology (J. van Mill and G. M. Reed,
eds., 1990), Lawson and Mislove asked whether the process of taking duals of a topology
terminates after finitely many steps with two topologies that are de Groot dual to each
other.
The complete solution of this question was first presented by Kovár at the Prague
Topological Symposium in 2001. In fact, he have proved that τ dd = τ dddd for any
topology τ . It follows that at most 4 topologies can arise from iterating the de Groot
dual.
At the Prague Topological Symposium in 2001, another related problem that which
topology is double de Groot dual was raised.
The main purpose of this note is to give some equivalent conditions under which
a topology is double de Groot dual. Our work improves some corresponding results of
Kovár.