Transcript Interoperability in Information Systems

Information Systems and
Categories: Sketches and Models
for Database Modelling
Nick Rossiter
Research Conference, Informatics,
Northumbria University, 15th May
2003
http://computing.unn.ac.uk/staff/CGNR1/
[email protected]
Motivation
• Interoperability
– Working together of information systems
– Difficult area particularly with heterogeneous models
– Formal basis lacking
• Work by NR/MH/DAN has involved:
– Looking at a sound formal basis
– Formalisation of object-relational model
– Using category theory
Motivation 2
• Most recent work (Northumbria, seminar
Nov 2002):
– Has been well received
– Shows that four levels are needed for
addressing data
– Provides Godement calculus for manipulating
across levels
Overview of Presentation
1. Database theory is underpinned by the term
model.
2. Unfortunately model does not have a universal
meaning.
3. Explore some meanings
4. Look at interoperability representations
5. Introduce Dolittle diagrams
6. Look at concept of model and sketch in category
theory
7. Future work may try and unify last two (5+6)
A Definition of Model in the
Database World
• Philosophical areas – for example, in
interoperability
• Database Model: a representation of policies in a
structured form according to some perceived view
of reality e.g.
–
–
–
–
–
Relational model – world is tabular
Hierarchical model – world is tree-like
Security model – world is task-based
Object model – world is based on o-o paradigm
ER model – world is graph-based
Another Database Definition for
Model
• A model comprises:
– A data structure
– A language for manipulating the structure
– A collection of rules governing acceptable states of the
structure
• On this basis:
– ER is not a model (no general manipulation language)
– Relational is a model (e.g. data structure = table,
manipulation language = SQL, rules = referential
integrity)
Also Design Models
• Examples of design models:
– ER
– UML
• Always graphically based.
• Often provide a route to a basic model for
implementation, population and
manipulation
Modelling a Whole System
• Most models are aimed at data definition
level (schema).
• Full system has multiple levels:
– One below the schema – the data values
– Two above – constructs available and concepts
to be employed
Mappings in complete system
Concepts
MetaMeta
Policy
Constructs
Meta
Organize
Schema Types
Classify
Instantiate
Named Data Values
Downward arrows are intension-extension pairs
Category Theory: Comparing one
System with Another
CC
P
CS

CC
P´
O
SM

CS´
O´
DT
I

SM´
I´
DT´
,,  are natural transformations (comparing functors)
Godement Calculus
• Rules showing:
– composition of functors and natural
transformations is associative
– natural transformations can be composed with
each other
• For example:
• (I´O´)  = I´(O´ );
•   = ( O) o (I´ );
(OP)
= ( O)P
 = P o (O´ )
Analogous Levels for
Interoperability
Level
Category
Architecture
1. data values Objects (identity iddt
arrows)
2. named
values
3. classified
values
4. contrasted
representation
Category
DT
Functor
C: DT
SM
* o * (* is
Natural
transformation
dual of )
Category Theory: Detail - Example of
modelling Relationships – the Pullback
ls x m
S
l
s
sxm

S XIMG M

rs x m 
s x m *m
M
s
(s)-1
m
(m)-1
W/IMG
Pullback showing fuller collection of arrows
Dolittle Diagram of S and M in Context of IMG
S = source, M = medium, IMG = image, W = world
Logic available: product, join, project, existential and universal
quantifiers, select, insert, units of adjunction and co-adjunction
Constraints available: cardinality, membership class
Recent Publications in this Area
• Rossiter, N, From Classical to Quantum Databases with Applied
Pullbacks, 78th Meeting Peripatetic Seminar on Sheaves and Logic,
Institut de Recherche Mathématique Avancée, Strasbourg University
15-16 February (2003).
• Rossiter, N, Nelson, D A, & Heather, M A, Formalizing Types with
Ultimate Closure for Middleware Tools in Information Systems
Engineering, 5th ICEIS, Angers, France 23-26 April 8pp (2003).
• Rossiter, N, & Heather, M, Four-level Architecture for Closure in
Interoperability, EFIS2003, Fifth International Workshop on
Engineering Federated Information Systems, Coventry, UK, 17-18 July
6pp (2003).
• Heather, M A, & Rossiter, B N, The Anticipatory and Systemic
Adjointness of E-Science Computation on the Grid, Computing
Anticipatory Systems, Proceedings CASYS`01, Liège, Dubois, D M,
(ed.), AIP Conference Proceedings 627 565-574 (2002).
Other Work with Databases and
Categories
• Michael Johnson, Robert Rosebrugh and RJ
Wood, Entity-Relationship-Attribute
Designs and Sketches, TAC 10(3) 94-111.
– sketches for design (class structure)
– models for states (objects) where model is used
in categorical sense
– lextensive category (finite limits, stable disjoint
finite sums) for query language
Sketch
• Developed also in databases by:
– Zinovy Diskin, Boris Cadish: Algebraic Graph-Based Approach to
Management of Multidatabase Systems, NGITS’95 69-79 (1995).
• Sketch originally from Charles Ehresmann.
• Many different sorts of sketch – 12 kinds listed in
Charles Wells, Sketches, Outline with References,
at http://www.cwru.edu/artsci/math/wells/pub/papers.html#sketch
– For instance Finite Product (FP) is much used but it has
no cocones (sums)
• Most suitable appears to be Finite Discrete (FD)
sketch D = (E, L, R, S)
•
•
•
•
finite graph E (data structure)
set of diagrams L in E (constraints)
Finite set R of discrete cones in D (relationships)
Finite set S of discrete cocones in D (attributes)
Model in Categories
• Model (M) – graph homomorphism
• M:DC
• M maps:
–
–
–
–
takes any node in E to a set of values (populates)
L  commutative diagrams
R  limit cones
S  co-limit cocones
• C is a target category (extension)
• preserve products and co-products in state
• Evaluate:
Future Research
– Use of sketch as construction for two bottom levels of
architecture (schema, values)
– Feasibility of building in Dolittle diagram for logic
• Then if outcome positive:
– Either Add top two levels (constructs, concepts) to
sketch to give 4-level architecture with adjointness
connecting the levels as in recent publications
– Or Extend sketch construction to 4-levels (through
repeated sketch-model constructions, transitive closure)
• Else if outcome negative:
– Use fundamental categorical levels (named object,
category, functor, nat trans) for 4-levels as in recent
publication and develop from there
Database Group
• Progressing Open Database Project
– Development of open source software
– Based on fundamental view of relational model
• Developing work on previous slide to:
– Specify formally object-relational model
– Try realising this formalisation with the Open Database
Project
– Advance interoperability with sounder foundations
Some Publications in Other Areas
Security in Multi-agency
Services
• Aljareh, S, & Rossiter, N, A Task-based Security Model to
facilitate Collaboration in Trusted Multi-agency Networks,
ACM Symposium on Applied Computing (SAC) 2002,
Madrid, 744-749 March (2002).
• Aljareh, S, & Rossiter, N, Towards Security in Multiagency Clinical Information Services, Health Informatics
Journal 8(2) 96-104 (2002).
• Aljareh, S, Dobson, J, & Rossiter, N, Satisfaction of Health
Record Security Principles through Collaborative
Protocols, NI'2003, 8th International Congress in Nursing
Informatics, Brazil, 5pp, 20-25 June (2003).
Natural Computing (Quantum)
• Heather, M A, & Rossiter, B N, Locality, Weak or
Strong Anticipation and Quantum Computing I.
Non-locality in Quantum Theory, International
Journal Computing Anticipatory Systems 13 307326 (2002).
• Heather, M A, & Rossiter, B N, Locality, Weak or
Strong Anticipation and Quantum Computing. II.
Constructivism with Category Theory,
International Journal Computing Anticipatory
Systems 13 327-339 (2002).