1. A Natural Basis for Interoperability
Nick Rossiter, Mike Heather and David Nelson
I-ESA’06
Northumbria University, University of Sunderland
06/12/2018

2. Information Systems Very diverse Usually multilevel
A stand-alone piece of information
Is valueless
Needs to be typed
Needs to be related
Needs to be placed in context
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3. Example of Context This is Herring Gull argenteus (subspecies)
Trinomial – 3-level name is:
Larus argentatus argenteus
Kingdom: Metazoa ((=Animalia) multicellular animals)
Phylum: Chordata (chordates)
Class: Aves (birds)
Order: Charadriiformes (gulls and shore birds)
Family: Laridae (gulls, terns)
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4. Interoperability An area where context is paramount is
the ability to request and receive services between various systems and use their functionality.
More than data exchange.
Implies a close integration
Various kinds dependent on ambition:
E.g. syntactic, semantic, structural and organisational
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5. Motivation/Problems Linking of Different Systems (Current/legacy)
Homogeneous models
Difficult enough
Different viewpoints in modelling
E.g. library system
A fine could be:
A relational table
A column in a table
A value in an income ledger
Inconsistent use of modelling features
Systems that achieve interoperability in such circumstances are ranked
As semantically interoperable
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6. Motivation/Problems 2 Heterogeneous Models Far more difficult
In addition to different semantic viewpoints
Diverse modelling constructions
Data structures
Objects, relations, records
Process
Business process, procedures, methods
More recent models are semantically richer
More scope for variation in style
Systems that achieve interoperability in such circumstances are ranked
As structurally (or organisationally) interoperable
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7. Demands for Interoperability
Business needs
Data warehousing
Web warehousing
GRID
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8. Attempted Solutions RDF (Resource Description Framework)
Triples (uri – e.g. resource/property/statement)
From W3C (XML basis)
MOF/MDA (Meta Object Facility/Model Driven Architecture)
Meta Meta is better-better!
Relates classes in different systems
From OMG (UML basis but claimed to be extensible)
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9. Attempted Solutions 2 Ontologies Defines meaning of data
Being
Defines meaning of data
Like a dictionary
But is usually much more
Everything is defined in context
Multi-level definitions
No clear consensus
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10. Formal Basis For preceding techniques These are partial:
Some set theoretic justifications
These are partial:
Emphasis on a level
Contrived multi-level
Above all – lack concept of naturality
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11. Categories Category Theory Developed from 1940s
Many pure mathematicians
Eilenberg, Mac Lane, Kan, Lawvere, Barr, Wells, Johnstone
Much improved presentation since 1970s
Saunders Mac Lane “Categories for the Working Mathematician” 2nd ed Springer (2000)
Barr & Wells “Category Theory for Computing Science” 3rd ed CRM (1999).
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12. Applied Categories Physics including quantum studies Databases
John Baez
Databases
Bob Rosebrugh, Michael Johnson, Zinovy Diskin, Lellahi & Spyratos
Business process
Arthur ter Hofstede
Computer program semantics
Much work e.g. Cambridge
Programs to Support Category Theory
OCaml (ENRIA, France)
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13. Abstract Nonsense One might ask "Why category theory?“
Category theory is known as highly abstract mathematics.
Some call it abstract nonsense.
It chases abstract arrows and diagrams, proves nothing about those arrows and diagrams, rarely talks about what arrows are for and often concepts go beyond one's imagination.
However, when this 'abstract nonsense' works, it is like magic. One may discover a simple theorem actually means very deep things and some concepts beautifully unify and connect things which are unrelated before.
[Tatsuya Hagino. A Categorical Programming Language. PhD Thesis, University of Edinburgh, 1987]
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14. The Simplest Category Discrete Category Identity arrows (objects) only
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15. A Not Very Useful Category
6 arrows
Not connected
Does not conflict
with axioms
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16. Basic Category Illustrates 2 axioms when connections made.
Composition:
h = g o f
Associativity:
r o (q o p) =
(r o q) o p
Also unit law
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17. Cartesian Closed Category
Basis of much
Computing Science
Research in CT C
P+P+P
Has identity, products,
limits, coproducts
PxPxP
Identity functor
1C: C - C
Initial object PxPxP
provides handle on category
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18. Functors Map from one category to another Preserve composition
E.g. F: C  D
Preserve composition
Various kinds
Identity (map category to itself)
Free (add structure)
Underlying/Forgetful (remove structure)
Adjoint (two-way relationship)
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19. Natural Transformations
Map from one functor to another
E.g. : F  G
Functors must be of same variance
Source and target categories must be of same type
No further levels are needed
Comparison of natural transformations is a natural transformation
E.g. :   
An arrow in a category is defined in context as unique up to natural isomorphism
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20. Informal Requirements for IS Architecture
Concepts
MetaMeta Policy
Meta Organize
Classify Instantiate
Constructs
Schema Types
Named Data Values
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Downward arrows are intension-extension pairs

21. Formalising the Architecture
Requirements:
mappings within levels and across levels
bidirectional mappings
closure at top level
open-ended logic
relationships (product and coproduct)
Choice: Category theory as used in mathematics as a workspace for relating different constructions
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22. Figure 2: Interpretation of Levels as Natural Schema in General
blue – category, red - functor, green - natural transformation
Figure 2: Interpretation of Levels as Natural Schema in General
Terms
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23. Figure 3: Example for Comparison of Mappings in two Systems
(Organisational interoperability)
Figure 3: Example for Comparison of Mappings in two Systems
Categories: CPT concepts, CST constructs, SCH schema, DAT data,
Functors: P policy, O org, I instance,
Natural transformations: , , 
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24. black - objects
Figure 4: Defining the Four Levels with Contravariant Functors and Intension-Extension (I-E) Pairs
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25. Figure 5: Examples of Levels in the Four Level Architecture
Cross-over arrows indicate contravariant mapping
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26. If functors are adjoint, there is a unique relationship between
them (a natural bijection).
Figure 6: Composition of Adjoints is Natural
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27. Six Expressions for Adjunctions
One for each functor and its dual:
1) P, P' ; 2) O, O'; 3) I, I'
One for each pair of adjacent functors and its dual
4) OP, P'O'; 5) OI, I'O'
One for all three functors composed together and its dual
6) IOP, P'O'I'
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28. Simple expression for P, P'
Adjoint 1
<P, P', cpt cst >: CPT  CST
P maps from category CPT to CST
P' maps from category CST to CPT (dual of P)
 is the unit of adjunction, measuring here the change in cpt
through application of P and P' in turn
 is the co unit of adjunction, measuring here the change in cst through application of P' and P in turn
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29. More complex expression for
IOP, P'O'I' -- Adjoint 6
IOP maps from category CPT to DAT
P'O'I' maps from category DAT to CPT (opposite direction
to IOP)
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30. is the counit of adjunction, measuring here the changes in
is the unit of adjunction, measuring here the changes in cpt, cst and sch through application of P, O, I, I', O' and P' in turn
is the counit of adjunction,
measuring here the changes in
dat, sch and cst through
application of I', O',P', P, O
and I in turn,
When  and  signify no change,
then special case for relationship -- equivalence
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31. Godement Calculus Manipulates categorical diagrams
Is a natural calculus
Provides rules showing:
composition of functors and natural transformations is associative
natural transformations can be composed with each other
Developed by Godement in 1950s
Has Interchange laws
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32. Figure 8: Organisational Interoperability in terms of Godement:
Comparison of Three Systems
Figure 8: Organisational Interoperability in terms of Godement:
Variable Policy
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33. Equations (Figure 8) for Godement Calculus from Simmons
Equations (6) interchange, (7)-(8) associativity, (9) permutation, (10) different paths (composition)
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34. Technical Conditions for Interoperability
That our categories obey the rules of category theory
every triangle in the diagram commutes (composition)
order of evaluating arrows is immaterial (associativity)
identity arrows are composable with other arrows
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35. Anticipated Problems 1 Type Information
ICEIS 2005
Anticipated Problems 1 Type Information
Semantic annotation needed
To obtain metameta types from implicit sources
Needs open architecture
Agents have potential
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36. Anticipated Problems 2 Composition Failure
Partial functions
Most categories are based on total functions
In real world many mappings are partial
not all of the source objects participate in a relationship (mapping)
Composition breaks down in a ‘total function’ category if a partial function occurs
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37. Use of Category Theory as a Standard for Interoperability
Formal basis – rigor, predictability
Handles
Data structures (categories)
Processes (functors)
Manipulation (Godement calculus)
Satisfies
Naturality (natural transformations, adjoints)
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38. Summary
Formal four-level architecture promising for tackling interoperability:
Use of category theory in natural role
Structure and relations through arrows (identity, category, functor, natural transformation)
Manipulate through Godement calculus
Suitable as a standard
Problems:
Composition failure (particularly with partial functions)
Need semantic annotation
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39. Prospects – PhD students
Robert Warrender (Sunderland) – testing 4-level ct architecture for relational and o-o databases
Dimitris Sisiaridis (Northumbria)– using 4-level ct architecture for security
Tim Reichert (Heilbronn/Northumbria) – using languages such as Qi for realising interoperability with ct. Development of tool for demonstrating technique.
06/12/2018

40. Recent/Future Publications
Rossiter, Nick, & Heather, Michael, Conditions for Interoperability, 7th International Conference on Enterprise Information Systems (ICEIS), Florida, USA, May 2005, (2005)
Rossiter, Nick, Heather, Michael, & Nelson, David, A Natural Basis for Interoperability, I-ESA’06, Interoperability for Enterprise Software and Applications Conference, University of Bordeaux, March 2006, 12pp, Springer (2006).
Rossiter, Nick, & Heather, Michael, Free and Open Systems Theory, EMCSR-2006, 18th European Meeting on Cybernetics and Systems Research, University of Vienna, April 2006, 6pp (2006).
06/12/2018