1. Conditions for Interoperability
Nick Rossiter
Michael Heather
School of Informatics, Engineering and Technology
Northumbria University
05/05/2019

2. Interoperability Interoperability More than data exchange.
the ability to request and receive services between various systems and use their functionality.
More than data exchange.
Implies a close integration
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3. Motivations Diversity of modelling techniques
Data warehousing requires heterogeneous systems to be connected
Semantic Web/RDF/Ontologies
GRID
MOF/MDA
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4. Figure 1: Classical ANSI/SPARC Architecture
for Databases
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5. Suitability of Classical Architecture
Levels are not independent of each other
No universal closure of types
Need for interoperability:
Orthogonal type architecture
Formal mappings between the levels of the architecture
Natural closure of architecture
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6. 1st step – Identify Architecture Components and 2-way Mappings
Concepts
MetaMeta Policy
Meta Organize
Classify Instantiate
Constructs
Schema Types
Named Data Values
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Downward arrows are intension-extension pairs

7. Formalising the Architecture
Requirements:
mappings within levels and across levels
bidirectional mappings
closure at top level
open-ended logic
relationships (product and coproduct)
Candidate: category theory as used in mathematics as a workspace for relating different constructions
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8. Choice: category theory
Requirements:
mappings within levels and across levels
arrows: function, functor, natural transformation
bidirectional mappings
adjunctions
closure at top level
four levels of arrow, closed by natural transformation
open-ended logic
Heyting intuitionism
relationships (product and coproduct)
Cartesian-closed categories (like 2NF): pullback and pushout
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9. Figure 2: More Detailed Interpretation of Levels
in Category Theory: Natural Schema
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10. Forms of Interoperability
Semantic:
agreed concepts
a common framework of constructs
schema and data vary
e.g. working within a relational framework
Organisational:
agreed concepts (but open ended)
constructs, schema and data vary
e.g. working within an object framework
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11. 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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12. Four Levels are Sufficient
In category theory:
objects are identity arrows
categories are arrows from object to object
functors are arrows from category to category
natural transformations are arrows from functor to functor
An arrow between natural transformations is a composition of natural transformations, not a new level
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13. Figure 4: Alternative Interpretation of Levels in the Architecture
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14. 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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15. Figure 5: Godement Calculus in Barr and Wells (1990)
1st ed., p.96
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16. Equations (Figure 5) for Godement Calculus from Barr and Wells (1990)
Equations (1)-(4): interchange, associativity and permutativity
Equation (5): different paths
o vertical composition
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17. Figure 6: Godement in Simmons, Lecture Notes
on Category Theory, section 3.8
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18. Figure 7: Commuting Diagram in Simmons, Lecture
Notes on Category Theory, section 3.8
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19. Application Semantic Interoperability Organisational Interoperability
Agreed concepts and constructs
Constant policy for mapping from concepts to constructs
Figure 5 – Barr & Wells approach
Organisational Interoperability
Agreed (but open ended) concepts
Variable policy for mapping from concepts to constructs
Figure 6 – Simmons approach
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20. Figure 8: Semantic Interoperability in terms
of Godement Calculus.
Constant Policy
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21. Figure 9: Organisational Interoperability
in terms of Godement Calculus.
Variable Policy
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22. Equations (Figure 6) for Godement Calculus from Simmons
Equations (6) interchange, (7)-(8) associativity, (9) permutation, (10) different paths
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23. 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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24. 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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25. 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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26. Figure 10: Punctured Commuting Diagram
ICEIS 2005
Figure 10: Punctured Commuting Diagram
After Freyd (1990)
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27. Figure 11: Punctured Commuting Diagram for Library Example
ACC = accessions, STK = stock, ISS = issues, CAT = catalogue
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28. Possible Advances 1: Develop New Category
Develop category of partial (lifted) functions
Lellahi & Spyratos (FIDE)
Enormous effort in basic category theory
Category theory is founded on total functions
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29. Possible Advances 2: Sketches
Use sketches
Relax composition rules for selected diagrams
Map graph-based sketch onto a category
Work by Rosebrugh, Diskin
Appealing for initial productivity
intuitively similar to ER modelling
But on fringes of category theory and lack flexibility and natural closure
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30. Preferred Advance Avoid partial functions
Avoid such functions in design by greater use of roles
Convert all such functions into total ones:
map null relationships onto initial object (bottom)
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31. Figure 12: Non-punctured Commuting Diagram for Library Example
ACC = accessions, STK = stock, ISS = issues, CAT = catalogue
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32. Summary
Formal four-level architecture promising for tackling interoperability:
Use of category theory in natural role
Structure through arrows (identity, category, functor, natural transformation)
Manipulate through Godement calculus
Problems:
Composition failure (particularly with partial functions)
Need semantic annotation
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