A Natural Basis for Interoperability

1. A Natural Basis for Interoperability Nick Rossiter, Mike Heather and David Nelson I-ESA’06 Northumbria University, University of Sunderland david.nelson@sunderland.ac.uk nick.rossiter@unn.ac.uk http://computing.unn.ac.uk/staff/CGNR1/

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

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)

4. Interoperability • An area where context is paramount is • Interoperability • 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

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

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

7. Demands for Interoperability • Business needs • Data warehousing • Web warehousing • GRID

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)

9. Attempted Solutions 2 • Ontologies • Being • Defines meaning of data • Like a dictionary • But is usually much more • Everything is defined in context • Multi-level definitions • No clear consensus

10. Formal Basis • For preceding techniques • Some set theoretic justifications • These are partial: • Emphasis on a level • Contrived multi-level • Above all – lack concept of naturality

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).

12. Applied Categories • Physics including quantum studies • 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)

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]

14. The Simplest Category Discrete Category Identity arrows (objects) only

15. A Not Very Useful Category 6 arrows Not connected Does not conflict with axioms

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

17. Cartesian Closed Category 1C Basis of much Computing Science Research in CT C P+P+P PxPxP Has identity, products, limits, coproducts Identity functor 1C: C - C Initial object PxPxP provides handle on category

18. Functors • Map from one category to another • 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)

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

20. Informal Requirements for IS Architecture MetaMetaPolicy Meta Organize Classify Instantiate Concepts Constructs Schema Types Named Data Values 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

22. blue – category, red - functor, green - natural transformation Figure 2: Interpretation of Levels as Natural Schema in General Terms

23. (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: , , 

24. black - objects Figure 4: Defining the Four Levels with Contravariant Functors and Intension-Extension (I-E) Pairs

25. Figure 5: Examples of Levels in the Four Level Architecture Cross-over arrows indicate contravariant mapping

26. If functors are adjoint, there is a unique relationship between them (a natural bijection). Figure 6: Composition of Adjoints is Natural

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'

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

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)

30. 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

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

32. Comparison of Three Systems Figure 8: Organisational Interoperability in terms of Godement: Variable Policy

33. Equations (Figure 8) for Godement Calculus from Simmons Equations (6) interchange, (7)-(8) associativity, (9) permutation, (10) different paths (composition)

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

35. Anticipated Problems 1Type Information • Semantic annotation needed • To obtain metameta types from implicit sources • Needs open architecture • Agents have potential

36. Anticipated Problems 2Composition 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

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)

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

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.

40. Recent/Future Publications • Rossiter, Nick, & Heather, Michael, Conditions for Interoperability, 7th International Conference on Enterprise Information Systems (ICEIS), Florida, USA, 25-28 May 2005, 92-99 (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).