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Four-level Architecture for Closure in Interoperability

Explore the four-level architecture for achieving closure in interoperability, using category theory and Godement calculus. Discover how this approach provides an architecture for universal interoperability and allows for the composition of mappings at any level.

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Four-level Architecture for Closure in Interoperability

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  1. Four-level Architecture for Closure inInteroperability EFIS, 17th-18th July 2003 Nick Rossiter & Michael Heather Informatics, Northumbria University http://computing.unn.ac.uk/staff/CGNR1/ nick.rossiter@unn.ac.uk

  2. Interoperability • Interoperability: the ability to request and receive services between various systems and use their functionality. • More than data exchange. • Implies a close integration

  3. Motivations • Diversity of modelling techniques • Data warehousing requires heterogeneous systems to be connected • Semantic Web/RDF/Ontologies • GRID • MOF/MDA

  4. Mappings in Systems are two-way MetaMetaPolicy Meta Organize Classify Instantiate Concepts Constructs Schema Types Named Data Values Downward arrows are intension-extension pairs

  5. 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

  6. 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

  7. Comparing one System with Another CCCSSMDT CCCS´SM´DT´ P O I    P´ O ´ I ´ ,,  are natural transformations (comparing functors)

  8. 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´); (OP) = (O)P •   = (O) o (I´ );  = P o (O´)

  9. 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

  10. Analogous Levels for Interoperability

  11. Pullbacks are used extensively for database relationships Here of S and M in Context of IMG S = source, M = medium, IMG = image, W = world

  12. Discussion • Category theory shows that: • four levels are ideal for interoperability • more than four yields no benefits • less than four gives only local interoperability • Categorical approach provides: • an architecture for universal interoperability • a calculus (Godement) for composing mappings at any level • adjunctions for evaluating two-way mappings

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