Conditions for Interoperability Nick Rossiter Michael Heather School of Informatics, Engineering and Technology Northumbria University nick.rossiter@unn.ac.uk http://computing.unn.ac.uk/staff/CGNR1/ 05/05/2019

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 05/05/2019

Motivations Diversity of modelling techniques Data warehousing requires heterogeneous systems to be connected Semantic Web/RDF/Ontologies GRID MOF/MDA 05/05/2019

Figure 1: Classical ANSI/SPARC Architecture for Databases 05/05/2019

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 05/05/2019

1st step – Identify Architecture Components and 2-way Mappings Concepts MetaMeta Policy Meta Organize Classify Instantiate Constructs Schema Types Named Data Values 05/05/2019 Downward arrows are intension-extension pairs

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 05/05/2019

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 05/05/2019

Figure 2: More Detailed Interpretation of Levels in Category Theory: Natural Schema 05/05/2019

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 05/05/2019

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: , ,  05/05/2019

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 05/05/2019

Figure 4: Alternative Interpretation of Levels in the Architecture 05/05/2019

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 05/05/2019

Figure 5: Godement Calculus in Barr and Wells (1990) 1st ed., p.96 05/05/2019

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 05/05/2019

Figure 6: Godement in Simmons, Lecture Notes on Category Theory, section 3.8 05/05/2019

Figure 7: Commuting Diagram in Simmons, Lecture Notes on Category Theory, section 3.8 05/05/2019

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 05/05/2019

Figure 8: Semantic Interoperability in terms of Godement Calculus. Constant Policy 05/05/2019

Figure 9: Organisational Interoperability in terms of Godement Calculus. Variable Policy 05/05/2019

Equations (Figure 6) for Godement Calculus from Simmons Equations (6) interchange, (7)-(8) associativity, (9) permutation, (10) different paths 05/05/2019

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 05/05/2019

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 05/05/2019

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 05/05/2019

Figure 10: Punctured Commuting Diagram ICEIS 2005 Figure 10: Punctured Commuting Diagram After Freyd (1990) 05/05/2019

Figure 11: Punctured Commuting Diagram for Library Example ACC = accessions, STK = stock, ISS = issues, CAT = catalogue 05/05/2019

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 05/05/2019

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 05/05/2019

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) 05/05/2019

Figure 12: Non-punctured Commuting Diagram for Library Example ACC = accessions, STK = stock, ISS = issues, CAT = catalogue 05/05/2019

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 05/05/2019