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/ 06/12/2018

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 06/12/2018

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) 06/12/2018

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 06/12/2018

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 06/12/2018

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 06/12/2018

Demands for Interoperability Business needs Data warehousing Web warehousing GRID 06/12/2018

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) 06/12/2018

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 06/12/2018

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 06/12/2018

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). 06/12/2018

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) 06/12/2018

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] 06/12/2018

The Simplest Category Discrete Category Identity arrows (objects) only 06/12/2018

A Not Very Useful Category 6 arrows Not connected Does not conflict with axioms 06/12/2018

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 06/12/2018

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 06/12/2018

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) 06/12/2018

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 06/12/2018

Informal Requirements for IS Architecture Concepts MetaMeta Policy Meta Organize Classify Instantiate Constructs Schema Types Named Data Values 06/12/2018 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) Choice: Category theory as used in mathematics as a workspace for relating different constructions 06/12/2018

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 06/12/2018

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: , ,  06/12/2018

black - objects Figure 4: Defining the Four Levels with Contravariant Functors and Intension-Extension (I-E) Pairs 06/12/2018

Figure 5: Examples of Levels in the Four Level Architecture Cross-over arrows indicate contravariant mapping 06/12/2018

If functors are adjoint, there is a unique relationship between them (a natural bijection). Figure 6: Composition of Adjoints is Natural 06/12/2018

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' 06/12/2018

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 06/12/2018

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) 06/12/2018

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 06/12/2018

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 06/12/2018

Figure 8: Organisational Interoperability in terms of Godement: Comparison of Three Systems Figure 8: Organisational Interoperability in terms of Godement: Variable Policy 06/12/2018

Equations (Figure 8) for Godement Calculus from Simmons Equations (6) interchange, (7)-(8) associativity, (9) permutation, (10) different paths (composition) 06/12/2018

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 06/12/2018

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 06/12/2018

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 06/12/2018

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) 06/12/2018

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 06/12/2018

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

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). 06/12/2018