Categories for Information Nick Rossiter 1st March 2006 School of Computing, Engineering and Information Sciences Northumbria University nick.rossiter@unn.ac.uk http://computing.unn.ac.uk/staff/CGNR1/ 09/04/2019

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 09/04/2019

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) 09/04/2019

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 09/04/2019

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 09/04/2019

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 09/04/2019

Demands for Interoperability Business Needs Data warehousing Web Warehousing GRID 09/04/2019

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) 09/04/2019

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 09/04/2019

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 09/04/2019

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). 09/04/2019

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) 09/04/2019

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] 09/04/2019

The Simplest Category Discrete Category Identity arrows (objects) only 09/04/2019

A Not Very Useful Category 6 arrows Not connected Does not conflict with axioms 09/04/2019

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 09/04/2019

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 09/04/2019

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) 09/04/2019

Natural Transformations Map from one functor to another E.g. : F  G Functors must be of same variance 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 09/04/2019

Informal Requirements for IS Architecture Concepts MetaMeta Policy Meta Organize Classify Instantiate Constructs Schema Types Named Data Values 09/04/2019 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) Choice: Category theory as used in mathematics as a workspace for relating different constructions 09/04/2019

Blue – category, red functor, green nat trans Figure 2: More Detailed Interpretation of Levels in Category Theory: Natural Schema 09/04/2019

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 09/04/2019

Comparison of Three Systems Figure 9: Organisational Interoperability for use with Godement Calculus. Variable Policy 09/04/2019

Equations (Figure 6) for Godement Calculus from Simmons Equations (6) interchange, (7)-(8) associativity, (9) permutation, (10) different paths (composition) 09/04/2019

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 09/04/2019

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 09/04/2019

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 09/04/2019

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 Problems: Composition failure (particularly with partial functions) Need semantic annotation 09/04/2019

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. 09/04/2019

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). Also leading to journal paper as part of set-up. EU Athena initiative 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). 09/04/2019