Information Systems and Categories: Sketches and Models for Database Modelling

1. Information Systems and Categories: Sketches and Models for Database Modelling Nick Rossiter Research Conference, Informatics, Northumbria University, 15th May 2003 http://computing.unn.ac.uk/staff/CGNR1/ nick.rossiter@unn.ac.uk

2. Motivation • Interoperability • Working together of information systems • Difficult area particularly with heterogeneous models • Formal basis lacking • Work by NR/MH/DAN has involved: • Looking at a sound formal basis • Formalisation of object-relational model • Using category theory

3. Motivation 2 • Most recent work (Northumbria, seminar Nov 2002): • Has been well received • Shows that four levels are needed for addressing data • Provides Godement calculus for manipulating across levels

4. Overview of Presentation • Database theory is underpinned by the term model. • Unfortunately model does not have a universal meaning. • Explore some meanings • Look at interoperability representations • Introduce Dolittle diagrams • Look at concept of model and sketch in category theory • Future work may try and unify last two (5+6)

5. A Definition of Model in the Database World • Philosophical areas – for example, in interoperability • Database Model: a representation of policies in a structured form according to some perceived view of reality e.g. • Relational model – world is tabular • Hierarchical model – world is tree-like • Security model – world is task-based • Object model – world is based on o-o paradigm • ER model – world is graph-based

6. Another Database Definition for Model • A model comprises: • A data structure • A language for manipulating the structure • A collection of rules governing acceptable states of the structure • On this basis: • ER is not a model (no general manipulation language) • Relational is a model (e.g. data structure = table, manipulation language = SQL, rules = referential integrity)

7. Also Design Models • Examples of design models: • ER • UML • Always graphically based. • Often provide a route to a basic model for implementation, population and manipulation

8. Modelling a Whole System • Most models are aimed at data definition level (schema). • Full system has multiple levels: • One below the schema – the data values • Two above – constructs available and concepts to be employed

9. Mappings in complete system MetaMetaPolicy Meta Organize Classify Instantiate Concepts Constructs Schema Types Named Data Values Downward arrows are intension-extension pairs

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

11. 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´)

12. Analogous Levels for Interoperability

13. Category Theory: Detail - Example of modelling Relationships – the Pullback Dolittle Diagram of S and M in Context of IMG S = source, M = medium, IMG = image, W = world Logic available: product, join, project, existential and universal quantifiers, select, insert, units of adjunction and co-adjunction Constraints available: cardinality, membership class

14. Recent Publications in this Area • Rossiter, N, From Classical to Quantum Databases with Applied Pullbacks, 78th Meeting Peripatetic Seminar on Sheaves and Logic, Institut de Recherche Mathématique Avancée, Strasbourg University 15-16 February (2003). • Rossiter, N, Nelson, D A, & Heather, M A, Formalizing Types with Ultimate Closure for Middleware Tools in Information Systems Engineering, 5th ICEIS, Angers, France 23-26 April 8pp (2003). • Rossiter, N, & Heather, M, Four-level Architecture for Closure in Interoperability, EFIS2003, Fifth International Workshop on Engineering Federated Information Systems, Coventry, UK, 17-18 July 6pp (2003). • Heather, M A, & Rossiter, B N, The Anticipatory and Systemic Adjointness of E-Science Computation on the Grid, Computing Anticipatory Systems, Proceedings CASYS`01, Liège, Dubois, D M, (ed.), AIP Conference Proceedings 627 565-574 (2002).

15. Other Work with Databases and Categories • Michael Johnson, Robert Rosebrugh and RJ Wood, Entity-Relationship-Attribute Designs and Sketches, TAC 10(3) 94-111. • sketches for design (class structure) • models for states (objects) where model is used in categorical sense • lextensive category (finite limits, stable disjoint finite sums) for query language

16. Sketch • Developed also in databases by: • Zinovy Diskin, Boris Cadish: Algebraic Graph-Based Approach to Management of Multidatabase Systems, NGITS’95 69-79 (1995). • Sketch originally from Charles Ehresmann. • Many different sorts of sketch – 12 kinds listed in Charles Wells, Sketches, Outline with References, at http://www.cwru.edu/artsci/math/wells/pub/papers.html#sketch • For instance Finite Product (FP) is much used but it has no cocones (sums) • Most suitable appears to be Finite Discrete (FD) sketch D = (E, L, R, S) • finite graph E (data structure) • set of diagrams L in E (constraints) • Finite set R of discrete cones in D (relationships) • Finite set S of discrete cocones in D (attributes)

17. Model in Categories • Model (M) – graph homomorphism • M : D  C • M maps: • takes any node in E to a set of values (populates) • L  commutative diagrams • R  limit cones • S  co-limit cocones • C is a target category (extension) • preserve products and co-products in state

18. Future Research • Evaluate: • Use of sketch as construction for two bottom levels of architecture (schema, values) • Feasibility of building in Dolittle diagram for logic • Then if outcome positive: • Either Add top two levels (constructs, concepts) to sketch to give 4-level architecture with adjointness connecting the levels as in recent publications • Or Extend sketch construction to 4-levels (through repeated sketch-model constructions, transitive closure) • Else if outcome negative: • Use fundamental categorical levels (named object, category, functor, nat trans) for 4-levels as in recent publication and develop from there

19. Database Group • Progressing Open Database Project • Development of open source software • Based on fundamental view of relational model • Developing work on previous slide to: • Specify formally object-relational model • Try realising this formalisation with the Open Database Project • Advance interoperability with sounder foundations

20. Some Publications in Other Areas

21. Security in Multi-agency Services • Aljareh, S, & Rossiter, N, A Task-based Security Model to facilitate Collaboration in Trusted Multi-agency Networks, ACM Symposium on Applied Computing (SAC) 2002, Madrid, 744-749 March (2002). • Aljareh, S, & Rossiter, N, Towards Security in Multi-agency Clinical Information Services, Health Informatics Journal 8(2) 96-104 (2002). • Aljareh, S, Dobson, J, & Rossiter, N, Satisfaction of Health Record Security Principles through Collaborative Protocols, NI'2003, 8th International Congress in Nursing Informatics, Brazil, 5pp, 20-25 June (2003).

22. Natural Computing (Quantum) • Heather, M A, & Rossiter, B N, Locality, Weak or Strong Anticipation and Quantum Computing I. Non-locality in Quantum Theory, International Journal Computing Anticipatory Systems 13 307-326 (2002). • Heather, M A, & Rossiter, B N, Locality, Weak or Strong Anticipation and Quantum Computing. II. Constructivism with Category Theory, International Journal Computing Anticipatory Systems 13 327-339 (2002).