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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/ [email protected] 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 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 Overview of Presentation 1. Database theory is underpinned by the term model. 2. Unfortunately model does not have a universal meaning. 3. Explore some meanings 4. Look at interoperability representations 5. Introduce Dolittle diagrams 6. Look at concept of model and sketch in category theory 7. Future work may try and unify last two (5+6) 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 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) 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 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 Mappings in complete system Concepts MetaMeta Policy Constructs Meta Organize Schema Types Classify Instantiate Named Data Values Downward arrows are intension-extension pairs Category Theory: Comparing one System with Another CC P CS CC P´ O SM CS´ O´ DT I SM´ I´ DT´ ,, are natural transformations (comparing functors) 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´ ); • = ( O) o (I´ ); (OP) = ( O)P = P o (O´ ) Analogous Levels for Interoperability Level Category Architecture 1. data values Objects (identity iddt arrows) 2. named values 3. classified values 4. contrasted representation Category DT Functor C: DT SM * o * (* is Natural transformation dual of ) Category Theory: Detail - Example of modelling Relationships – the Pullback ls x m S s ls x m S XIMG M rs x m s x m *m M s (s)-1 m (m)-1 W/IMG Pullback showing fuller collection of arrows 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 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). 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 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) Model in Categories • Model (M) – graph homomorphism • M:DC • 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 • Evaluate: Future Research – 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 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 Some Publications in Other Areas 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 Multiagency 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). 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 307326 (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).