Some Fundamental Questions in Databases

1. Some Fundamental Questions in Databases Nick Rossiter (with Michael Heather) Informatics Research Conference, Northumbria University

2. Questions • Why so many database models? • What is their rationale? • Are there undiscovered new models? • Is there an ultimate data model? • Do all models have to be reductionist? Informatics Research Conference, Northumbria University

3. Meaning of Database Model • A model is a representation of reality according to some perceived view. • A database model for representing this view has: • a structure • a manipulation language • rules for controlling the structure Informatics Research Conference, Northumbria University

4. Classical Database Models 1 Informatics Research Conference, Northumbria University

5. Classical Database Models 2 Informatics Research Conference, Northumbria University

6. Classical Relationships • Relationships are often performed in a separate process: • Entity-Relationship Modelling • Unified Modelling (UML) • Normalisation is needed to verify schema design: • particularly to relate key and non-key attributes. • The levels, mappings and relationships all have to be integrated in a consistent database design. Informatics Research Conference, Northumbria University

7. Developing non-classical Areas • The new developing areas are: • quantum computation, exploiting quantum mechanics principles in physics, • nanoscale chemistry, • bio- and molecular-computing processing as in genetics • Collectively referred to as natural computing • Natural computing is: • Real-world processing • does not rely on any model. • Data can be input neat • without any reductionist pre-processing. Informatics Research Conference, Northumbria University

8. X + f* Non-classical Database Design • Not layered (Theory of Categories) • Use Dolittle approach (push me-pull you creature of High Lofting): • A database design is a topos -- a Dolittle diagram subsuming the pullback/pushout relationships as: Cartesian Closed Category Informatics Research Conference, Northumbria University

9. What is f*? • f* is an examination and re-indexing functor • organises the data into a key for storage and applies a query for interrogation of the database. • puts together a key by concatenation as in the relational model. • looks up information for retrieval by inspecting the key. • In quantum theory: • the key (X) is entanglement, • the colimit (+) is superposition, • In genetics it is a DNA strand. Informatics Research Conference, Northumbria University

10. Enriched Pullback • In terms of the Dolittle diagram: • f* is the same operation in classical and natural computation. • What then corresponds to the database schema in natural computing? • The pullback diagram contains many more arrows than in the Dolittle diagram. • This enriched diagram satisfies our needs. Informatics Research Conference, Northumbria University

11. Pullback of S and M in Context of IMG S = source, M = medium, IMG = image, W = world Informatics Research Conference, Northumbria University

12. Contents of Enriched Dolittle • The Dolittle diagram relates binary categorial limits (X) and colimits (+) for types • Logic is Heyting -- intuitionistic logic. • Godement calculus can be used for composing arrows across levels • pullback functor (f* or ): • emulates the join operation of databases • Other arrows represent: • projection ; membership  • existential  or  quantification • universal  or  quantification Informatics Research Conference, Northumbria University

13. Higher-level Arrows for Semantics/DB Design • Originality with the unit of adjunction  • Example:  gives properties of relationship onto limit •  = 0, no creativity, mapping S to S XIMG M is 1:1 •  = 1, maximum creativity, mapping S to S  IMG M is from S to cartesian product of S  M. • Style with the counit  of adjunction. • Example:  gives properties of relationship onto colimit •  = 1, preservation of style, each S is found exactly once in S XIMG M •  = 0, loss of style, each S occurs in S  IMG M maximum number of times (S  M). Informatics Research Conference, Northumbria University

14. Normalisation and Defeasance • Not captured at the data model level • except for 1NF in some models • A complex topic jargon-wise • Perhaps related to defeasibility or nonmonotonic reasoning: • A kills B implies A has committed murder • A kills B AND B attacked A implies A has committed manslaughter • strengthening of antecedent changes conclusion Informatics Research Conference, Northumbria University

15. Integration of Normalisation within the model • Many normalisation definitions say: • A table is normalised if • condition X holds • and additional conditions B, C,, D etc do not hold • Normalisation failure is due to defeasance • Exploring incorporation of defeasance into topos Informatics Research Conference, Northumbria University

16. Non-classical Database Model Informatics Research Conference, Northumbria University

17. Questions • Why so many database models? • What is their rationale? • Are there undiscovered new models? • Is there an ultimate data model? • Do all models have to be reductionist? Informatics Research Conference, Northumbria University