Some Fundamental Questions in Databases

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Some Fundamental Questions in Databases

• Nick Rossiter
• (with Michael Heather)

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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?

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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

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Classical Database Models 1
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Classical Database Models 2
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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.
  

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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.

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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
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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.

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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.

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Pullback of S and M in Context of IMG

S source, M medium, IMG image, W world
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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

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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
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• ? 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).

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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

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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

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Non-classical Database Model
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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?