Categories for Information

1
Categories for Information

• Nick Rossiter
• 1st March 2006
• School of Computing, Engineering and Information
  Sciences
• Northumbria University
• nick.rossiter_at_unn.ac.uk
• http//computing.unn.ac.uk/staff/CGNR1/

2
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

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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)
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Interoperability

• An area where context is paramount is
• Interoperability
• 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

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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

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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

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Demands for Interoperability

• Business Needs
• Data warehousing
• Web Warehousing
• GRID

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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)

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Attempted Solutions 2

• Ontologies
• Being
• Defines meaning of data
• Like a dictionary
• But is usually much more
• Everything is defined in context
• Multi-level definitions
• No clear consensus

10
Formal Basis

• For preceding techniques
• Some set theoretic justifications
• These are partial
• Emphasis on a level
• Contrived multi-level
• Above all lack concept of naturality

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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).

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Applied Categories

• Physics including quantum studies
• 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)

13
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

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The Simplest Category
Discrete Category Identity arrows (objects)
only
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A Not Very Useful Category
6 arrows Not connected Does not conflict with
axioms
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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
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Cartesian Closed Category
1C
Basis of much Computing Science Research in CT
C
PPP
PxPxP
Has identity, products, limits, coproducts
Identity functor 1C C -? C
Initial object PxPxP provides handle on category
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Functors

• Map from one category to another
• 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)

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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

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Informal Requirements for IS Architecture
Concepts

• MetaMeta Policy
• Meta
  Organize
• 
  
• Classify
  Instantiate

Constructs
Schema Types
Named Data Values
Downward arrows are intension-extension pairs
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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

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Blue category, red functor, green nat trans
Figure 2 More Detailed Interpretation of Levels
in Category Theory Natural Schema
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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

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Comparison of Three Systems
Figure 9 Organisational Interoperability for
use with Godement Calculus. Variable Policy
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Equations (Figure 6) for Godement Calculus from
Simmons Equations (6) interchange, (7)-(8)
associativity, (9) permutation, (10) different
paths (composition)
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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

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Anticipated Problems 1Type Information

• Semantic annotation needed
• To obtain metameta types from implicit sources
• Needs open architecture
• Agents have potential

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Anticipated Problems 2Composition 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

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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

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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.

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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-ESA06, 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).