Using the Theory of Institutions for Topographic Knowledge Representation

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Using the Theory of Institutions for Topographic
Knowledge Representation

• Patrick Browne, School of Computing, Dublin
  Institute of Technology, Kevin Street, Dublin 8,
  Ireland. Patrick.Browne_at_comp.dit.ie
• Prof. Mike Jackson, School of Computing and
  Information, University of Central England in
  Birmingham, Perry Barr, Birmingham BT42 2SU, UK.
  Mike.Jackson_at_uce.ac.uk

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Modeling and unifying diversity

• Diversity is pervasive in topographic information
  We seek to represent diverse concepts in a
  unified, formal, and holistic way.
• Unification involves the thematic, cartographic,
  temporal, metric, directional, topological,
  network, set-theoretic and logical aspects of
  topographic objects.
• For this research we consider unification as the
  integration of the various mathematical
  formalisms that represent different aspects of
  topographic objects.

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Modeling and unifying diversity

• This research uses formal techniques for the
  unification of topographic knowledge. In
  particular, we argue that the Theory of
  Institutions as proposed by Goguen and Burstall
  is a suitable formalism to facilitate this
  unification.

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Example geometry lineage are two diverse
concepts that need to blended.

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Example inserting geometry
There are three possible cases which need to be
considered when inserting a node into a
2-complex 1) New node coincides with an existing
node 2) New node falls on an existing line 3)
New node falls within a triangle

How does insert impinge on temporal records,
database state, identity of topographic object,
network connectivity?
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Example inserting geometry
There are three possible cases which need to be
considered when inserting a node into a
2-complex 1) New node coincides with an existing
node. (not shown) 2) New node falls on an
existing line. (top example) 3) New node falls
within a triangle (bottom example)
How does insert impinge on temporal records,
database state, identity of topographic object,
network connectivity?
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Example roads, buildings, and regions

Consider a line. From the properties of metric
spaces it has a length.
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Example roads, buildings, and regions

Consider a line. Lets call it a road. From graph
theory we have a path
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Example roads, buildings, and regions
Consider a line. Lets add a field with an area
and a topology.
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Example roads, buildings, and regions
Lets add an administrative district and some
houses
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Example roads, buildings, and regions
Lets divide the field in two by inserting a new
fence. We need to delete the old area and add
two new areas. What about adjacency relation
between fields?
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Example roads, buildings, and regions
T1
T2
We now have a picture of the world at T1 and
T2. Not only have some objects changed but some
spatial relationships have changed.
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Example queries

• Is there a route from A to B? (now is assumed)
• Was there a route from a to be in 2000?
• Does the route in 1 pass through the
  administrative district ?
• Does the route in 1 pass touch the administrative
  district?
• What fields were adjacent to F1 in 2004?

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Conclusion

• We need a good way to organise this information.
• Currently UML and OCL are used to represent
  spatial standards (OGC).
• Various representations are used by the
  geo-research community.
• We propose the use of the theory of institutions
  from Gougan and Burstall.

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Land Registration

• House Details
• Built
• Current owner

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Informal Setting
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Informal Setting (detail)
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Expected outcomes

• a) A multifaceted representation in CafeOBJ,
  using a set of reasonable temporal,
  topological, thematic, and metric models. These
  sub-models will be representative rather than
  optimal.
• b) These sub-models constructed in a) need to be
  unified. We need to identify the properties that
  facilitate unification
• c) A model for update including insert, delete,
  and the propagation of change.
• d) Querying
• e) We need to prove certain properties of the
  above.

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Approach

• Category Theory (CT) CT can capture
  relationships between individual specifications
  and also between models of those specifications.
• Theory of Institutions (TOI) Institutions
  offer a categorical abstract model theory which
  formalizes the intuitive notion of a logical
  system, including syntax, semantics, and
  satisfaction between them.
• Algebraic Specification This is a well
  established computer science technique with
  strong links to category theory. TOI has extended
  the semantic range of algebraic specifications.

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Approach Theory of Institutions
A change in signature induces model changes, but
satisfaction relation is maintained. (a bit like
design by contract)
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Approach Institutions
Institution mappings constitute the fundamental
tool for encoding an institution into another,
for importing concepts and results from one
institution to another, for multi-logic
environments.
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Approach Institutions
H hidden A algebra O order M many S
sorted RWL rewriting logic
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Approach

• Existing spatial-temporal models will be selected
  with a view to producing a set of consistent
  algebras based on those models.
• Parts of these models will be merged, producing a
  single system.
• The single system will offer different
  perspectives and ways of reasoning about the
  topographic objects of e.g. structural or
  behavioural. This will enable use to use a set of
  proof mechanism e.g. induction and co-induction.

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Results to date

• We have adapted existing algebras (lambda
  calculus, propositional logic, interval logic,
  and temporal logic) from the algebraic literature
  to CafeOBJ.
• We are working on a database institution from
  Goguen written in BOBJ.
• We are working both G-Maps and Simplicial
  Complexes for the topology.
• We will use the CafeOBJ type systems for the
  thematic information (perhaps with a CASL
  spatial-temporal ontology).

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Results to date

• Merging temporal intervals, could be used to
  calculate the lifetime of a complex object.
• ceq mergeMax(L1, L2) (min(start(L1),
  start(L2)) max(finish(L1), finish(L2))) if
  (start(L2) lt finish(L1)) and (start(L1) lt
  finish(L2)) .
• ceq mergeMax(L1, L2) nil if not ((start(L2) lt
  finish(L1)) and (start(L1) lt finish(L2)))).
• These intervals overlap, can be merged
• These intervals dont overlap, cant be merged.

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Results to date

• Merging temporal intervals, useful for
  identifying the time that parts need to exist to
  form a whole.
• ceq mergeMin(L1, L2) (max(start(L1),start(L2))
  min(finish(L1), finish(L2))) if (start(L2) lt
  finish(L1)) and (start(L1) lt finish(L2)) .
• ceq mergeMin(L1, L2) nil if not ((start(L2) lt
  finish(L1)) and (start(L1) lt finish(L2))) .
• These intervals overlap, can be merged
• These intervals dont overlap, cant be merged.

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Results to date pushout of algebras
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Results to date general colimits
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Results to date pullbacks, limits
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Interesting Questions
Can we apply CT directly to model update?
DB at T1
Delete
Insert
DB at T2
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Interesting Questions
Can we apply CT directly to model the geo-domain?
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Some remaining technical questions

• The relationship between colimits and adjunctions
  for joining parameterized specifications.
  Colimits seem to deal with actualization of
  parameters, while adjunction deals with certain
  satisfaction conditions.
• The Grothendieck construction seems to handle
  inter-institutional relationships. Seems to be at
  higher level with institutions as its objects?
• The thesis should make these concepts useful to
  the geo-community. Apart from the structuring
  capability of CT it may be of some use for
  modelling geo-objects (similar to the way OO
  provides class hierarchies)