mr. patrick browne [email protected] school of computing dublin institute of technology...
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Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Patrick Browne, Mike Jackson
Towards a Unified Spatial-Temporal Data Model and Query Language for Geographical Information Systems
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Rationale
• It is generally acknowledge that geo-data requires the database functionality that is readily available for conventional tabular data.
• The general approach to this research was inspired by a paper by Herring, Egenhofer and Frank[1].
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Aims
• The aim of this research is to develop a formal data model and query language called the Topographic Data Model (TDM), which will facilitate the management of topographic information.
• Formal proofs together with a prototype will validate the research.
• Both map construction (insert) and map usage (query) will be considered.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Modeling Diversity
• Diversity is pervasive in topographic information (TI). Diverse concepts must be unified in a formal and holistic way.
• This diversity could lead to a very large PhD. A recent paper on spatial temporal DB [2] had eight authors! Hence model building blocks must be representative and small rather than exhaustive and complete.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Unifying Diversity
• For unification of the thematic, temporal, metric, directional, topological, and network, aspects we need:– Well-defined local theories (or local logics). – Well-defined protocols that allow local theories
to communicate.– A mechanism to merge two or more local
theories.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Expected outcomes
• A prototype spatial temporal language to manage topographical data.
• A mechanism for composing existing theories and algebras.
• A mechanism for allowing theories to communicate.• A mechanism for checking the correctness of theories.• Directions for future research e.g. ontologies for
geographic information.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
TDM: the proposed model
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
The Approach
• Analysis of current spatial and temporal models: – Open GIS Consortium OGC[3].
– ST-Complexes, Worboys[4].
• Current models can be analyzed from the perspective of: – Operations: Algebraic Specification[5], Institutions[6].
– Unification: Category Theory[7,14], Institutions[6].
– Communication: Information Flow[8,14]
– Structural Consistency: Formal Concept Analysis[9,14].
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
The Approach
• Existing models will be analysed, producing a set of consistent algebras.
• Current models will be merged, producing a single algebra.
• The initial unified TDM will generate problems that do not arise when existing models are considered in isolation.
• Hence, it will be necessary to re-engineer the emerging TDM.
• The final model will have a broader expressive range than its components.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Results to date
• Category theory (CT) is a suitable meta-technique. CT is an abstract formalism that studies objects and morphism between objects. A category can be described as a graph with rules for composition.
• CT pros:– More expressive than OO theory, most branches of
mathematics and computer science can be described or specified in categorical terms.
• CT cons:– Difficult to learn, about 30% done.– Difficult to apply, need to think categorically.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Techniques
• Category theory(CT): Though central to research CT is very abstract and needs to be augmented with additional formalisms and tools.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Techniques
• Algebraic Specification(AS): This is a well established computer science technique with strong links to category theory.
• Institutions: Offer a categorical abstract model theory which formalizes the intuitive notion of logical system, including syntax, semantics, and satisfaction between them. Most modern algebraic specification languages such as CASL, CafeOBJ are constructed around institutions
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Techniques
• Information Flow(IF): Examines how information about some components carries information about other component of a system. IF has formal definitions of channel, classification and infomorphism. Category theory is a central concept in IF.
• Formal Concept Analysis(FCA): Is a mathematical technique for conceptual data analysis and knowledge processing. It formalizes the terms context, concept and concept hierarchy.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Tools
• Promising modelling tools include: – CafeOBJ[4], CASL[10] and SPECWARE[11].
These tools are based on institutions and use fundamental concepts from category theory.
• Work has been done using SPECWARE and CafeOBJ.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Merging Algebras with Colimits
• The following example from Smith[11] illustrates how algebras can be joined in a meaningfull way. Here CT is used in the context of algebraic specifications. Specifications are considered as objects in a category called Spec. The Specware code is included in the notes.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Merging Algebras with Colimits
Colimit operation
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Merging Algebras with Colimits
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Worboys ST-Complex[4]
A triangulated road segment and two houses.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Worboys ST-Complex[4]
• There are three possible cases which need to be considered when inserting a node into a 2-complex (or triangulated network):
1) New node coincides with an existing node
2) New node falls on an existing line
3) New node falls within a triangle
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Worboys ST-Complex[4]
lo
px
t0
t1 Inserting a node on a line in a triangulated
network.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Worboys ST-Complex[4]
Inserting a node in an existing triangle
l0
l1 l2
p0 p1
p2
px
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
The Euclidean plane in CaféOBJ[12]
obj R2 ispr FLOAT * (sort Float to Real) .sort Vect .op 0 : -> Vect .op <_,_> : Real Real -> Vect .op _+_ : Vect Vect -> Vect .op -_ : Vect -> Vect .op _*_ : Real Vect -> Vect .vars a b a’ b’ k : Real .eq 0 = < 0 , 0 > .eq < a , b > + < a’ , b’ > = < a + a’ , b + b’ > .eq k * < a , b > = < k * a , k * b > .eq - < a , b > = < - a , - b > .endo
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Topographic Data Model (TDM)
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Whats next?
• More Study required in:– Category Theory
– Algebraic Specifications techniques and tools
– Institutions
– Information Flow
– Formal Concept Analysis
– Functional Query Languages[13]
• The representitive algebras for unification need to be selected and developed.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
Risks
• A wide range of techniques and tools could allow work to become unfocused, but …
– Category Theory does provide the central underlying theory.
– Algebraic specifications in the form of Institutions and tools will provide the main representation.
– The building blocks will be representative and small rather than exhaustive and complete.
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
References
[1] Herring, Egenhofer and Frank called “Using category theory to model GIS applications". 4th Int. Sym. On Spatial Data Handling, Zurich, Switzerland, 1990.
[2] Tripod: A Comprehensive System for the Management of Spatial and Aspatial Historical Objects, Tony Grifflths, Alvaro A.A. Fernandes, Norman W. Paton, Keith T. Mason, Bo Huang, Mike Worboys, Chris Johnson, John Stell in GlS’Ol,2001, Atlanta, Georgia, USA.
[3] The Open GIS Consortium develops open publicly available geo-processing specifications. See http://www.opengis.org
[4] M.F. Worboys, "A Unified Model for Spatial and Temporal Information." The Computer Journal Vol 37 Number 1 The British Computer Society.
[5] Foundations of Algebraic Specification and Formal Program Development by D. Sannella and A. Tarlecki, publication under preparation 2002.
[6] Institutions: Razvan Diaconescu at Logics of Formal Software Specification Languages, PhD Summer School 2004, Slovakia.
[7] Basic Category Theory for Computer Scientists: Pierce, B., The MIT Press (1991)[8] Information Flow: Barwise and Seligman, Cambridge University Press 1997, ISBN 0-
521-58386-1
Mr. Patrick [email protected]
School of ComputingDublin Institute of Technology
References
[9] Formal Concept Analysis: Ganter and Willie, Springer-Verlag 1999, ISBN 3-540-62771-5
[10] CASL http://www.cofi.info. Last accessed 7-7-2004
[11] Douglas R. Smith. Software Development by Refinement Technical Paper from Kestrel Institute, Palo Alto, California 94304 USA.
[12] Category-based Constraint Logic, Razvan Diaconescu, Japan Advanced Institute for Science and Technology, 1999
[13] Comprehending Queries, PhD Thesis, 1999, Torsten Grust, University of Konstanz, Germany.
[14] The Information Flow Framework http://suo.ieee.org/IFF/ Last accessed 4-7-2004