Patrick Browne, Mike Jackson Towards a Unified Spatial-Temporal Data Model and Query Language for Geographical Informa

1. Patrick Browne, Mike Jackson Towards a Unified Spatial-Temporal Data Model and Query Language for Geographical Information Systems

2. 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].

3. 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.

4. Modeling Diversity • Diversity is pervasive in topographicinformation (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.

5. 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.

6. 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.

7. TDM: the proposed model

8. 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].

9. The Approach • Existing modelswill be analysed, producing a set of consistent algebras. • Current modelswill 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.

10. 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.

11. Techniques • Category theory(CT): Though central to research CT is very abstract and needs to be augmented with additional formalisms and tools.

12. 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

13. 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.

14. 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.

15. 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.

16. Merging Algebras with Colimits Colimit operation

17. Merging Algebras with Colimits

18. Worboys ST-Complex[4]

19. 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

20. Worboys ST-Complex[4]

21. Worboys ST-Complex[4]

22. The Euclidean plane in CaféOBJ[12] obj R2 is pr 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

23. Topographic Data Model (TDM)

24. 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.

25. 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.

26. 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

27. 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