1. Provably Correct and Complete Transaction Rules for GIS
Gerhard Gröger
Lutz Plümer
University of Vechta, Germany Institute for Environmental Sciences

2. Maps - a Vector Data Model for GIS
Maps - a formal, graph theoretical concept for GIS
A Map M(V,E,F) is a plane (non-inter-secting) Graph G(V,E) with a set F of Faces as Atomic Areal Objects and an explicit representation of face-edge incidences
Additional Assumptions:
Not separable
Vertex degree  2
No. of faces  2
Straight edges
Faces are bounded by simple cycles of edges
Maps should not be confused with Cartographic Maps
Example: I II
OUT A B C D
Faces:
I: ABC
II: ACD
OUT: ADCB

3. Updates of Maps
Problem: How to preserve the map properties when an update occurs? Þ Integrity Checking using Transaction Rules
Design Goals for Transaction Rules:
Correctness: Consistency of maps is preserved
Efficiency: The number of objects to be checked should be minimal
Completeness: You can do anything you need for your application
Convenience: An user’s request which is specified only partially should be completed consistently
Existing GIS: Integrity checking is hidden in code, not formalized and not transparent
Database Theory: Methods for deriving Integrity Constraints auto- matically (Simplification Method, Nicolas 1982 et al.) are not efficient for maps and do not consistently complete an incomplete request
We have formally specified Transaction Rules which are provably Correct and Complete, Efficient and Convenient

4. Transaction Rules: Applications
Applications Changing Topology (e.g. Digitizing)
TR1: Initial Transaction: Adding a face to the empty plane
TR2: Final Transaction: Deletion of a face yielding the empty plane
TR3: Insertion of a polyline splitting an existing face
TR4: Deletion of a polyline merging two existing faces
Applications Preserving Topology (e.g. Cartographic Generalization, Homogenization)
TR5: Changing geometry of a set of vertices preserving topology
Special cases:
Changing geometry of a single vertex
Changing geometry of the internal vertices of a polyline

5. Deletion of a Polyline: Error Casev1 l vn f1 f4 f3 w1 w2
f2 = OUT
After deletion of the internal vertices of polyline l = (v1,...,vn) the window of face OUT is not a simple cycle

6. Transaction Rule 4: Deletion of a Polyline
Deletion of (the interior vertices of) a polyline l = v1,…vn merging its two incident faces f1 and f2
Condition:
1. degree(v1) > 2 and degree(vn) > 2
2. l is a polyline incident to the faces f1 and f2
3. Apart from v1,…,vn, no vertex incident to f1 or f2 is incident to both f1 and f2
Action:
Delete the internal vertices and all edges of l
Replace all occurrences of f1 or f2 by f, where f is a new face

7. Correctness and Completeness of TR1, ..., TR4
Theorem (Correctness): The Transaction Rules TR3 and TR4 preserve the consistency of maps
Proof: Jordan’s CurveTheorem is used to justify the restrictions
Theorem (Completeness): Any map can be generated / deleted by a finite number of applications of TR1, ...,TR4
Proof: Crucial point: How to select a polyline the deletion of which preserves simple cycles?
Jordan’s Curve Theorem: Any simple closed curve partitions the plane into two disjoint connected components, one of which is bounded and one not bounded

8. Changing Geometry preserving Topology: Vertices, Edges and Faces relevant for checking
Relevant modified vertex
Relevant modified edge
Relevant nonmodified vertex
Relevant nonmodified edge
Irrelevant vertex
Irrelevant edge
Relevant face/area
Example: Change of geometry of vertices v1 and v2

9. Transaction Rule 5: Changing Geometry Preserving Topology
Change of geometry of a set VC = {v1,…,vn } of vertices preserving topology
Condition:
1. The coordinates of all relevant modified vertices are pair-wise distinct
2. The coordinates of all relevant modified vertices are distinct to all relevant nonmodified vertices
3. The relevant modified edges are pair-wise nonintersecting
4. No relevant modified edge intersects a relevant nonmodified edge
5. Each relevant modified vertex lies in the relevant area
6. No relevant edge lies in the interior of a relevant face
Action:
Change the geometry of each vi Î VC accordingly

10. Changing Geometry Preserving Topology: Error Cases and Detecting Conditions
OUT v3 v3 v3 v1 v3 v1 v1 v2 v2 v2 v2
a) Original map
b) Intersection detected by Condition 4 of TR 5
c) Vertex v1 lies not in the rele-vant area (Con-dition 5 of TR 5)
d) Interior of relevant face (OUT) contains a relevant edge (Condition 6 of TR 5)
Legend:
Relevant area
Relevant nonmodified vertex
Relevant nonmodified edge
Relevant modified vertex
Relevant modified edge

11. Conclusion
A set of transaction rules for maps has been presented which is
specified formally
provably correct (consistency is preserved) and
provably complete with respect to important application scenarios (Digitizing, Cartographic Generalization, Homogenization)
Methods for automatically deriving such rules (Nicolas et al.) are not suitable for GIS, since general geometrical/topological knowledge (e.g. Jordan’s Curve Theorem) must be incorporated
Implementation is immediate in active databases
Uncertainty of spatial data: Confidence Intervals are handled using buffer techniques