Qualitative representations of the geospatial world

1. Qualitative representations of the geospatial world Tony Cohn School of Computing The University of Leeds a.g.cohn@leeds.ac.uk http://www.comp.leeds.ac.uk/ Particular thanks to: EPSRC, EU, Leeds QSR group and ...

2. Contents • Brief survey of qualitative spatial/spatio-temporal representations and reasoning • Motivation • Some qualitative spatial representations • Spatial Change • Challenges

3. The geospatial world • Huge amounts of metric and symbolic data • Very diverse ontologically • Natural and man made objects • Processes at many different time scales • Many different kinds of objects • Different spatial scales • Different representations, languages, standards,… • Abstraction, analysis, mining, comparison, querying, integration…

4. Qualitative spatial/spatio-temporal representations • Naturally provides abstraction • Well developed calculi, languages, (often) semantics • Complementary to metric representations • Provide foundation for geospatial ontologies and reasoning

5. Some Challenges (Summary) • Vagueness and uncertainty • Space and time • Efficiency/expressiveness • Combining calculi for different spatial aspects • Choosing/designing appropriate representations and ontologies, at the appropriate level of granularity, and moving between these • Integrating ontologies • Combining qualitative and quantative representations • Interfacing with the human user; “cognitive semantics” • Modelling is hard

6. What is QR? (1) • QR (about physical systems) • symbolic, not analogical • continuous scalar quantities mapped to finite discrete space (qualitative quantity space) • e.g... á-, 0, +ñ • model situation by relationships between these quantities • relative size; arithmetical relationships, ... • de Kleer, Kuipers, Forbus,…

7. What is QR? (2) • relevant distinctions only • e.g. empty/full ... • - 0 + • Ambiguity • Not a replacement for Quantitative reasoning

8. What is QSR? (1) • Develop QR representations specifically for space • Richness of QSR derives from multi-dimensionality • Consider trying to apply temporal interval calculus in 2D: < = m o s d f • Can work well for particular domains -- e.g. envelope/address recognition (Walischewski 97)

9. What is QSR? (2) • Many aspects: • ontology, topology, orientation, distance, shape... • spatial change • Vagueness and uncertainty • reasoning mechanisms • pure space v. domain dependent

10. “Poverty Conjecture” (Forbus et al, 86) • “There is no purely qualitative, general purpose kinematics” • Of course QSR is more than just kinematics, but... • 3rd (and strongest) argument for the conjecture: • “No total order: Quantity spaces don’t work in more than one dimension, leaving little hope for concluding much about combining weak information about spatial properties''

11. “Poverty Conjecture” (2) • transitivity: key feature of qualitative quantity space • can this be exploited much in higher dimensions ?? • “we suspect the space of representations in higher dimensions is sparse; that for spatial reasoning almost nothing weaker than numbers will do”. • Challenge: to provide calculi which allow a machine to represent and reason qualitatively with spatial entities of higher dimension, without resorting to the traditional quantitative techniques.

12. Why QSR? • Traditional QR spatially very inexpressive • Potential applications of QSR in: • Natural Language Understanding • GIS/GIScience • Visual Languages • Biological systems • Robotics • Multi Modal interfaces • Event recognition from video input • Spatial analogies • ...

13. Ontology of Space • extended entities (regions)? • points, lines, boundaries? • mixed dimension entities? • Open/closed/regular/non regular regions? • Multi-piece (disconnected)? Interior connected? • What is the embedding space? • connected? discrete? dense? dimension? Euclidean?... • What entities and relations do we take as primitive, and what are defined from these primitives? Challenge 2: the diversity of spatial ontology

14. Mereology • Theory of parthood (Simons 87) • In fact, many theories • What principles should hold? • E.g. Weak supplementation principle: If x is a proper part of y, then there should be some other proper part z of y not identical with x. (not all mereologies obey this principle)

15. Mereotopology • Combining mereology and topological notions • Usually built from a primitive binary conection relation, C(x,y) • Reflexive and symmetric • Several different interpretations in the literature • Can define many relations from C(x,y)

16. Defining relations using C(x,y) (1) • DC(x,y) ºdf¬C(x,y) x and y are disconnected • P(x,y) ºdf"z [C(x,z) ®C(y,z)] x is a part of y • PP(x,y) ºdfP(x,y) Ù¬P(y,x) x is a proper part of y • EQ(x,y) ºdfP(x,y) ÙP(y,x) x and y are equal • alternatively, an axiom if equality built in

17. Defining relations using C(x,y) (2) • O(x,y) ºdf9z[P(z,x) ÙP(z,y)] • x and y overlap • DR(x,y) ºdf¬O(x,y) • x and y are discrete • PO(x,y) ºdfO(x,y) Ù¬P(x,y) Ù ¬P(y,x) • x and y partially overlap

18. Defining relations using C(x,y) (3) • EC(x,y) ºdfC(x,y) Ù¬O(x,y) • x and y externally connect • TPP(x,y) ºdfPP(x,y) Ù 9z[EC(z,y) ÙEC(z,x)] • x is a tangential proper part of y • NTPP(x,y) ºdfPP(x,y) Ù ¬TPP(x,y) • x is a non tangential proper part of y

19. RCC-8 • 8 provably jointly exhaustive pairwise disjoint relations (JEPD) DC EC PO TPP NTPP EQ TPPi NTPPi

20. C(x,y) is very expressive • Can also define: • Holes, dimension, one pieceness • Topological functions • Boolean functions (sum, complement, intersection) • …

21. An alternative basis: 9-intersection model (9IM) • 29 = 512 combinations • 8 relations assuming planar regular point sets • potentially more expressive • considers relationship between region and embedding space • Variant models discrete space (16 relations) (Egenhofer & Sharma, 93)

22. “Dimension extended” method (DEM) • In the case where array entry is ‘¬’, replace with dimension of intersection: 0,1,2 • 256 combinations for 4-intersection • Consider 0,1,2 dimensional spatial entities • 52 realisable possibilities (ignoring converses) • (Clementini et al 93, Clementini & di Felice 95)

23. The 17 different L/A relations of the DEM

24. Mereology and Topology • Which is primal? (Varzi 96) • Mereology is insufficient by itself • can’t define connection or 1-pieceness from parthood 1. generalise mereology by adding topological primitive 2. topology is primal and mereology is sub theory 3. topology is specialised domain specific sub theory Challenge: choosing primitives and inter-relating primitives in different theories

25.  Baarle-Nassau/Baarle-Hertog

26. Between Topology andMetric representations • What QSR calculi are there “in the middle”? • Orientation, convexity, shape abstractions… • Some early calculi integrated these • we will separate out components as far as possible • Some example calculi in next few slides • Mostly defined using algebraic techniques rather than logics, or only semi-formally. Challenge: finding expressive but efficient “semi-metric” calculi.

27. Orientation • Naturally qualitative: clockwise/anticlockwise orientation • Need reference frame • deictic: x is to the left of y (viewed from observer) • intrinsic: x is in front of y • (depends on objects having fronts) • absolute: x is to the north of y • Most work 2D • Most work considers orientation between points or wrt directed lines Challenge: combining region based mereotopology with point based orientation calculi.

28. Qualitative Positions wrt oriented lines • pos(p,li) = + iff p lies to left of li • pos(p,li) = 0 iff p lies on li • pos(p,li) = - iff p lies to right of li l1 l2 +-- --- ++- +-+ l3 --+ +++ -++ Note: 19 positions (7 named) -- 8 not possible

29. Star Calculus (Renz and Ligozat) If more than 2 intersecting lines used for defining sectors, then easy to define a coordinate system and thus a geometry.

30. Qualitative Shape Descriptions • boundary representations • axial representations • shape abstractions • synthetic: set of primitive shapes • Boolean algebra to generate complex shapes Challenge: finding useful qualitative shape calculi

31. boundary representations | É • Hoffman & Richards (82): label boundary segments: • curving out É • curving in Ì • straight | • angle outward > • angle inward < • cusp outward  • cusp inward Á • Meathrel & Galton (2001) provide a hierarchical, unbounded representation calculus • Generalises all previous approaches > > • Ì • Ì < > | É • Ì > >

32. Using Convex Hull to describe shape • conv(x) + C(x,y) • topological inside • geometrical inside • “scattered inside” • “containable inside” • ...

33. Expressiveness of conv(x) • Constraint language of EC(x) + PP(x) + Conv(x) • can distinguish any two bounded regular regions not related by an affine transformation • Davis et al (97) • intractable (at least as hard as determining whether set of algebraic constraints over reals is consistent • Davis et al (97)

34. Mereogeometries • Region Based Geometry (RBG) • 2nd order axiomatisation • P(x,y) + Sphere(x) • Categorical • (Region based version of Tarski’s geometry) • Borgo and Masolo (06) • Analysis of several other systems (eg de Laguna) • Four shown to be strongly semantically equivalent • Some work on on constraint systems • Less expressive but more tractable

35. Qualitative Spatio-temporal representations • Many temporal calculi • Temporal modal logics, Allen’s calculus… • How to combine? • Ontology of space-time (3+1D v. 4D) • Computational issues • Capturing interactions between time and space • continuity Challenge: finding useful qualitative spatio-temporal calculi

36. Decidable Spatiotemporal modal logics(Wolter & Zakharyashev) • Combine point based temporal logic with RCC8 • temporal operators: Since, Until • can define: Next (O), Always in the future ¤+, Sometime in the future ¦+ • ST0: allow temporal operators on spatial formulae • satisfiability is PSPACE complete • Eg ¬ ¤+P(Kosovo,Yugoslavia) • Kosovo will not always be part of Yugoslavia • can express continuity of change (conceptual neighbourhood) • Can add Boolean operators to region terms • E.g. EQ(UK,GB+N.Ireland)

37. Spatiotemporal modal logic (contd) • ST1: allow O to apply to region variables (iteratively) • Eg ð ¤+P(OEU,EU) • The EU will never contract • satisfiability decidable and NP complete • ST2: allow the other temporal operators to apply to region variables (iteratively) • finite change/state assumption • satisfiability decidable in EXPSPACE • P(Russia, ¦+ EU) • all points in Russia will be part of EU (but not necessarily at the same time)

38. Metatheoretic results: decidability • Topology not decidable (Grzegorczyk, 51): • Boolean algebra is decidable • add: closure operation or EC results in undecidability • can encode arbitrary statements of arithmetic • Decidable subsystems? • Constraint language of “RCC8” (Bennett 94) • Modal/intuitionistic encoding • Other decidable languages? • Constraint language of RCC8 + Conv(x) (Davis et al, 97) • Modal logics of place • àP: “P is true somewhere else” (von Wright 79) • Some spatio-temporal logics • (See below)

39. Reasoning by Relation Composition • R1(a,b), R2(b,c) • R3(a,c)? • In general R3 is a disjunction • Ambiguity

40. Composition tables are quite sparse • cf poverty conjecture

41. Composition Tables and Constraints • Reasoning using composition tables is a constraint based approach to reasoning • Finite set of JEPD relations (e.g. RCC-8) • Composition table gives constraints amongst these relations • Given a set of ground, possibly disjunctive facts • For each triple of objects, check if constraints are satisfied • If all combinations of triples are consistent wrt the composition table, then path consistent

42. Spatial Change • Challenge: Want to be able to reason over time about spatial entities • continuous deformation, motion • c.f.. traditional Qualitative simulation (e.g. QSIM: Kuipers, QPE: Forbus,…) • Equality change law • transitions from time point instantaneous • transitions to time point non instantaneous + - 0

43. Kinds of spatial change (1) • Topological changes in ‘single’ spatial entity: • change in dimension • usually by abstraction/granularity shift • e.g. road: 1D Þ 2D Þ 3D • change in number of topological components • e.g. breaking a cup, fusing blobs of mercury • change in number of tunnels • e.g. drilling through a block of wood • change in number of interior cavities • e.g. putting lid on container

44. Kinds of spatial change (2) • Topological changes between spatial entities: • e.g. change of RCC/4IM/9IM/… relation change in position, size, shape, orientation, granularity • may cause topological change

45. Continuity Networks/Conceptual Neighbourhoods • What are next qualitative relations if entities transform/translate continuously? • E.g. RCC-8 • If uncertain about the relation what are the next most likely possibilities? • Uncertainty of precise relation will result in connected subgraph (Freksa 91) • Can be used as basis of a qualitative simulation algorithm

46. What exactly is qualitative continuity? • No spatial leaps • No pinching • No temporal gaps • Can we formally prove the non existence of the missing links in the conceptual neighbourhood from a formal definition of qualitative continuity?

47. Continuity of Multiple Component Histories

48. Conceptual Neighbourhoods for other calculi • Virtually every calculus with a set of JEPD relations has presented a CN. • E.g.

49. Vagueness • Ubiquitous in geographic phenomena • Hills, valleys, forests, rivers, lakes … • Even man made artifacts (walls, roofs,…) • Can’t avoid, must develop techniques to handle • Eg: • The tree is near the summit of the mountain. • The mountain is far from the sea. • ² The tree is not near to the sea. • Challenge: representing vagueness in a useful way (we can still make inferences)

50. Modal Supervaluation Logic • We can define modal operators which take account of how the truth of propositions may vary according to different senses of the concepts that it contains. • U — is unequivocally true. • S — is true in some sense. • U¬(Near(x,y) Æ Far(x,y)) • (Pond(x) ! S(Lake(x))) • Applications in ontology • e.g. current geo-ontology projects • Reified approach with key parameters (e.g. width, depth, flow for river/lake)