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Tony Cohn, The University of Leeds 2007 NESC-3-07 0.0 13:35 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...

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Tony Cohn, The University of Leeds 2007 NESC-3-07 1.0 13:35 Contents Brief survey of qualitative spatial/spatio-temporal representations and reasoning Motivation Some qualitative spatial representations Spatial Change Challenges

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Tony Cohn, The University of Leeds 2007 NESC-3-07 2.0 13:35 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…

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Tony Cohn, The University of Leeds 2007 NESC-3-07 3.0 13:35 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 4.0 13:35 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 5.0 13:35 What is QR? (1) QR (about physical systems) symbolic, not analogical continuous scalar quantities mapped to finite discrete space (qualitative quantity space) e.g..., 0, + model situation by relationships between these quantities relative size; arithmetical relationships,... de Kleer, Kuipers, Forbus,…

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Tony Cohn, The University of Leeds 2007 NESC-3-07 6.0 13:35 Not a replacement for Quantitative reasoning What is QR? (2) relevant distinctions only e.g. empty/full... - 0 + Ambiguity

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Tony Cohn, The University of Leeds 2007 NESC-3-07 7.0 13:35 Develop QR representations specifically for space Richness of QSR derives from multi-dimensionality Consider trying to apply temporal interval calculus in 2D: What is QSR? (1) = < m o d f s Can work well for particular domains -- e.g. envelope/address recognition (Walischewski 97)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 8.0 13:35 What is QSR? (2) Many aspects: ontology, topology, orientation, distance, shape... spatial change Vagueness and uncertainty reasoning mechanisms pure space v. domain dependent

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Tony Cohn, The University of Leeds 2007 NESC-3-07 9.0 13:35 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 dont work in more than one dimension, leaving little hope for concluding much about combining weak information about spatial properties''

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Tony Cohn, The University of Leeds 2007 NESC-3-07 10.0 13:35 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. 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.

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Tony Cohn, The University of Leeds 2007 NESC-3-07 11.0 13:35 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...

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Tony Cohn, The University of Leeds 2007 NESC-3-07 12.0 13:35 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 13.0 13:35 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)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 14.0 13:35 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)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 15.0 13:35 Defining relations using C( x,y ) (1) DC (x,y) df ¬C( x,y) x and y are disconnected P (x,y) df z [ C( x,z) C( y,z)] x is a part of y x PP (x,y) df P (x,y) ¬ P (y,x) x is a proper part of y EQ (x,y) df P (x,y) P (y,x) x and y are equal alternatively, an axiom if equality built in

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Tony Cohn, The University of Leeds 2007 NESC-3-07 16.0 13:35 Defining relations using C( x,y ) (2) O (x,y) df 9z[P(z,x ) P (z,y)] x and y overlap DR (x,y) df ¬ O (x,y) x and y are discrete PO (x,y) df O (x,y) ¬ P (x,y) ¬ P (y,x) x and y partially overlap

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Tony Cohn, The University of Leeds 2007 NESC-3-07 17.0 13:35 Defining relations using C( x,y ) (3) EC (x,y) df C (x,y) ¬ O (x,y) x and y externally connect zzx TPP (x,y) df PP (x,y) 9 z[ EC (z,y) EC (z,x)] x is a tangential proper part of y NTPP (x,y) df PP (x,y) ¬ TPP (x,y) x is a non tangential proper part of y

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Tony Cohn, The University of Leeds 2007 NESC-3-07 18.0 13:35 RCC-8 DC EC PO TPP NTPP EQ TPPi NTPPi u 8 provably jointly exhaustive pairwise disjoint relations (JEPD)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 19.0 13:35 C(x,y) is very expressive Can also define: Holes, dimension, one pieceness Topological functions Boolean functions (sum, complement, intersection) …

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Tony Cohn, The University of Leeds 2007 NESC-3-07 20.0 13:35 An alternative basis: 9-intersection model (9IM) 2 9 = 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)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 21.0 13:35 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)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 22.0 13:35 The 17 different L/A relations of the DEM

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Tony Cohn, The University of Leeds 2007 NESC-3-07 23.0 13:35 Mereology and Topology Which is primal? (Varzi 96) Mereology is insufficient by itself cant 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 24.0 13:35 Baarle-Nassau/Baarle-Hertog

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Tony Cohn, The University of Leeds 2007 NESC-3-07 25.0 13:35 Between Topology and Metric 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.

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Tony Cohn, The University of Leeds 2007 NESC-3-07 26.0 13:35 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.

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Tony Cohn, The University of Leeds 2007 NESC-3-07 27.0 13:35 Qualitative Positions wrt oriented lines pos(p,l i ) = + iff p lies to left of l i pos(p,l i ) = 0 iff p lies on l i pos(p,l i ) = - iff p lies to right of l i l1l1 l2l2 l3l3 +-- ++- +++ -++ --+ --- +-+ Note: 19 positions (7 named) -- 8 not possible

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Tony Cohn, The University of Leeds 2007 NESC-3-07 28.0 13:35 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.

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Tony Cohn, The University of Leeds 2007 NESC-3-07 29.0 13:35 boundary representations axial representations shape abstractions synthetic: set of primitive shapes Boolean algebra to generate complex shapes Qualitative Shape Descriptions Challenge: finding useful qualitative shape calculi

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Tony Cohn, The University of Leeds 2007 NESC-3-07 30.0 13:35 Hoffman & Richards (82): label boundary segments: curving out curving in straight angle outward > angle inward < cusp outward Â cusp inward Á Meathrel & Galton (2001) provide a hierarchical, unbounded representation calculus Generalises all previous approaches boundary representations > > > < > | >

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Tony Cohn, The University of Leeds 2007 NESC-3-07 31.0 13:35 conv (x) + C (x,y) topological inside geometrical inside scattered inside containable inside... Using Convex Hull to describe shape

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Tony Cohn, The University of Leeds 2007 NESC-3-07 32.0 13:35 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)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 33.0 13:35 Mereogeometries Region Based Geometry (RBG) 2 nd order axiomatisation P(x,y) + Sphere(x) Categorical (Region based version of Tarskis 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 34.0 13:35 Qualitative Spatio-temporal representations Many temporal calculi Temporal modal logics, Allens 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 35.0 13:35 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 ¦ + ST 0 : 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)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 36.0 13:35 Spatiotemporal modal logic (contd) ST 1 : allow O to apply to region variables (iteratively) Eg ð ¤ + P( O EU,EU) The EU will never contract satisfiability decidable and NP complete ST 2 : 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)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 37.0 13:35 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 P: P is true somewhere else (von Wright 79) Some spatio-temporal logics (See below)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 38.0 13:35 Reasoning by Relation Composition R1(a,b), R2(b,c) R3(a,c)? u In general R3 is a disjunction u Ambiguity

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Tony Cohn, The University of Leeds 2007 NESC-3-07 39.0 13:35 Composition tables are quite sparse cf poverty conjecture

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Tony Cohn, The University of Leeds 2007 NESC-3-07 40.0 13:35 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 41.0 13:35 Spatial Change Challenge: Want to be able to reason over time about spatial entities 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 +

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Tony Cohn, The University of Leeds 2007 NESC-3-07 42.0 13:35 Kinds of spatial change (1) Topological changes in single spatial entity: change in dimension usually by abstraction/granularity shift e.g. road: 1D 2D 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 43.0 13:35 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

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Tony Cohn, The University of Leeds 2007 NESC-3-07 44.0 13:35 Continuity Networks/ Conceptual Neighbourhoods 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 What are next qualitative relations if entities transform/translate continuously? E.g. RCC-8

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Tony Cohn, The University of Leeds 2007 NESC-3-07 45.0 13:35 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?

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Tony Cohn, The University of Leeds 2007 NESC-3-07 46.0 13:35 Continuity of Multiple Component Histories

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Tony Cohn, The University of Leeds 2007 NESC-3-07 47.0 13:35 Conceptual Neighbourhoods for other calculi Virtually every calculus with a set of JEPD relations has presented a CN. E.g.

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Tony Cohn, The University of Leeds 2007 NESC-3-07 48.0 13:35 Vagueness Ubiquitous in geographic phenomena Hills, valleys, forests, rivers, lakes … Even man made artifacts (walls, roofs,…) Cant 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) Challenge: representing vagueness in a useful way (we can still make inferences)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 49.0 13:35 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)

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Tony Cohn, The University of Leeds 2007 NESC-3-07 50.0 13:35 Indeterminate boundaries/vague regions: egg-yolk calculus Using RCC8: 601 jointly exhaustive, pairwise disjoint relations 40 natural clusters Can specify that hill and valley are vague regions which touch, without specifying the boundary Can also be used to represent locational uncertainty as well as boundary indeterminacy...

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Tony Cohn, The University of Leeds 2007 NESC-3-07 51.0 13:35 Recap Surprisingly rich languages for qualitative spatial representation symbolic representations Topology, orientation, distance,... hundreds of distinctions easily made Static reasoning: composition, constraints, 0-order logic Dynamic reasoning: continuity networks/conceptual neighbourhood diagrams

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Tony Cohn, The University of Leeds 2007 NESC-3-07 52.0 13:35 Discussion Topics Modelling Choice of language/representation(s) Granularity of representation How best to handle vagueness and uncertainty Space and time Integration of representations (incl. semantics) Do we need new/more languages? Inference and Computation What kinds of tasks? (prediction, simulation, consistency checking, ontology integration, change of granularity/abstraction…) Integration with quantitative representations/computation Bridging the research/application gap

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