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
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Contents
Brief survey of qualitative spatial/spatio-temporal
representations and reasoning
Motivation
Some qualitative spatial representations
Spatial Change
Challenges
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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
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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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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
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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
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What is QR? (2)
 relevant distinctions only
e.g. empty/full ...

0
+
 Ambiguity
*
+
0
-
+
+
0
-
0
0
0
0
0
+
+
+
0
-
+
+
+
?
0
+
0
-
?
-
Not a replacement for Quantitative reasoning
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Tony Cohn, The University of Leeds 2007
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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)
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Tony Cohn, The University of Leeds 2007
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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
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“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''
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Tony Cohn, The University of Leeds 2007
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“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.
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Tony Cohn, The University of Leeds 2007
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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
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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
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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
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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
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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
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
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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
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Defining relations using C(x,y) (3)
EC(x,y) df C(x,y) ¬O(x,y)
x and y externally connect
TPP(x,y) df PP(x,y) 9z[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
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RCC-8
8
provably jointly exhaustive pairwise disjoint
relations (JEPD)
DC
EC
PO
TPP NTPP
EQ TPPi NTPPi
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Tony Cohn, The University of Leeds 2007
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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
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An alternative basis: 9-intersection model (9IM)

boundary(y)
interior(y) exterior(y)
boundary(x)
¬

¬
interior(x)



exterior(x)
¬

¬
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)
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Tony Cohn, The University of Leeds 2007
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“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
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The 17 different L/A relations of the DEM
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Tony Cohn, The University of Leeds 2007
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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
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Baarle-Nassau/Baarle-Hertog
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Tony Cohn, The University of Leeds 2007
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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
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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
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with point based orientation calculi.
Tony Cohn, The University of Leeds 2007
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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
l1 p lies to right of li
l2
+--
---
+++-+
l3
--+
+++
-++
Note: 19 positions (7 named) -- 8 not possible
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Tony Cohn, The University of Leeds 2007
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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
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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
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Tony Cohn, The University of Leeds 2007
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boundary representations
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
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Tony Cohn, The University of Leeds 2007
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>
Using Convex Hull to describe shape
conv(x) + C(x,y)
 topological inside
geometrical inside
“scattered inside”
“containable inside”
...
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Tony Cohn, The University of Leeds 2007
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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
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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 NESC-3-07
more tractable
33.0
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Tony Cohn, The University of Leeds 2007
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
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Tony Cohn, The University of Leeds 2007
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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)
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Tony Cohn, The University of Leeds 2007
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Spatiotemporal modal logic (contd)
ST1: allow O to apply to region variables
(iteratively)
Eg ¤+P(O EU,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)
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Tony Cohn, The University of Leeds 2007
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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
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Reasoning by Relation Composition
R1(a,b), R2(b,c)
R3(a,c)?
 In
general R3 is a disjunction
 Ambiguity
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Tony Cohn, The University of Leeds 2007
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Composition tables are quite sparse
DC
EC
DC
?
EC
DR,PO,
PP
DR,PO,
PPi
DR,PO,
PPi
TPP
DC
DR,PO,
TPP,TPi
DR,
PO,
PPi
DR
NTPP
DC
DC
PO
TPPi
DR,PO,
PPi
NTPPi DR,PO,
PPi
EQ
DC
PO
TPP
NTPP TPPi
DR,PO, DR,PO, DR,
DC
PP
PP
PO,
PP
DR,PO, EC,PO, PO,
DR
PP
PP
PP
?
PO,PP PO,
DR,
PP
PO,
PPi
DR,PO, PP
NTPP DR,PO,
PP
TPP,TPi
NTPPi EQ
DC
DC
DC
EC,PO,
PPi
PO,PPi
DR,
PO,
PPi
DR,
PO,
PPi
DR,PO, NTPP
NTPP DR,PO, ?
PP
PP
PO,PPi PO,TPP PO,
PPi
NTPPi
,TPi
PP
PO,PPi PO,PPi O
NTPPi
NTPPi
EC
PO
TPP
•cf13:35
poverty conjecture
Tony Cohn, The University of Leeds 2007
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NTPP TPPi
DC
PO
TPP
NTPP
TPPi
NTPPi
NTPPi EQ
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
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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,…)
-
+
0
Equality change law
transitions from time point instantaneous
transitions to time point non instantaneous
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Tony Cohn, The University of Leeds 2007
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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
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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
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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
13:35 be used as basis of a qualitative simulation algorithm
Tony Cohn, The University of Leeds 2007
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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
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Continuity of Multiple Component Histories
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Tony Cohn, The University of Leeds 2007
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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
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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)
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Tony Cohn, The University of Leeds 2007
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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
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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
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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
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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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Tony Cohn, The University of Leeds 2007
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