Transcript PowerPoint Presentation - Towards a Naive Geography

Towards a Naive Geography
Pat Hayes & Geoff Laforte
IHMC
University of West Florida
Ontology
“All the things you are…”
Upper-level ontology standardization effort now under way.
Top levels form a lattice (more or less) based on about a
dozen (more or less) orthogonal distinctions:
(abstract/concrete) (dependent/independent)
(individual/plurality) (essential/non-essential)
(universal/particular) (occurrent/continuant)…
Most of these don’t have anything particularly to do with
geography, but they seem to apply to geography as much as to
everything else.
Ontology
Some particularly ‘geographical’ concepts
Continuant
Occurrent
physical entity with space-like parts
physical entity with time-like parts
(Can some things be both?)
Location piece of physical space
Terrain piece of geographical space (consisting of
locations suitably related to each other.)
History
spatio-temporal region (the ‘envelope’ of a
continuant or occurrent.)
Ontology
Many tricky ontological issues don’t seem to arise in
geographical reasoning.
What happens to the hole in a bagel when you take the bagel into a railway
tunnel? Is a carpet in the room or part of the room? (What about the paint?)
Is doing nothing a kind of action? Is a flame an object or a process?
On the other hand, maybe they do...
Ontology
some personal opinions
Some issues are basically tamed
Holes, surfaces, boundaries; Dimension; Qualitative spatiotemporal
reasoning.
Some others aren’t
Blurred things, indistinctness; tolerances and granularity. (heap paradox...been
around for a while.)
Distributive properties: textures, roughness, etc.
Geographical Inference
Should apply to maps, sentences and databases.
Valid = truth-preserving
Interpretation = a way the world could be, if the representation
is true of it
Semantics a la Tarski , a brief primer

Specify the syntax


Interpretation is defined recursively


Expressions have immediate ‘parts’
I(e) = M(t, I(e1),…,I(en) )
Structural agnosticism yields validity

Interpretation is assumed to have enough structure to define
truth…..but that’s all.
Simple maps have no syntax (worth a damn…)
= Oil well
= Town
Different tokens of same symbol mean different things
Indexical?? ( “This city…”)
Bound variable?? ( “The city which exists here…”)
Existential assertion? ( “A city exists here…”)
Different tokens of same symbol mean different things
Indexical?? ( “This city…”)
Bound variable?? ( “The city which exists here…”)
Existential assertion? ( “A city exists here…”)
Located symbol = location plus a predicate
The map location is part of the syntax
The map location is part of the syntax
I(e)=M(t, I(e1),…,I(en) ) …. where n = 1
The interpretation of a symbol of type t located at p is given by
M(t, I(p) ) = M(t)( I(p) )
M(triangle) = Oil-well
M(circle) = Town
The map location is part of the syntax
I(e)=M(t, I(e1),…,I(en) ) …. where n = 1
The interpretation of a symbol of type t located at p is given by
M(t, I(p) ) = M(t)( I(p) )
M(triangle) = Oil-well
M(circle) = Town
But what is I(p) ?
For that matter, what is p, exactly ?
What is I(p) ?
For that matter, what is p ?
Need a way to talk about spaces and locations
1. Geometry (not agnostic; rules out sketch-maps)
2. Topology (assumes continuity)
3. Axiomatic mereology
(more or less…)
What is I(p) ?
For that matter, what is p ?
Assume that space is defined by a set of locations (obeying
certain axioms)
… map and terrain are similar
… tread delicately when making assumptions
What is I(p) ?
For that matter, what is p ?
A location can be any place a symbol can indicate, or where a thing might
be found (or any piece of space defining a relation between other pieces
of space)
surface patches, lines, points, etc...
Different choices of location set will give different ‘geometries’ of the
space.
Note, do not want to restrict to ‘solid’ space (unlike most axiomatic
mereology in the literature.)
Sets of pixels on a finite screen
All open discs in R2 (or R3 or R4 or…)
All unions of open discs
The closed subsets of any topological space
The open subsets
The regular (= ‘solid’) subsets
All subsets
All finite sets of line segments in R2
All piecewise-linear polygons
… and many more …
Assume basic relation of ‘covering’ p<q
p<p
p<q & q<r implies p<r
p<q & q<p implies p=q
Every set S of locations has a unique minimal covering location
(p e S) implies p< ^S
((p e S) implies p<q) implies q< ^S
(Mereologists usually refuse to use set theory...but we have no mereological
sensibility :-)
Can define many useful operations and properties:
‘Everywhere’:
Overlap:
Sum:
Complement:
forall p (p<^L)
pOq =df exists r ( r<p and r<q )
p+q =df ^{p,q}
¬p =df ^{q: not pOq}
… but not (yet) all that we will need:
Boundary? Direction?
There is a basic tension between continuity and syntax
What are the subexpressions of a spatially extended symbol in a continuum?
Set of sub-locations is clear if it covers no location of a symbol; it is maximally
clear if any larger location isn’t clear.
Immediate subexpressions are minimal covers of maximally clear sets.
Sets of subexpressions of a finite map are well-founded (even in a
continuum.)
What is I(p) ?
Part of the meaning of an interpretation must be the projection function
from the terrain of the interpretation to the map:
What is I(p) ?
But interpretation mappings go
from the map to the interpretation:
…and they may not be invertible.
What is I(p) ?
covering inverse of function between location spaces:
/f(p) = ^{q : f(q) = p }
f
/f
I(p) =?= /projectionI(p)
What is I(p) ?
For locations of symbols, the covering inverse of the projection function
isn’t an adequate interpretation:
What is I(p) ?
For locations of symbols, the covering inverse of the projection function
isn’t an adequate interpretation.
I(p) is a location covered by the covering inverse:
I(p) < /projection(p)
Which is really just a fancy way of saying:
projection(I(p))=p
Some examples
London tube map
Terrain is ‘Gill space’: minimal sets of elongated rectangles joined at pivots
Projection takes rectangle to spine (and adds global fisheye distortion)
Some examples
Linear route map
Terrain is restriction of R2 to embedded road graph.
Projection takes non-branching segment to (numerical description of)
length and branch-point to (description of) direction.
Some examples
Choropleth Map
Terrain is restriction of underlying space to maximal regions
Projection preserves maximality.
(Actually, to be honest, it requires boundaries.)
Adjacency requires boundaries
Need extra structure to describe ‘touching’
(Asher : C)
We want boundaries to be locations as well…
bdp
‘b is part of the boundary of p’
Adjacency requires boundaries
bdp
Define full boundary of p to be ^{b : b d p }
Boundary-parts may have boundaries...
... but full boundaries don’t.
Adjacency is defined to be sharing a common boundary part:
pAq =df exist b (b d p and b d q )
Axioms for boundaries
( b d p & c<b ) implies c d p
( b d p & p<q ) implies ( b d p or b<q )
( -->adjacency analysis)
Homology axiom:
not ( c d ^{b : b d p } )
Boundaries define paths
Examples of boundary spaces
Pixel regions with linear boundaries joined at edge and corners
Pixel regions with interpixels
Subsets of a topological space with sets of limit points
Circular discs with circular arcs in R2
Piecewise linear regions with finite sets of line-segments and points in
R2
Need to consider edges between pixels as boundary locations.
Or, we can have both interpixels and lines as boundaries.
Maps and sentences
Since map surface and interpretation terrain are similar, axiomatic theory
applies to both.
Terrain spatial ontology applies to map surface, so axiomatic theory of
terrain is also a theory of map locations.
A theory which is complete for the relations used in a map is expressive
enough to translate map content, via
I(p) < /projectionI(p)
Maps and Sentences
n
Goal is to provide a coherent account of how geographical
information represented in maps can be translated into
logical sentences while preserving geographical validity.
n
Almost there... current work focussing on adjacency and
qualitative metric information.