Towards a Naive Geography

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Towards a Naive Geography

• Pat Hayes Geoff Laforte
• IHMC
• University of West Florida

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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 dont have
anything particularly to do with geography, but
they seem to apply to geography as much as to
everything else.
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Ontology
Some particularly geographical
concepts Continuant physical entity with
space-like parts Occurrent 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.)
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Ontology
Many tricky ontological issues dont 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...
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Ontology
some personal opinions
Some issues are basically tamed Holes, surfaces,
boundaries Dimension Qualitative spatiotemporal
reasoning. Some others arent Blurred things,
indistinctness tolerances and granularity. (heap
paradox...been around for a while.) Distributive
properties textures, roughness, etc.
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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

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Semantics a la Tarski , a brief primer

• Specify the syntax
• Expressions have immediate parts
• Interpretation is defined recursively
• I(e) M(t, I(e1),,I(en) )
• Structural agnosticism yields validity
• Interpretation is assumed to have enough
  structure to define truth..but thats all.

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Simple maps have no syntax (worth a damn)
Oil well Town
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Different tokens of same symbol mean different
things
Indexical?? ( This city) Bound variable?? (
The city which exists here) Existential
assertion? ( A city exists here)
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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
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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
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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 ?
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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)
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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
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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.)
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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
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Assume basic relation of covering pltq pltp
pltq qltr implies pltr pltq qltp implies
pq Every set S of locations has a unique minimal
covering location (p e S) implies plt
S ((p e S) implies pltq) implies qlt
S (Mereologists usually refuse to use set
theory...but we have no mereological sensibility
-)
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Can define many useful operations and
properties Everywhere forall p (pltL)
Overlap pOq df exists r ( rltp and rltq
) Sum pq df p,q Complement p df q
not pOq but not (yet) all that we will
need Boundary? Direction?
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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 isnt clear. Immediate subexpressions
are minimal covers of maximally clear sets. Sets
of subexpressions of a finite map are
well-founded (even in a continuum.)
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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

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What is I(p) ?

• But interpretation mappings go
• from the map to the interpretation
• and they may not be invertible.

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What is I(p) ?

• covering inverse of function between location
  spaces
• /f(p) q f(q) p

f
/f
I(p) ? /projectionI(p)
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What is I(p) ?

• For locations of symbols, the covering inverse of
  the projection function isnt an adequate
  interpretation

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What is I(p) ?

• For locations of symbols, the covering inverse of
  the projection function isnt an adequate
  interpretation.
• I(p) is a location covered by the covering
  inverse
• I(p) lt /projection(p)
• Which is really just a fancy way of saying
• projection(I(p))p

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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)
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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.
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Some examples
Choropleth Map Terrain is restriction of
underlying space to maximal regions Projection
preserves maximality. (Actually, to be honest, it
requires boundaries.)
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Adjacency requires boundaries
Need extra structure to describe
touching (Asher C) We want boundaries to be
locations as well b d p b is part
of the boundary of p
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Adjacency requires boundaries
b d p Define full boundary of p to be b b d
p Boundary-parts may have boundaries... ...
but full boundaries dont. Adjacency is defined
to be sharing a common boundary part pAq df
exist b (b d p and b d q )
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Axioms for boundaries
( b d p cltb ) implies c d p ( b d p pltq )
implies ( b d p or bltq ) ( --gtadjacency
analysis) Homology axiom not ( c d b b d
p )
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Boundaries define paths
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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
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Need to consider edges between pixels as boundary
locations. Or, we can have both interpixels and
lines as boundaries.
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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) lt
/projectionI(p)
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Maps and Sentences

• Goal is to provide a coherent account of how
  geographical information represented in maps can
  be translated into logical sentences while
  preserving geographical validity.
• Almost there... current work focussing on
  adjacency and qualitative metric information.