1. Towards a Naive Geography
Pat Hayes & Geoff Laforte
IHMC
University of West Florida
This was a presentation to NIMA made in 2003 at a PI meeting, to report on progress of our project.

2. 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.
The audience was not well versed in axiomatic ontology, so needed some background motivation. Hence the rather high-level suyrvey here. The 'upper-level' effort referred to is what eventually morphed into the SUMO project.

3. 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.)

4. 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...

5. 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.

6. 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
Again, broad overview of background for a non-semantic audience.

7. 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 that’s all.

8. Simple maps have no syntax (worth a damn…)
This is where the actual content begins, in trying to make sense of what a 'map syntax' could be, when a bare-bones map seems to have nothing remotely like a syntactic structure, but is just 'marks on a surface'.
= Oil well = Town

9. Different tokens of same symbol mean different things
Indexical?? ( “This city…”)
Bound variable?? ( “The city which exists here…”)
Existential assertion? ( “A city exists here…”)
First look at what the marks mean. They arent like names in a logical formalism. So what are they like?

10. 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
We decided we need a new notion, since WHERE the mark occurs seems like an essential part of the map structure (= syntax, though it’s not like a conventional syntax)

11. 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
This enables us to write Tarski-style semantic equations, which was the whole point.

12. 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 ?
Now we need to start describing space on the map and on the territory.

13. For that matter, what is p ?
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…)

14. For that matter, what is p ?
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
'tread delicately' is a warning from experience here.

15. For that matter, what is p ?
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.)
'solid' means a regular set in topology, often assumed by axiomatizers. But this rules out lines and surfaces as 'places', which is hopelessly restrictive for describing maps.

16. … and many more … 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 …

17. 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 :-)
The unique minimal cover is the key mathematical tool here.

18. … but not (yet) all that we will need: Boundary? Direction?
Can define many useful operations and properties:
‘Everywhere’: forall p (p<^L)
Overlap: pOq =df exists r ( r<p and r<q )
Sum: p+q =df ^{p,q}
Complement: ¬p =df ^{q: not pOq}
… but not (yet) all that we will need:
Boundary? Direction?

19. 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.)
Defining a notion of subexpression for continuous symbols was one of our most difficult challenges.

20. 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:

21. What is I(p) ? But interpretation mappings go
from the map to the interpretation:
…and they may not be invertible.
The fact that the map projection is going the 'wrong way' for a semantic notion was an early observation which motivated much of the subsequent development.

22. What is I(p) ?
covering inverse of function between location spaces:
/f(p) = ^{q : f(q) = p } f
In particular, it motivated the minimal covering axioms, and this is why. /f
I(p) =?= /projectionI(p)

23. What is I(p) ?
For locations of symbols, the covering inverse of the projection function isn’t an adequate interpretation:
Does the road go through the town? The map symbology says it does, but when we apply the magnification implicit in the inverse projection, this does not follow from the spatial constraints alone. So the map syntax cannot be defined SOLELY in terms of inverse projections.

24. 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
As so often happens, the result of an extended analysis is something very simple. I guess this ought to be reassuring.

25. 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)
One of the nice things we discovered is that this analysis applies to a wide variety of kinds of 'map', including some very 'symbolic' map types. We invented the term 'Gill space' in honor of the typographer Eric Gill, whose sans-serif letters were developed for use on the tube maps.

26. 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.

27. Some examples Choropleth Map
Terrain is restriction of underlying space to maximal regions
Projection preserves maximality.
(Actually, to be honest, it requires boundaries.)

28. 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’

29. 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 don’t.
Adjacency is defined to be sharing a common boundary part:
pAq =df exist b (b d p and b d q )

30. 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 } )

31. Boundaries define paths

32. 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

33. Need to consider edges between pixels as boundary locations.
Or, we can have both interpixels and lines as boundaries.

34. 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)

35. 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.