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 dont 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 physical entity with space-like parts Occurrent physical entity with time-like parts (Can some things be both?) Location piece of physical space Terrainpiece of geographical space (consisting of locations suitably related to each other.) Historyspatio-temporal region (the envelope of a continuant or occurrent.)

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...

Ontology 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. some personal opinions

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 l Specify the syntax l Expressions have immediate parts Interpretation is defined recursively I(e) = M(t, I(e 1 ),…,I(e n ) ) l Structural agnosticism yields validity l Interpretation is assumed to have enough structure to define truth…..but thats all.

= Oil well = Town Simple maps have no syntax (worth a damn…)

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

I(e)=M(t, I(e 1 ),…,I(e n ) ) …. 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(e 1 ),…,I(e n ) ) …. 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 S) implies p< ^S ((p 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: 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?

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.)

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 } What is I(p) ? I(p) =?= /projection I (p) f /f

For locations of symbols, the covering inverse of the projection function isnt an adequate interpretation: 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) < /projection(p) Which is really just a fancy way of saying: projection(I(p))=p What is I(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… b p b is part of the boundary of p

Adjacency requires boundaries b p Define full boundary of p to be ^{b : b 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 p and b q )

Axioms for boundaries ( b p & c<b ) implies c p ( b p & p<q ) implies ( b p or b<q ) ( -->adjacency analysis) Homology axiom: not ( c ^{b : b 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) < /projection I (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.